On cohomological Hall algebras of quivers : generators

TL;DR

This paper investigates the structure of the cohomological Hall algebra associated with quivers, demonstrating its purity, computing its Poincare polynomials, and proposing a conjectural relation to Yangians, with implications for Kac polynomials.

Contribution

It introduces a family of algebra generators for the cohomological Hall algebra and conjectures its equivalence to the Yangian of Maulik and Okounkov.

Findings

Proves the purity of the cohomological Hall algebra.

Computes Poincare polynomials in terms of Kac polynomials.

Establishes a variant of Okounkov's conjecture related to Kac polynomials.

Abstract

We study the cohomological Hall algebra Y of a lagrangian substack of the moduli stack of representations of the preprojective algebra of an arbitrary quiver Q, and their actions on the cohomology of Nakajima quiver varieties. We prove that Y is pure and we compute its Poincare polynomials in terms of (nilpotent) Kac polynomials. We also provide a family of algebra generators. We conjecture that Y is equal, after a suitable extension of scalars, to the Yangian introduced by Maulik and Okounkov. As a corollary, we prove a variant of Okounkov's conjecture, which is a generalization of the Kac conjecture relating the constant term of Kac polynomials to root multiplicities of Kac-Moody algebras.

Equations

$$\pi :\mathfrak{M}(v,w)\to {\mathfrak{M}}_0(v,w)$$

$$L_w=\underset{v}{⨁}{H}_{\mathrm{top}}(\mathfrak{L}(v,w)),$$

$$\underset{v}{⨁}{K}^{{\mathbb{C}}^{\ast }\times G(w)}(\mathfrak{L}(v,w)).$$

$$\underset{v}{⨁}{H}_{\ast }^{{\mathbb{C}}^{\ast }\times G(w)}(\mathfrak{L}(v,w)),$$

$$F_w=\underset{v}{⨁}{H}_{\ast }^{T\times G(w)}(\mathfrak{M}(v,w)).$$

$${\Lambda }^L(v)\subseteq {\Lambda }^1(v)\subseteq {\Lambda }^0(v),$$

$$\sum_{v}P(\Lambda^{\flat}(v),q)\,q^{\langle v,v\rangle+\tau}z^{v}=(1-q^{-1})^{-\tau}\operatorname{Exp}\nolimits\Big{(}\sum_{v}\frac{A^{\flat}(q^{-1})}{1-q^{-1}}z^{v}\Big{)}$$

$$Y^{♭}=\underset{v}{⨁}{H}_{\ast }^{T\times G(v)}({\Lambda }^{♭}(v)),Y=\underset{v}{⨁}{H}_{\ast }^{T\times G(v)}(Rep(\Pi ,v)).$$

$${\Lambda }_{(v)}=\{0\}\times Rep(\mathbb{C}{Q}^{\ast },v)\subset {\Lambda }^0(v).$$

$$Rep(\Pi ,v)/G(v)\to Rep(\mathbb{C}Q,v)/G(v).$$

$$\mathbf{Y}(0)=H_{T\times G(\infty )}^{\ast }.$$

$$Y_K^{♭}=Y^{♭}{\otimes }_{\mathbb{k}}K,{\mathbf{Y}}_K^{♭}={\mathbf{Y}}^{♭}{\otimes }_{\mathbb{k}}K.$$

$$\mathfrak{M}(v,w{)}^{\#}\subset \mathfrak{M}(u,y{)}^{{\mathbb{C}}^{\ast }}$$

$$\mathbb{Y}={\mathbb{Y}}_{+}\otimes {\mathbb{Y}}_0\otimes {\mathbb{Y}}_{-},\mathbf{g}={\mathbf{g}}_{+}\oplus {\mathbf{g}}_0\oplus {\mathbf{g}}_{-}.$$

$$M_j^G(X)=M_{j+\epsilon (l_i-g)}(X{\times }_G{U}_i)$$

$$M_{\ast }(X/G)=M_{\ast +\epsilon g}^G(X).$$

$$\mathrm{deg}([Y/G])=\epsilon \dim (Y/G)=\epsilon \dim Y-\epsilon g.$$

$$f^{♡}:M_{\ast }^H(X//G)\to M_{\ast +\epsilon g}^{H\times G}(X).$$

$$M_G^{\ast }=M_G^{\ast }(\text{Spec}(k)).$$

$$M_G^{\ast }(X)\to M_{\epsilon \dim X-\ast }^G(X),\alpha \mapsto \alpha \cap [X].$$

$$A_{\ast }^G(Y_1\cap Y_2)\to A_{\ast }^G(Y_1)\oplus A_{\ast }^G(Y_2)\to A_{\ast }^G(Y_1\cup Y_2)\to 0.$$

$$f^{\ast }:M_{\ast }^G(X)\to M_{\ast +\epsilon d}^G(Z).$$

$$\displaystyle\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 11.28056pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\\&\crcr}}}\ignorespaces{\hbox{\kern-8.41667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 15.1863pt\raise 5.74306pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.38196pt\hbox{$\scriptstyle{g^{\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 34.29167pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern-11.28056pt\raise-18.41666pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.30556pt\hbox{$\scriptstyle{f^{\prime}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 34.29167pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 30.88193pt\raise-18.41666pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.0625pt\raise-27.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-7.01389pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 15.7363pt\raise-31.64581pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{g}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.41667pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.41667pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{X.}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$$

$$(f,f^{\prime }{)}^{!}:M_{\ast }^G(Y)\to M_{\ast +\epsilon d}^G(W).$$

$$|(X/G)({\mathbb{F}}_q)|=|X({\mathbb{F}}_q)|/|G({\mathbb{F}}_q)|.$$

$$E(H_{\ast }^G(X);x,y)=\sum _{p,q,i}(-1{)}^i{\dim }_{\mathbb{C}}({\mathrm{gr}}_F^p{\mathrm{gr}}_{p+q}^W(H_i^G(X){\otimes }_{\mathbb{Q}}\mathbb{C}))x^p{y}^q,$$

$$\sum _i(-1{)}^i{\dim }_{\mathbb{Q}}(H_i(X/G))q^{i/2}=\sum _i{\dim }_{\mathbb{Q}}(H_{2i}(X/G))q^i=|(X/G)({\mathbb{F}}_q)|$$

$$E(H_{\ast }(X{\times }_G{U}_i);x,y)=P_{U_i/G}(xy)P_X(xy).$$

$$E(H_{\ast }(X/G);x,y)=P_X(xy)/P_G(xy)=E(H_{\ast }(X);x,y)/E(H_{\ast }(G);x,y).$$

$$P_X(t^2)/P_G(t^2)=E(H_{\ast }(X/G);t,t)=\sum _i{\dim }_{\mathbb{Q}}(H_i(X/G))t^i=\sum _i(-1{)}^i{\dim }_{\mathbb{Q}}(H_i(X/G))t^i.$$

Full text

1. 1 Introduction

2. 2 Equivariant homology

3. 2.1 Definition

4. 2.2 Equivariant homology and homology of Artin stacks

5. 2.3 Properties

6. 2.4 Mosaic

7. 2.5 Purity and polynomial count

8. 3 The semi-nilpotent variety

9. 3.1 Quivers

10. 3.2 The moment map and the preprojective algebra

11. 3.3 The group action

12. 3.4 Definition

13. 3.5 The stratification

14. 3.6 The one vertex quiver

15. 3.7 Quiver varieties

16. 3.8 Hecke correspondences

17. 4 The homology of the semi-nilpotent variety

18. 4.1 The Lagrangian quiver variety

19. 4.2 The homology of the Lagrangian quiver variety

20. 4.3 Purity

21. 4.4 Torsion-freeness

22. 4.5 Kirwan Surjectivity for fixed point quiver varieties

23. 5 The algebra $\operatorname{Y}\nolimits^{\flat}$

24. 5.1 Definition

25. 5.2 Comparison of $\operatorname{Y}\nolimits^{0}$ and $\operatorname{Y}\nolimits^{1}$

26. 5.3 Dimension

27. 5.4 Generators

28. 5.5 The algebra $\mathbf{Y}^{\flat}$

29. 5.6 The representation of $\mathbf{Y}^{\flat}$ in the homology of quiver varieties

30. A

31. A.1 Proof of Proposition 3.1

32. A.2 Proof of Lemma 3.9

33. A.3 Proof of Proposition 3.11

34. A.4 Kirwan surjectivity for local quiver varieties

35. A.5 Non-injectivity of the shuffle realization map for small tori

1. Introduction

In the mid 90s, H. Nakajima associated to each quiver $Q$ and each pair $(v,w)$ of dimension vectors for $Q$ a projective morphism

[TABLE]

where $\mathfrak{M}(v,w)$ is a smooth quasi-projective symplectic variety and $\mathfrak{M}_{0}(v,w)$ is a (usually singular) affine variety. The varieties $\mathfrak{M}(v,w)$ and $\mathfrak{M}_{0}(v,w)$ have several remarkable geometric properties : the variety $\mathfrak{M}(v,w)$ has a hyperkähler structure, it possess a resolution of the diagonal, its cohomology is generated by algebraic cycles, the varieties $\mathfrak{M}(v,w)$ and $\mathfrak{M}_{0}(v,w)$ carry a natural action of the reductive group $G(w)\times T$, where $T$ is a certain torus whose rank depends on $Q$ and $G(w)$ preserves the symplectic form, etc. These varieties include the Hilbert schemes of points on $\mathbb{A}^{2}$ or on resolutions of Kleinian surface singularities, the moduli spaces of instantons on $\mathbb{A}^{2}$ and the resolutions of Slodowy slices in the nilpotent cone of $\mathfrak{gl}_{n}$.

Quiver varieties have played an important role in geometric representation theory. For instance, when the quiver $Q$ contains no edge loops it can be regarded as an orientation of the generalized Dynkin diagram of a Kac-Moody algebra $\mathfrak{g}_{Q}$, and Nakajima constructed an action of $\mathfrak{g}_{Q}$ on the space

[TABLE]

where $\mathfrak{L}(v,w)=\pi^{-1}(0)$ is a Lagrangian subvariety of $\mathfrak{M}(v,w)$, see [31]. Here and below all (co)homology groups are taken with rational coefficients. The resulting module is identified with the integrable irreducible highest weight module of highest weight $\sum_{i}w_{i}\Lambda_{i}$ where the $\Lambda_{i}$’s are the fundamental weights of $\mathfrak{g}_{Q}$. In a similar vein, when the quiver $Q$ is of finite type, Nakajima constructed a representation of the quantum affine algebra of $\mathfrak{g}_{Q}$ on the space

[TABLE]

The resulting module is called a universal standard module, and it is a geometric analog of the global Weyl modules. A cohomological version of this construction, due to Varagnolo [42], yields an action of the Yangian of $\mathfrak{g}_{Q}$ on the space

[TABLE]

and the resulting module is again the universal standard module. Nakajima’s and Varagnolo’s actions in equivariant K-theory and Borel-Moore homology extend to the case of arbitrary quivers with no edge loops, but the precise nature of the algebra which acts, or of the structure of the resulting module, are not well understood in general. There are a few notable exceptions : the algebras associated to the Jordan quiver are known as the elliptic Hall algebra or quantum toroidal algebra of $\mathfrak{gl}_{1}$ and the affine Yangian of $\mathfrak{gl}_{1}$ respectively. They play an important role in the study of the Alday-Gaiotto-Tachikawa (AGT) correspondence in string theory, see [40], [27], [35]. Similarly, the algebra associated to affine quivers are the quantum toroidal algebras and affine Yangians, see [32].

Since we have no algebraic candidate for the symmetry algebra of the equivariant K-theory and Borel-Moore homology of quiver varieties when the quiver is not of finite, affine or Jordan type, one may hope to describe this algebra in another way, for instance as a Hall algebra. This was the raison d’être of the cohomological Hall algebras in equivariant K-theory and equivariant Borel-Moore homology introduced in [38], [39], [40] (in the particular case of quivers with only one vertex).

The aim of this paper is to introduce and study, for an arbitrary quiver, a certain convolution algebra $\operatorname{Y}\nolimits$ acting on the equivariant Borel-Moore homology groups of arbitrary Nakajima quiver varieties $F_{w}$, where

[TABLE]

This algebra contains the algebra considered by Varagnolo. To be more precise, set $\Bbbk$ equal to the $T$-equivariant cohomology ring $H^{*}_{T}$ of the point. We will introduce three $\Bbbk$-algebras $\operatorname{Y}\nolimits,$ $\operatorname{Y}\nolimits^{0}$ and $\operatorname{Y}\nolimits^{1}$, all acting on $F_{w}$, and which all become isomorphic after extension of scalars to the fraction field $K$ of $\Bbbk$. Our algebras $\operatorname{Y}\nolimits^{\flat}$ are in some sense the largest algebras which act on the homology of quiver varieties by means of some Hecke correspondences. These correspondences are required to be lagrangian if $\flat\in\{0,1\}$.

Before stating more precisely our results, let us introduce some notation. Let $Q=(I,\Omega)$ be an arbitrary quiver. We will call real the vertices of $I$ without $1$-cycles and imaginary the other vertices. Let $v$ be a dimension vector and let $Rep(\mathbb{C}Q,v)/G(v)$ be the moduli stack of complex representations of $Q$ of dimension $v$. Let $\Pi$ be the preprojective algebra of $Q$ and let $Rep(\Pi,v)/G(v)$ be the moduli stack of complex $\Pi$-representations of dimension $v$, which is isomorphic to the cotangent stack of $Rep(\mathbb{C}Q,v)/G(v)$.

Geometry of Lagrangian substacks. In [3] we defined Lagrangian substacks $\Lambda^{0}(v)/G(v)$ and $\Lambda^{1}(v)/G(v)$ of the quotient stack $Rep(\Pi,v)/G(v)$, by using some semi-nilpotency condition. See also [2]. If $Q$ has no $1$-cycle and no oriented cycle, then $\Lambda^{0}(v)$ and $\Lambda^{1}(v)$ both coincide with Lusztig’s nilpotent variety $\Lambda^{L}(v)$. We always have

[TABLE]

but the inclusions are strict in general. The torus $T$ acts on $Rep(\Pi,v)/G(v)$ and $\Lambda^{\flat}(v)/G(v)$ for $\flat\in\{0,1\}$. We prove the following in Theorem 3.2, Proposition 4.6, and Theorem 5.4 (under some mild condition on the torus $T$).

Theorem A**.**

For any quiver $Q=(I,\Omega)$, any $v\in\mathbb{N}^{I}$ and any $\flat\in\{0,1\}$, we have

1. $\mathrm{(a)}$

$H^{T\times G(v)}_{*}(\Lambda^{\flat}(v))$* is pure and even.* 2. $\mathrm{(b)}$

$H^{T\times G(v)}_{*}(\Lambda^{\flat}(v))$* is free as a $\Bbbk$-module.* 3. $\mathrm{(c)}$

$H^{T\times G(v)}_{*}(\Lambda^{\flat}(v))$* is torsion-free as a $H^{*}_{T\times G(v)}$-module.* 4. $\mathrm{(d)}$

The Poincaré polynomial $P(\Lambda^{\flat}(v),q)=\sum_{i}\dim_{\mathbb{Q}}(H^{T\times G(v)}_{2i}(\Lambda^{\flat}(v)))\,q^{i}$ is given by

[TABLE]

where $\tau$ is the rank of $T$, $A^{\flat}(t)$ is the $\flat$-nilpotent Kac polynomial and $\operatorname{Exp}\nolimits$ is the plethystic exponential.

Statement (d) of Theorem A is a consequence of (a) and of the computation, conducted in [3], of the number of rational points of $\Lambda^{\flat}(v)$ over finite fields

Cohomological Hall algebras. The construction of $\operatorname{Y}\nolimits^{\flat}$ is given in terms of the cohomological Hall algebras of the above stacks. More precisely, if $\flat\in\{0,1\}$ we have

[TABLE]

For any dimension vector $v$, we set

[TABLE]

This is always an irreducible component of $\Lambda^{0}(v)$. It is also an irreducible component of $\Lambda^{1}(v)$ in case $v$ is supported on a subquiver without oriented cycles. From a geometric perspective, the quotient stack $\Lambda_{(v)}/G(v)$ is the zero section the cotangent bundle map

[TABLE]

Note that we have $H^{T\times G(v)}_{*}(\Lambda_{(v)})\simeq H^{*}_{T\times G(v)}$. Motivated by the analogy with Yangians, for each $\flat\in\{\emptyset,0,1\}$, we consider a slight extension $\mathbf{Y}^{\flat}$ of $\operatorname{Y}\nolimits^{\flat}$ by adding a loop Cartan part equal to

[TABLE]

Then, we write

[TABLE]

Finally, we say that an imaginary vertex is elliptic or isotropic if it carries a single $1$-loop and hyperbolic if it carries more than one $1$-loop. We denote by $I^{\operatorname{r}\nolimits},$ $I^{\operatorname{e}\nolimits}$ and $I^{\operatorname{h}\nolimits}$ respectively the real, elliptic and hyperbolic vertices of $I$. Let $\{\delta_{i}\,;\,i\in I\}\subset\mathbb{N}^{I}$ be the basis of delta functions. In Propositions 5.1, 5.2, 5.8 and Theorem 5.18 we prove the following.

Theorem B**.**

For any $\#\in\{\emptyset,0,1\}$ and any $\flat\in\{0,1\}$ we have the following.

1. $\mathrm{(a)}$

There is an associative $\mathbb{Z}\times\mathbb{N}^{I}$-graded $\Bbbk$-algebra structure on $\operatorname{Y}\nolimits^{\#}$. 2. $\mathrm{(b)}$

There is a representation of $\operatorname{Y}\nolimits^{\#}$ on $F_{w}$ for each $w$. 3. $\mathrm{(c)}$

The diagonal action of $\operatorname{Y}\nolimits^{\flat}$ on $\bigoplus_{w}F_{w}$ is faithful. 4. $\mathrm{(d)}$

There are $K$-algebra isomorphisms $\operatorname{Y}\nolimits^{1}_{K}\simeq\operatorname{Y}\nolimits^{0}_{K}\simeq\operatorname{Y}\nolimits_{K}.$ 5. $\mathrm{(e)}$

The $K$-algebra $\operatorname{Y}\nolimits_{K}$ is generated by the subspaces $H^{T\times G(v)}_{*}(\Lambda_{(v)})\otimes_{\Bbbk}K$ where $v$ runs over the set $\{\delta_{i},l\delta_{j}\;;\;i\in I,j\in I^{\operatorname{h}\nolimits}\,,\,l>1\}.$ The $K$-algebra $\mathbf{Y}_{K}$ is generated by $\mathbf{Y}(0)$ and the collection of fundamental classes $\{[\Lambda_{(\delta_{i})}],[\Lambda_{(l\delta_{j})}]\;;\;i\in I,j\in I^{\operatorname{h}\nolimits},l>1\}.$

There is also a more precise result providing a family of generators for $\operatorname{Y}\nolimits^{1}$, see Proposition 5.12. In particular, when $Q$ has no vertex carrying a $1$-cycle we have $\operatorname{Y}\nolimits^{1}=\operatorname{Y}\nolimits^{0}$ and these coincide with the $\Bbbk$-algebra constructed by Varagnolo, see Remark 5.10. In general, the $\Bbbk$-algebras $\operatorname{Y}\nolimits,$ $\operatorname{Y}\nolimits^{0}$ and $\operatorname{Y}\nolimits^{1}$ are different integral forms inside the $K$-algebra $\operatorname{Y}\nolimits_{K}$. This choice of integral form is responsible for the different types of Kac polynomials entering the Poincaré polynomial formulas in Theorem A(c). The proof of statements (a) and (b) follows the approach in [40]. See also [43]. Part (d) is a consequence of the localization theorem. The proof of (e) is more involved. First, we use the algebra $\operatorname{Y}\nolimits^{1}$ and the geometry of $\Lambda^{1}(v)$, in particular, the crystal structure studied in [2], to reduce ourselves to a one vertex quiver. Then, we deal separately with the real, elliptic and hyperbolic cases. A crucial step in the argument is the following version of the Kirwan surjectivity for local quiver varieties. Let $\mathfrak{M}(u,y)$ be a quiver variety, and let

[TABLE]

be a local quiver subvariety of $\mathfrak{M}(u,y)$, that is a fixed point quiver variety associated to the $\mathbb{C}^{*}$-action as in (4.3).

Theorem**.**

The cohomology ring $H^{*}_{T\times G(w)}(\mathfrak{M}(v,w)^{\#})$ is generated by the Chern classes of the universal and tautological bundles.

T. Nevins and K. McGerty recently gave a proof of Kirwan surjectivity for the quiver varieties $\mathfrak{M}(v,w)$ themselves, see [25]. The above theorem does not seem to directly follow from it. Our proof bears some similarity to but is independent from theirs, see Theorem 4.8 and Appendix A.4. In fact, the same proof works for the fixed point quiver varieties associated to any cocharacter which scales the symplectic form by a nontrivial factor. As a corollary, we obtain (see Corollary 4.9 for notations)

Corollary 1.1**.**

Let $\sigma:\mathbb{C}^{*}\to T\times G(w)$ be a cocharacter acting nontrivially on the symplectic form of $\mathfrak{M}(v,w)$. Then the fixed point quiver varieties $\mathfrak{M}[\rho]\subset\mathfrak{M}(v,w)$ associated to the $\mathbb{C}^{*}$-action induced by $\sigma$ are connected.

This includes several cases of interest, such as the handsaw and graded quiver varieties.

Comparison with Kontsevich-Soibelman COHAs. We introduced K-theoretic Hall algebras in [39], [38] to study the moduli stack of preprojective representations of a one vertex quiver. Then, we studied its cohomological version in [40]. Independently, Kontsevich and Soibelman introduced some COHA associated to various Calabi-Yau categories in [23]. It has been recently proved by Davison in [11] that the COHA of the moduli stack of preprojective representations of quivers that we used is isomorphic to a particular case of the COHA of Kontsevich-Soibelman, viewing the moduli stack of preprojective representations of the quiver as a particular 2 dimensional Calabi-Yau category.

Comparison with Maulik-Okounkov Yangians. In [27], the authors defined and studied by some totally different means another associative algebra acting on the (co)homology of Nakajima quiver varieties associated to an arbitrary quiver $Q=(I,\Omega)$. Their construction, which stems from ideas in symplectic geometry, hinges on the notion of stable enveloppe to produce a quantum $R$-matrix, and then on the RTT formalism to define an associative $\mathbb{Z}\times\mathbb{Z}^{I}$-graded $\Bbbk$-algebra $\mathbb{Y}=\mathbb{Y}_{Q}$ acting on the space $F_{w}$ for any dimension vector $w$. Taking a quasi-classical limit, they also define a classical $R$-matrix and a $\mathbb{N}\times\mathbb{Z}^{I}$-graded Lie algebra ${{\bf g}}={{\bf g}}_{Q}$. If $Q$ is of finite type, then ${{\bf g}}_{Q}$ is the semisimple Lie algebra $\mathfrak{g}_{Q}$ associated with $Q$, and $\mathbb{Y}_{Q}$ is the Yangian of the same type. In general, the $\Bbbk$-algebra $\mathbb{Y}_{Q}$ is a deformation of the enveloping algebra of the Lie algebra of polynomial loops ${{\bf g}}[u]$. Their construction provides triangular decompositions for these algebras

[TABLE]

In [41], we compare our cohomological algebra $\operatorname{Y}\nolimits$ with the positive half $\mathbb{Y}_{+}$. Set $\mathbb{Y}_{+,K}=\mathbb{Y}_{+}\otimes_{\Bbbk}K$. We make the following conjecture.

Conjecture**.**

There is a unique $K$-algebra isomorphism $\mathbb{Y}_{+,K}\simeq\operatorname{Y}\nolimits_{K}$ which intertwines the respective actions of $\mathbb{Y}_{+,K}$ and $\operatorname{Y}\nolimits_{K}$ on $F_{w}\otimes_{\Bbbk}K$ for any $w$.

In [41] we prove one half of the above conjecture.

Theorem C**.**

There is a unique embedding $\operatorname{Y}\nolimits_{K}\subset\mathbb{Y}_{+,K}$ which intertwines the actions of $\mathbb{Y}_{+,K}$ and $\operatorname{Y}\nolimits_{K}$ on $F_{w}\otimes_{\Bbbk}K$ for any $w$.

By Theorem B, the proof of the above theorem boils down to checking that certain generalized Hecke correspondences corresponding to generators of $\operatorname{Y}\nolimits^{1}$ occur with a non zero and constant coefficient in a suitable stable enveloppe. Again, generalized Hecke correspondences associated to real, elliptic or hyperbolic vertices behave in very different ways and we have to treat each case separately.

Okounkov’s conjecture. Let us finish this introduction by mentioning one other motivation for this work. In [36], A. Okounkov conjectured that the graded dimensions of the root spaces ${{\bf g}}[\alpha]\subset{{\bf g}}$ are, after a suitable grading shift, precisely given by the Kac polynomial $A_{\alpha}(t)$. This is a generalization of the Kac conjecture, proved in [20], stating that for quivers with no $1$-cycles the multiplicity of the root $\alpha$ in the Kac-Moody algebra $\mathfrak{g}_{Q}$ is equal to the constant term $A_{\alpha}(0)$. Indeed, one expects to have an isomorphism $\mathfrak{g}_{Q}\simeq{{\bf g}}_{Q}[0]$, where the grading in ${{\bf g}}_{Q}$ is counted from middle dimension up. In other words, the Lie algebra ${{\bf g}}_{Q}$ is a graded extension of $\mathfrak{g}_{Q}$, whose character is conjecturally given by the full Kac polynomials rather than their constant terms. Our Theorem A(d) proves a variant of the above conjecture : the graded character of the cohomological Hall algebras $\operatorname{Y}\nolimits^{1}$ and $\operatorname{Y}\nolimits^{0}$ are encoded by the full nilpotent Kac polynomials $A^{0}_{\alpha}(t)$ and $A^{1}_{\alpha}(t)$. Note that, contrary to that of Maulik and Okounkov, our construction does not directly provide a construction of an underlying Lie algebra. See some recent work of Davison and Meinhardt in that direction in [12]. Moreover, we expect the discrepancy between the various types of Kac polynomials involved to correspond to different gradings on the same Lie algebra. Indeed, note that we have $A_{\alpha}(1)=A^{1}_{\alpha}(1)=A^{0}_{\alpha}(1)$, see [3].

Plan of the paper.

• §2 is essentially a reminder of some standard properties of equivariant Borel-Moore homology and equivariant Chow groups. We use both cohomological theories simultaneously throughout the paper. We also provide a proof of the folklore fact that the counting polynomial and equivariant Poincaré polynomials of a pure $G$-variety $X$ are equal if $X$ is of polynomial count (and $G$ is a product of general linear groups).

• In §3 we introduce the varieties ${\operatorname{M}\nolimits}(v)$ of representations of the preprojective algebra $\Pi$ of an arbitrary quiver $Q=(I,\Omega)$ as well as its Lagrangian subvarieties $\Lambda^{\flat}(v)$ for $\flat\in\{0,1\}$. In §3.5 we describe the stratification of $\Lambda^{1}(v)$ responsible for the crystal structure, see [2]. This is used in our proof of Theorem B in order to reduce ourselves to one vertex quivers. The geometry of $\Lambda^{0}(v)$ for one vertex quivers is the subject of §3.6 where the real, elliptic and hyperbolic cases are dealt with in details. After recalling the definition of quiver varieties, we study in §3.8 some generalized Hecke correspondences. We prove that they are Lagrangian local complete intersections in Proposition 3.16, and that they are irreducible in the hyperbolic one-vertex case in Proposition 3.17.

• In §4 we prove Theorem A. The purity statement is proved using a compactification of $\Lambda^{\flat}(v)$ constructed out of Lagrangian quiver varieties. Statements (b) and (d) follow, using the point count computations in [3]. We also consider the structure of $H_{*}^{T\times G(v)}(\Lambda^{\flat}(v))$ as a $H^{*}_{T\times G(v)}$-module and show that it is torsion free using some recent work of B. Davison on dimensional reduction. Finally, we briefly state the Kirwan surjectivity result for fixed point quiver varieties.

• The algebras $\operatorname{Y}\nolimits^{\flat}$ are defined in §5, which also contains the proof of statements (a), (c) and (d) of Theorem B. The most delicate part is (d). We first use the crystal stratification of $\Lambda^{1}(v)$ to reduce ourselves to one-loop quivers in Proposition 5.8, then we use the compactification via quiver varieties and finally conclude using the Kirwan surjectivity for graded quiver varieties in Theorem 4.8 and Proposition 5.12. We also define the extension $\mathbf{Y}^{\flat}$ of $\operatorname{Y}\nolimits^{\flat}$, by adding a Cartan part, and construct an action of the resulting algebras on the spaces $F_{w}$ for any $w$, thus completing the proof of Theorem B.

• We have postponed to the appendix the proof of the Kirwan surjectivity result for local quiver varieties as well as some folklore homological and geometric results concerning preprojective algebras of arbitrary quivers.

Acknowledgements

We would like to thank T. Bozec, F. Charles, M. McBreen, B.  Davison, H. Nakajima, A. Negut and A. Okounkov for useful discussions and correspondences. Special thanks are due to B. Crawley-Boevey for providing us the proof of Proposition 3.1 and B. Davison for pointing out an omission in the hypothesis of Proposition 4.6.

2. Equivariant homology

In this section we set ${\operatorname{k}\nolimits}=\mathbb{C}$.

2.1. Definition

By a variety we mean a reduced scheme over an algebraically closed field. All schemes considered here will be reduced.

Given an algebraic variety $X/{\operatorname{k}\nolimits}$, let $H_{*}(X)$ be the Borel-Moore homology group (= the locally finite singular homology group) of $X$ and let ${A}_{*}(X)$ the Chow group of $X$, both with rational coefficients. In particular, we have $H_{i}(X)=H_{c}^{i}(X)^{\vee}$ where $H^{*}_{c}(X)$ is the cohomology group with compact support. By [17], the additive functor $A_{*}$ is an oriented Borel-Moore homology in the sense of [24, §5]. This means in particular that there are pullback and refined pullback maps for any local complete intersection morphisms, pushforward maps for proper morphisms, external products and Chern classes which satisfy all the properties mentioned there. See below for more details. The functor $H_{*}$ is also oriented Borel-Moore homology. In particular, locally finite singular homology groups admit refined pullback maps. Note that Fulton’s definition of the refined pullback maps uses only the standard operations (proper pushforward, smooth pullback, external products and specializations) whose construction can be found in [13] in the setting of $H_{*}$. See [24, §6] for more details. Let $A^{*}$ and $H^{*}$ be the corresponding oriented cohomology functors. Thus $A^{*}$ is the Chow cohomology group, or operational Chow group, introduced by Fulton and MacPherson. We write $M_{*}$ for either $H_{*}$ or ${A}_{*}$ and $M^{*}$ for either $H^{*}$ or ${A}^{*}$. Set $\varepsilon=2$ if $M_{*}=H_{*}$ and $\varepsilon=1$ if $M_{*}={A}_{*}$.

Now, assume that $X$ is equipped with the action of a linear algebraic group $G/{\operatorname{k}\nolimits}$. We always assume that $X$ is quasi-projective and that we have fixed a very ample $G$-linearized line bundle over $X$. Thus, the variety $X$ embeds equivariantly in a projective space with a linear $G$-action.

Let $H_{*}^{G}(X)$ and ${A}_{*}^{G}(X)$ be the equivariant Borel-Moore homology and the equivariant Chow group of $X$. Standard references for equivariant Borel-Moore homology and equivariant Chow groups are [4], [19] and [14, §2.2, §2.8]. We write $M_{*}^{G}$ to denote either $H_{*}^{G}$ or ${A}_{*}^{G}$. Let $M^{*}_{G}$ be the corresponding equivariant cohomology functor. More precisely $H^{*}_{G}(X)$ is the equivariant cohomology group while ${A}^{*}_{G}(X)$ is the equivariant operational Chow group defined in [14]. Let $g$ be the dimension of $G$. If $X$ is $n$-dimensional then for each integer $j\geqslant i$ we have

[TABLE]

where $U_{i}$ is an open set with a free $G$-action in a finite dimensional representation of $G$ such that the complement has codimension $\geqslant n-i$ and $l_{i}$ is the dimension of $U_{i}$. We call the quotient $X\times_{G}U_{i}$ an approximation to the Borel construction $X_{G}:=X\times_{G}EG$ of $X$. In general, it is only representable by an algebraic space. However, since we only consider the case where $G$ is a product of linear groups, we may choose the approximation such that $X\times_{G}U_{i}$ and $U_{i}/G$ are both ${\operatorname{k}\nolimits}$-varieties by [14, lem. 7, prop. 23].

2.2. Equivariant homology and homology of Artin stacks

The quotient stack of a $G$-variety $X$ by $G$ is the Artin stack associated with the groupoid $G\times X\to X\times X$. We denote it by the symbol $X/G$. We use the symbol $[X/G]$ for the fundamental class of $X/G$ in $M_{*}(X/G)$, and the symbol $X/\!\!/G$ for the categorical quotient.

A morphism of quotient stacks $f:X/G\to Y/H$ is safe if it is given by a compatible pair $(f^{\flat},f^{\sharp})$ consisting of a scheme homomorphism $f^{\flat}:X\to Y$ and a surjective algebraic group homomorphism $f^{\sharp}:G\to H$ with unipotent kernel. The relative dimension of $f$ is the relative dimension of $f^{\flat}$ minus the dimension of the kernel of $f^{\sharp}$. We only consider safe morphisms.

Assume that there is a stack isomorphism $X/G\simeq Y/H$. Let $g,h$ be the dimension of $G,H$. By [14, prop. 16] we have an isomorphism $M_{*+\varepsilon g}^{G}(X)\simeq M_{*+\varepsilon h}^{H}(Y).$ As a consequence, we can define the Borel-Moore homology or the Chow group of $X/G$ by

[TABLE]

Any $G$-invariant closed subset $Y$ of $X$ admits an equivariant fundamental class $[Y]$ in $M_{*}^{G}(X)$. For any $G$-invariant closed subset $Y$ of $X$, the fundamental class of $Y/G$ is the element $[Y/G]$ in $M_{*}(X/G)$ which coincides under (2.2) with the equivariant fundamental class of $Y$ in $M_{*}^{G}(X)$. If $Y$ is equidimensional then we have

[TABLE]

If $X$ has dimension $d$ then $M_{i}(X/G)=0$ if $i>\varepsilon(d-g),$ while $M_{\varepsilon(d-g)}(X/G)$ is spanned by the equivariant fundamental classes of the irreducible components of $X$.

Finally, if $X$ is an $H\times G$-variety such that the quotient map $f:X\to X/\!\!/G$ is an $H$-equivariant principal $G$-bundle, then we have a descent $M_{H}^{*}$-linear isomorphism

[TABLE]

2.3. Properties

The functors $M_{*}^{G}$, $M_{G}^{*}$ satisfy the same properties as $M_{*}$, $M^{*}$, for which we refer to [5], [13], [15], [24]. Let us recall briefly a few facts.

2.3.1. Poincaré duality

The cup product equips $M_{*}^{G}(X)$ with a graded ring structure, and there is a cap product $\cap$ making $M_{*}^{G}(X)$ into an $M^{*}_{G}(X)$-module. We abbreviate

[TABLE]

If $X$ is pure dimensional there is a map

[TABLE]

If $X$ is smooth the intersection product $\cap$ on $M_{*}^{G}(X)$ is induced by the product $\cup$ on $M^{*}_{G}(X)$ via this map, which is invertible.

2.3.2. Excision

If $Y\subseteq X$ is a $G$-invariant closed subset, there is the localization exact sequence ${A}_{*}^{G}(Y)\to{A}_{*}^{G}(X)\to{A}_{*}^{G}(X\setminus Y)\to 0.$

2.3.3. Mayer-Vietoris

If $Y_{1},$ $Y_{2}$ are $G$-invariant closed subsets of $X$, the Mayer-Vietoris exact sequence in equivariant Chow groups is

[TABLE]

2.3.4. Pullback

The functor $M_{*}^{G}$ has the same functoriality as $M_{*}$. See, e.g., [14, §2.3] for Chow groups. Recall that a closed immersion of $G$-varieties $i:Z\to X$ is called a regular embedding if the ideal sheaf of $Z$ in $X$ is locally generated by a regular sequence. A local complete intersection morphism, an l.c.i. morphism for short, is a morphism $f:Z\to X$ which admits a factorization as $f=q\circ i$, where $i$ is a regular embedding and $q$ is a smooth, quasi-projective morphism. For any l.c.i. morphism $f$ of relative dimension $d$ there is a pullback morphism which is a graded vector space homomorphism

[TABLE]

An affine space bundle of rank $d$ over $X$ is a map $f:Z\to X$ such that $X$ can be covered by open affine subsets $U_{i}$ with an isomorphism $f^{-1}(U_{i})\simeq\mathbb{A}^{d}\times U_{i}$ under which the map $f$ corresponds to the projection on the second factor. If the map $f$ is a $G$-equivariant affine space bundle then the pullback $f^{*}$ is surjective. If $f$ admits a $G$-equivariant section then $f^{*}$ is an isomorphism.

2.3.5. Refined pullback

Consider the following cartesian square of $G$-varieties

[TABLE]

If $f$ is an l.c.i. morphism of relative dimension $d$, we have a refined pullback homomorphism

[TABLE]

If no confusion is possible we abbreviate $f^{!}=(f,f^{\prime})^{!}$. We use the following properties of the refined pullback : the multiplicativity (i.e., $f^{!}(x\cap y)=(f^{\prime})^{*}(x)\cap f^{!}(y)$ for all $x\in M^{*}_{G}(Y),$ $y\in M_{*}^{G}(Y)$), the proper base change property, the functoriality with respect to smooth morphisms and the excess intersection formula as in [24, thm. 6.6.6, thm. 6.6.9] respectively. It follows from the excess intersection formula that $f^{!}=(f^{\prime})^{*}$ whenever the map $f^{\prime}$ is an l.c.i. morphism of the same relative dimension as $f$.

2.3.6. Transversality

Irreducible subvarieties $Y,$ $Z$ of a pure dimensional variety $X$ are dimensionally transverse if for every irreducible component $W$ of $Y\cap Z$ we have ${\operatorname{codim}\nolimits}W={\operatorname{codim}\nolimits}Y+{\operatorname{codim}\nolimits}Z$. They are generically transverse if every irreducible component of $Y\cap Z$ contains a point $x$ at which $X,Y,Z$ are smooth and the tangent spaces satisfy $T_{x}Y+T_{x}Z=T_{x}X$. Note that $Y$ and $Z$ are generically transverse if and only if they are dimensionally transverse and each irreducible component of $Y\cap Z$ is reduced and contains a smooth point of $X$.

Now, let $X$ be a smooth $G$-variety and $Y$, $Z$ be $G$-equivariant subvarieties. If $Y,$ $Z$ are generically transverse then the intersection product on $M_{*}^{G}(X)$ is such that $[Y]\cap[Z]=[Y\cap Z]$. More generally, if $Y,$ $Z$ are dimensionally transverse and l.c.i. at the generic point of every component of $Y\cap Z$ then we still have $[Y]\cap[Z]=[Y\cap Z]$.

2.3.7. The cycle map

There is a cycle map ${\operatorname{cl}\nolimits}:{A}_{*}^{G}(X)\to H^{G}_{2*}(X)$ defined in [14, §2.8]. It is a degree doubling homomorphism which is natural with respect to pullback by l.c.i. morphisms or pushforward by proper maps.

2.3.8. Chern classes

Equivariant vector bundles on $X$ have equivariant Chern classes in $M^{*}_{G}(X)$. For any $G$-equivariant vector bundle $\mathcal{E}$ let ${\operatorname{ch}\nolimits}(\mathcal{E})=\sum_{l\geqslant 0}{\operatorname{ch}\nolimits}_{l}(\mathcal{E})$ be its equivariant Chern character, with ${\operatorname{ch}\nolimits}_{l}(\mathcal{E})\in M^{\varepsilon l}_{G}(X)$ for each $l$.

2.4. Mosaic

Let us finish this section with a few reminders.

First, given a variety $X$ with an action of a linear algebraic group $G$, a $G$-equivariant cycle on $X$ is a $\mathbb{Q}$-linear combination of $G$-invariant closed subvarieties of $X$. It can be viewed as an element in $M_{*}^{G}(X)$. If $X$ is a symplectic manifold then a cycle on $X$ is Lagrangian if each component is a Lagrangian subvariety.

Next, for any closed subset $Y\subseteq X$ of a $G$-variety $X$ we say that a class in $M_{*}^{G}(X)$ is supported on $Y$ if it comes from an element in $M_{*}^{G}(Y)$, or, equivalently, if its restriction to $X\setminus Y$ is zero.

Finally, a commutative diagram is called a fiber diagram if it consists of squares which are cartesian.

2.5. Purity and polynomial count

Recall that $X$ is an algebraic variety $X/{\operatorname{k}\nolimits}$ and that ${\operatorname{k}\nolimits}=\mathbb{C}$. Using Deligne’s construction of mixed Hodge structure on relative cohomology, one can define a well-behaved MHS on $H_{c}^{*}(X)$. Since $H_{i}(X)$ is the dual of $H^{i}_{c}(X)$ for each degree $i$, it is equipped with a MHS as well, which is functorial for open immersions and proper maps. See, e.g., [37, §6.3.1]. We say that $H_{*}(X)$ is even if $H_{i}(X)=0$ whenever $i$ is odd, and that $H_{*}(X)$ is pure if the MHS is pure.

We say that $X$ has polynomial count if there is a polynomial $P(t)\in\mathbb{Q}[t]$ and an $R$-form $X_{R}$ of $X$ over a finitely generated subring $R\subseteq\mathbb{C}$ such that for any morphism $R\to\mathbb{F}_{q}$ to a finite field, the number of $\mathbb{F}_{q}$-points of the scheme $X_{\mathbb{F}_{q}}:=X_{R}\otimes_{R}\mathbb{F}_{q}$ is $P(q)$. We abbreviate $X(\mathbb{F}_{q})$ for $X_{\mathbb{F}_{q}}(\mathbb{F}_{q})$. Note that we have

[TABLE]

For $G=GL(n)$ we may choose $U_{i}$ to be the Stiefel variety $\operatorname{Mon}\nolimits({\operatorname{k}\nolimits}^{n},{\operatorname{k}\nolimits}^{w})$ consisting of injective linear maps ${\operatorname{k}\nolimits}^{n}\to{\operatorname{k}\nolimits}^{w}$ with $w$ large enough. In this case the variety $U_{i}/G$ is a Grassmanian, hence it has polynomial count. Since we only consider the case where $G$ is a product of linear groups, from now on we always assume that $U_{i}/G$ has polynomial count for all integers $i$.

Since Deligne defined a MHS for any simplicial variety, his construction holds as well for the Borel construction $X_{G}$. This yields a MHS on $H^{*}_{G}(X)$. Using the approximation to $X_{G}$ we also define a MHS on $H_{*}^{G}(X)$ such that for each $i$ the MHS on $H_{j}^{G}(X)$ is the one on $H_{j+2(l_{i}-g)}(X\times_{G}U_{i})$ for all $j\geqslant i$. Using the functoriality of MHS, see, e.g., [37, §5.5.1], one checks as in [14, def-prop. 1] that the MHS on $H_{j}^{G}(X)$ does not depend on the choice of the integer $i$.

The $E$-polynomial of $H_{*}^{G}(X)$ is indeed a power series. It is given by

[TABLE]

where the (increasing) filtration $(W_{p})$ is the weight filtration on $H_{i}^{G}(X)$ and the (decreasing) filtration $(F^{p})$ is the Hodge filtration on $H_{i}^{G}(X)\otimes\mathbb{C}$. We have the following [21, thm. 2.1.8].

Proposition 2.1** (Katz).**

Assume that $X$ has polynomial count with count polynomial $P(t)\in\mathbb{Z}[t]$. Then, the $E$-polynomial of $H_{*}(X)$ is given by $E(H_{*}(X);x,y)=P(xy)$. ∎

We deduce the following.

Corollary 2.2**.**

Assume that $H_{*}(X/G)$ is pure and $X$ has polynomial count. Then, the grading of $H_{*}(X/G)$ is even and its Poincaré polynomial is given by

[TABLE]

for all finite fields $\mathbb{F}_{q}$.

Proof.

Let $P_{X}(t)$ be the count polynomial of $X$. For each integer $i$ the approximation $X\times_{G}U_{i}$ has polynomial count with count polynomial $P_{U_{i}/G}(t)\,P_{X}(t)$, the product of the count polynomials of $U_{i}/G$ and $X$. From Proposition 2.1 we deduce that

[TABLE]

Taking the limit $i\to-\infty,$ from (2.1), (2.2) we get that the $E$-polynomial of $H_{*}(X/G)$ is

[TABLE]

Since $H_{*}(X/G)$ is pure, we deduce from (2.4) that it is also even, hence its Poincaré polynomial is

[TABLE]

Therefore, we have

[TABLE]

∎

3. The semi-nilpotent variety

Let ${\operatorname{k}\nolimits}$ be any algebraically closed field.

3.1. Quivers

Let $Q=(I,\Omega)$ be a finite quiver with set of vertices $I$ and set of arrows $\Omega$. Let $Q^{*}=(I,\Omega^{*})$ be the opposite quiver, with the set of arrows $\Omega^{*}=\{h^{*}\,;\,h\in\Omega\}$ where $h^{*}$ is the arrows obtained by reversing the orientation of $h$. The double quiver is $\bar{Q}=(I,\bar{\Omega})$ with $\bar{\Omega}=\Omega\sqcup\Omega^{*}$. We set $\varepsilon(h)=1$ if $h\in\Omega$ and $-1$ if $h\in\Omega^{*}$. Let $h^{\prime}$, $h^{\prime\prime}$ be the source and the target in $I$ of an arrow $h\in\Omega$. For each vertex $i$ let $\Omega_{i\bullet}$ be the set of arrows in $\Omega$ with source $i$ and $\Omega_{\bullet i}$ be the set of arrows with target $i$. Put $\Omega_{ij}=\Omega_{\bullet j}\cap\Omega_{i\bullet}$ and $\bar{\Omega}_{ij}=\Omega_{ij}\cup(\Omega_{ji})^{*}$. If $i\neq j$ we write

[TABLE]

Fix a tuple $v=(v_{i}\,;\,i\in I)$ in $\mathbb{Z}^{I}$. The Ringel bilinear form $\langle\bullet,\bullet\rangle$ on $\mathbb{Z}^{I}$ is given by

[TABLE]

Let $(v,w)=\langle v,w\rangle+\langle w,v\rangle$ be the Euler bilinear form. To avoid confusions we may write $(\bullet,\bullet)_{Q}$ for $(\bullet,\bullet)$. We set

[TABLE]

For each $i\in I$ let $\delta_{i}\in\mathbb{Z}^{I}$ denote the delta function at the vertex $i$.

Let ${\operatorname{k}\nolimits}Q$ be the path algebra of $Q$. A dimension vector of $Q$ is a tuple $v\in\mathbb{N}^{I}$. Let ${\operatorname{k}\nolimits}^{v}$ denote the $I$-graded vector space ${\operatorname{k}\nolimits}^{v}=\bigoplus_{i\in I}{\operatorname{k}\nolimits}^{v_{i}}$. We may abbreviate

[TABLE]

Let $Rep({\operatorname{k}\nolimits}Q,v)$ be the set of representations of ${\operatorname{k}\nolimits}Q$ in $V$, with its natural structure of affine ${\operatorname{k}\nolimits}$-variety. We abbreviate

[TABLE]

An element of $\operatorname{R}\nolimits(v)$ is a pair $\bar{x}=(x,x^{*})$ where $x=(x_{h}\,;\,h\in\Omega)$ belongs to $Rep({\operatorname{k}\nolimits}Q,v)$ and $x^{*}=(x_{h}\,;\,h\in\Omega^{*})$ belongs to $Rep({\operatorname{k}\nolimits}Q^{*},v)$. Since $\operatorname{R}\nolimits(v)$ decomposes as

[TABLE]

it has a canonical structure of a symplectic vector space.

3.2. The moment map and the preprojective algebra

The algebraic group $G(v)=\prod_{i\in I}GL(v_{i})$ acts by conjugation on $\operatorname{R}\nolimits(v)$, preserving the symplectic form. Let $\mathfrak{g}(v)$ be the Lie algebra of $G(v)$. The moment map for the action of $G(v)$ on $\operatorname{R}\nolimits(v)$ is the map $\mu\,:\,\operatorname{R}\nolimits(v)\to\mathfrak{g}(v)$ given by

[TABLE]

We define ${\operatorname{M}\nolimits}(v)=\mu^{-1}(0).$ The preprojective algebra is defined as

[TABLE]

Proposition 3.1**.**

Assume that $Q$ is not of finite Dynkin type. Then the algebra $\Pi$ is Calabi-Yau of homological dimension two, i.e., there are functorial isomorphisms

[TABLE]

for any finite-dimensional $\Pi$-modules $M$, $N$. Moreover, we have

[TABLE]

The above result is well-known to experts. We could not, however, locate a precise reference in this generality in the literature, and we give some details on the proof in the appendix. The restriction on the quiver is irrelevant to all the geometric constructions we need in this paper, as we can always embed a Dynkin quiver into a non-Dynkin one, and hence view a moduli space or stack of representations of a Dynkin quiver as a moduli space or stack of representations of a non-Dynkin quiver.

3.3. The group action

Consider the group

[TABLE]

The second product is over all pairs $(i,j)$ in $I\times I$ such that $i\neq j$ and $G_{\operatorname{dil}}=\mathbb{G}_{m}$. The group $G_{\Omega}$ acts on $\operatorname{R}\nolimits(v)$ as follows. The factor $G_{\operatorname{dil}}$ acts by dilation on the summand $Rep({\operatorname{k}\nolimits}Q^{*},v)$ of $\operatorname{R}\nolimits(v)$. Write

[TABLE]

Then $SP(2q_{i})$ and $GL(q_{ij})$ act in the obvious way on the first and the second summands. The zero set ${\operatorname{M}\nolimits}(v)$ is preserved by the action of the groups $G(v)$ and $G_{\Omega}$. Let $\theta,\theta^{*}$ be the cocharacters of $G_{\Omega}$ given by

[TABLE]

Thus $\theta(z)$ acts by multiplication by $z$ on the summand $Rep({\operatorname{k}\nolimits}Q,v)$ in $\operatorname{R}\nolimits(v)$, while $\theta^{*}(z)$ acts by multiplication by $z$ on the summand $Rep({\operatorname{k}\nolimits}Q^{*},v)$. Fix a closed connected subgroup ${{T}}$ of $G_{\Omega}$ which centralizes $\theta$. To simplify, we assume $T$ to be a subtorus

[TABLE]

where a tuple $(z_{h}\,,\,z)$ acts by $z_{h}$ on $x_{h}$ and by $z/z_{h}$ on $x_{h^{*}}$ for all $h\in\Omega$.

3.4. Definition

Fix an increasing flag $W$ of $I$-graded vector spaces in $V$

[TABLE]

Then, we consider the closed subset $\Lambda_{W}$ of ${\operatorname{M}\nolimits}(v)$ given by

[TABLE]

Up to conjugacy by an element of $G(v),$ the flag $W$ is completely determined by the sequence of dimension vectors

[TABLE]

The tuple $\nu=(\nu_{1},\dots,\nu_{r})$ is a composition of $v$, i.e., it is a tuple of dimension vectors with sum $v$. We write $\nu\vDash v$. We say that the flag $W$ is of type $\nu$. Then, we define

[TABLE]

where the dot denotes the $G(v)$-action on ${\operatorname{M}\nolimits}(v)$.

We say that a composition $\nu\vDash v$ is restricted if each $\nu_{p}$ is concentrated in a single vertex. Then, we also say that the flag $W$ is restricted. The semi-nilpotent variety $\Lambda^{0}(v)$ and the strongly semi-nilpotent variety $\Lambda^{1}(v)$ are the closed subsets of ${\operatorname{M}\nolimits}(v)$ given by

[TABLE]

where $\mu$ runs over the set of all compositions of $v$ and $\nu$ over the set of all restricted compositions of $v$. We have an obvious closed embedding

[TABLE]

The $T\times G(v)$-action on ${\operatorname{M}\nolimits}(v)$ yields a $T\times G(v)$-action on $\Lambda^{\flat}(v)$ for each $\flat$. In the next section we prove the following.

Theorem 3.2**.**

Assume that ${\operatorname{k}\nolimits}=\mathbb{C}$. Let $\flat$ be either 0 or 1.

• $\mathrm{(a)}$

$\Lambda^{\flat}(v)$* is a closed Lagrangian subvariety of $\operatorname{R}\nolimits(v)$, of dimension $d_{v}$,*

• $\mathrm{(b)}$

$H_{*}^{T\times G(v)}(\Lambda^{\flat}(v))$* is pure and even,*

• $\mathrm{(c)}$

${\operatorname{cl}\nolimits}:{A}_{*}^{T\times G(v)}(\Lambda^{\flat}(v))\to H_{2*}^{T\times G(v)}(\Lambda^{\flat}(v))$* is surjective,*

• $\mathrm{(d)}$

$M_{*}^{T\times G(v)}(\Lambda^{\flat}(v))$* is free as an $M^{*}_{T}$-module.*

Remark 3.3*.*

Our definition of the semi-nilpotent and strongly semi-nilpotent varieties is the same as in [3, §1.1]. If the quiver $Q$ has no oriented cycle which does not involve 1-loops, i.e., if any oriented cycle in $Q$ is a product of 1-loops, then the strongly semi-nilpotent variety and the semi-nilpotent one coincide.

Remark 3.4*.*

Switching the roles of $\Omega$ and $\Omega^{*}$ we get the notion of $*$ semi-nilpotent variety which is denoted by the symbol $\Lambda^{*,\flat}(v)$. To avoid confusion we may also indicate the quiver $Q$ and write $\Lambda^{\flat}(v)/Q$ for $\Lambda^{\flat}(v)$. Thus, we have a canonical isomorphism $\Lambda^{\flat}(v)/Q^{*}=\Lambda^{*,\,\flat}(v)/Q$. We have also an isomorphism $\Lambda^{\flat}(v)/Q\to\Lambda^{\flat}(v)/Q^{*}$ such that $(x,x^{*})\mapsto({}^{t}x,{}^{t}x^{*})$, where the upper script ${}^{t}\bullet$ indicates the transpose matrix.

Remark 3.5*.*

In the theorem it is essential to consider the $T\times G(v)$-equivariant Borel-Moore homology rather than usual Borel-Moore homology. For instance, let $Q$ be the quiver of type $A_{2}$ and set $v=(1,1)$. Then, we have $\Lambda^{\flat}(v)=\{(x,x^{*})\in{\operatorname{k}\nolimits}^{2}\,;\,x\,x^{*}=0\}$, hence

[TABLE]

which is neither even nor pure. The formulas in Theorem 5.4 below, with $T=1$, yield the following equality of formal series

[TABLE]

This example also shows that $H_{*}^{T\times G(v)}(\Lambda^{\flat}(v))$ is not, in general, free as an $H^{*}_{T\times G(v)}$-module.

Remark 3.6*.*

Let $(v)$ be the composition of $v$ with only one term. Then, we have

[TABLE]

3.5. The stratification

Let us introduce two stratifications of $\Lambda^{0}(v)$ by $G(v)$-invariant locally closed subsets

[TABLE]

To do this, first consider the path algebra ${\operatorname{k}\nolimits}\bar{Q}$ equipped with the $\mathbb{N}$-grading such that all arrows in $\Omega$ are of degree $1$ and all arrows in $\bar{\Omega}$ are of degree [math]. For $\bar{x}\in\Lambda^{0}(v)$ we consider the ${\operatorname{k}\nolimits}$-subalgebra $A_{\bar{x}}$ of $\operatorname{End}\nolimits(V)$ generated by 1 and the action of the elements $x_{h}$ with $h\in\bar{\Omega}$. We equip it with the filtration such that

[TABLE]

where the map $ev_{\bar{x}}$ takes a path $\sigma=h_{1}\cdots h_{s}$ to the element $x_{h_{1}}\cdots x_{h_{s}}$. We set

[TABLE]

Let $W_{\bar{x}}$ be the unique increasing flag of $I$-graded vector spaces

[TABLE]

where $W_{p}$ is the annihilator of $A_{\bar{x}}[\geqslant\!p]$. Given an increasing flag $W$ of $I$-graded vector spaces in $V$ of type $\nu$, we consider the open subsets $\Lambda_{W}^{\circ}\subseteq\Lambda_{W}$ and $\Lambda_{\nu}^{\circ}\subseteq\Lambda_{\nu}$ given by

[TABLE]

Note that the set $\Lambda_{W}$ may be reducible, and $\Lambda^{\circ}_{W}$ may not be dense in $\Lambda_{W}$.

Consider also the decreasing flag

[TABLE]

given by $F^{p}=A_{\bar{x}}[\geqslant\!p](V)$ for each $p$, and the sets

[TABLE]

Note that $F^{\bar{x}}$ is of type $\nu$ if we have $\nu_{p}=\dim(F^{p-1}/F^{p})$ for all $p$.

3.6. The one vertex quiver

In this section we consider in more details the case of the quiver $\operatorname{J}\nolimits(i,q)$ with vertex set $\{i\}$ and $q$ loops. Then, the dimension vector $v$ is an integer and we have $\Lambda^{0}(v)=\Lambda^{1}(v)$. We write $\Lambda$ for either $\Lambda^{0}$ or $\Lambda^{1}$.

3.6.1. The isotropic case

Assume that $q=1$. Then $Q$ is the Jordan quiver and the variety $\Lambda(v)$ is the set of commuting pairs $(x,x^{*})$ in $\mathfrak{g}(v)\times\mathfrak{g}(v)$ with $x$ nilpotent. We identify $\mathfrak{g}(v)\times\mathfrak{g}(v)$ with $\mathfrak{g}(v)\times\mathfrak{g}(v)^{*}$ via the trace map. For each partition $\nu\vdash v$ let $\nu^{*}$ be the partition dual to $\nu$ and let $O_{\nu}$ be the nilpotent $G(v)$-orbit of Jordan type $\nu^{*}$. Let $P\subseteq G(v)$ be the standard parabolic subgroup with block-type $\nu$. Let $U,$ $L$ be the unipotent radical of $P$ and its standard Levi complement. Let $\mathfrak{u}$, $\mathfrak{l}$, $\mathfrak{p}$ be their Lie algebras. Then $O_{\nu}$ is the $G(v)$-saturation of the unique dense $P$-orbit $\mathfrak{u}^{\circ}$ in the Lie algebra $\mathfrak{u}$ of $U$. In other words $O_{\nu}$ is the Richardson orbit associated with the standard parabolic $P$ of $G(v)$, see e.g., [6, thm. 7.1.1], [22, §3] for details (note that $O_{\nu}$ is usually labelled by the partition $\nu^{*}$). Then, we have

[TABLE]

We equip the set of partitions $\{\nu\,;\,\nu\vdash v\}$ with the anti-dominant ordering

[TABLE]

We define $\Lambda_{{\trianglelefteqslant}\,\nu}^{\circ}=\bigsqcup_{\mu\,{\trianglelefteqslant}\,\nu}\Lambda_{\mu}^{\circ}.$

Proposition 3.7**.**

Assume that $q=1$. Then, we have

• $\mathrm{(a)}$

$\Lambda_{\nu}^{\circ}$* is the conormal bundle to $O_{\nu}$,*

• $\mathrm{(b)}$

${\operatorname{Irr}\nolimits}(\Lambda(v))=\{\overline{\Lambda_{\nu}^{\circ}}\,;\,\nu\vdash v\}$,

• $\mathrm{(c)}$

*if $\nu\vdash v$ then *$\overline{\Lambda^{\circ}_{\nu}}\subseteq\Lambda_{\nu}\subseteq\Lambda_{{\trianglelefteqslant}\,\nu}^{\circ},$

• $\mathrm{(d)}$

$\Lambda_{W}$, $\Lambda_{\nu}$ are reducible in general,

• $\mathrm{(e)}$

the obvious map $G(v)\times_{P}\Lambda_{W}^{\circ}\to\Lambda^{\circ}_{\nu}$ is an isomorphism.

∎

3.6.2. The hyperbolic case

Assume that $q>1$. We equip the set of compositions $\{\nu\,;\,\nu\vDash v\}$ with the anti-dominant ordering.

Proposition 3.8**.**

Assume that $q>1$. Then, we have

• $\mathrm{(a)}$

$\Lambda_{W}$* is irreducible, $\Lambda_{\nu}$ is irreducible and Lagrangian in $\operatorname{R}\nolimits(v)$,*

• $\mathrm{(b)}$

${\operatorname{Irr}\nolimits}(\Lambda(v))=\{\Lambda_{\nu}\,;\,\nu\vDash v\}$,

• $\mathrm{(c)}$

if $\nu\vDash v$ then $\overline{\Lambda^{\circ}_{\nu}}=\Lambda_{\nu}\subseteq\Lambda_{{\trianglelefteqslant}\,\nu}^{\circ},$

• $\mathrm{(d)}$

the obvious map $G(v)\times_{P}\Lambda_{W}^{\circ}\to\Lambda_{\nu}^{\circ}$ is an isomorphism.

Proof.

Note first that $\Lambda^{0}(v)=\Lambda^{1}(v)$ since $Q$ has no oriented cycle which is not a product of $1$-loops. We write $\Lambda(v)=\Lambda^{0}(v)=\Lambda^{1}(v)$. By [2] the set $\Lambda^{\circ}_{\nu}$ is non-empty for any composition $\nu\vDash v$ and we have

[TABLE]

Let us now fix $\nu=(\nu_{1},\ldots,\nu_{r})$. It is clear that $\overline{\Lambda^{\circ}_{\nu}}\subset\Lambda_{\nu}$. Let us prove that $\Lambda_{\nu}$ is irreducible of dimension $qv^{2}$. Fix a flag $W$ of type $\nu$ and let $P$ be its stabilizer in $G(v)$. Since the map $G(v)\times_{P}\Lambda_{W}\to\Lambda_{\nu}$ is surjective, it is enough to show that $\Lambda_{W}$ is irreducible of dimension

[TABLE]

This is a consequence of the following two facts :

• (i)

every irreducible component of $\Lambda_{W}$ is of dimension $\geqslant d(\nu)$,

• (ii)

for any composition $\mu\neq\nu$ we have $\Lambda^{\circ}_{\mu}\not\subseteq\Lambda_{\nu}$.

Indeed, claim (i) implies that the closed subset $\Lambda_{\nu}\subseteq\Lambda(v)$ is of the form $\Lambda_{\nu}=\overline{\Lambda^{\circ}_{\nu}}\cup\overline{\Lambda^{\circ}_{\mu_{1}}}\cup\cdots\cup\overline{\Lambda^{\circ}_{\mu_{r}}}$ for some compositions $\mu_{1},\ldots,\mu_{r}$. Now, claim (ii) implies that $r=0$ and thus $\Lambda_{\nu}=\overline{\Lambda^{\circ}_{\nu}}$.

It remains to prove both claims. For part (i) we write

[TABLE]

View $\Lambda_{W}$ as a closed subvariety of the product

[TABLE]

via the map $\bar{x}\mapsto(\bar{x}|_{W_{1}}\,,\,\bar{x}|_{W_{2}}\,,\cdots).$ For each $(a,b)\in\mathbb{N}^{2}$ with $a<b$, consider the set

[TABLE]

It is the zero locus of $a^{2}+a(b-a)-1$ equations in a vector space of dimension $2qa^{2}+q(b-a)^{2}+2qa(b-a)$. Hence every irreducible component of $X(a,b)$ is of dimension

[TABLE]

The following result is proved in Appendix A.

Lemma 3.9**.**

For any $q>1$ and $v>0$ the variety ${\operatorname{M}\nolimits}(v)$ is irreducible and of dimension $(2q-1)v^{2}+1$. ∎

We deduce that the codimension of any irreducible component of ${\operatorname{M}\nolimits}(a,b)$ is

[TABLE]

in ${\operatorname{M}\nolimits}(a)\times{\operatorname{M}\nolimits}(b).$ Now observe that

[TABLE]

It follows that the codimension of any irreducible component of $\Lambda_{W}$ is $\leqslant\sum_{i\geqslant 0}c(\nu_{\leqslant i}\,,\,\nu_{\leqslant i+1})$ in ${\operatorname{M}\nolimits}(\nu_{\leqslant 1})\times\cdots\times{\operatorname{M}\nolimits}(v)$ and hence of dimension

[TABLE]

as wanted. This proves claim (i).

We now turn to claim (ii). To show that $\Lambda^{\circ}_{\mu}\not\subseteq\Lambda_{\nu}$ we will prove the existence of an element $\bar{x}\in\Lambda^{\circ}_{\mu}$ which satisfies the semi-nilpotency condition (3.7) with respect to a single flag (necessarily of type $\mu$). For this we will use the following basic construction. Pick distinct non-zero simple $\Pi$-modules $S_{1},\ldots,S_{k}$. We have $\operatorname{Hom}\nolimits(S_{i},S_{j})=\operatorname{Ext}\nolimits^{2}(S_{i},S_{j})=\{0\}$ for any $i\neq j$. Thus

[TABLE]

Choose $\xi_{1}\in\operatorname{Ext}\nolimits^{1}(S_{2},S_{1})\backslash\{0\}$ and denote by

[TABLE]

the corresponding short exact sequence. We have

[TABLE]

We may thus choose $\xi_{2}\in\operatorname{Ext}\nolimits^{1}(S_{3},M_{1})\backslash\{0\}$ and denote by

[TABLE]

the corresponding short exact sequence. Continuing this process we construct a sequence of extensions $0\to M_{l-1}\to M_{l}\to S_{1+1}\to 0$ for $l=1,\ldots,k-1$. We claim that $M_{k-1}$ has a unique composition sequence

[TABLE]

Arguing by induction we see that it is enough to prove that $M_{k-1}$ has a simple socle $S_{1}$. From the fact that the $S_{i}$’s are distinct we easily get that $\operatorname{Hom}\nolimits(S_{1},M_{k-1}/S_{1})=\{0\}$ and thus $\operatorname{Hom}\nolimits(S_{1},M_{k-1})={\operatorname{k}\nolimits}$. We now prove that $\operatorname{Hom}\nolimits(S_{l},M_{k-1})=\{0\}$ for any $l\neq 1$. As above, we have $\operatorname{Hom}\nolimits(S_{l},M_{k-1}/M_{l-1})=\{0\}$ and thus $\operatorname{Hom}\nolimits(S_{l},M_{k-1})=\operatorname{Hom}\nolimits(S_{l},M_{l-1})$. Because $\operatorname{Hom}\nolimits(S_{l},M_{l-2})=\{0\}$ and $S_{l}$ is simple, any map $S_{l}\to M_{l-1}$ is, up to a scalar, a section of the canonical map $M_{l-1}\to S_{l}$. But $M_{l-1}$ is a non-split extension of $S_{l}$ and $M_{l-2}$. Thus, such a section doesn’t exist. We conclude that $\operatorname{Hom}\nolimits(S_{l},M_{l-1})=\{0\}$ as wanted.

We may now prove the claim (ii) above. Fix $\mu=(\mu_{1},\ldots,\mu_{k})\neq\nu$. Because the ground field ${\operatorname{k}\nolimits}$ is infinite, we may choose distinct simple ${\operatorname{k}\nolimits}Q$-modules (and thus also $\Pi$-modules) $S_{1},\ldots,S_{k}$ of respective dimensions $\mu_{1},\ldots,\mu_{k}$ and build a $\Pi$-module $M_{k-1}$ as above. It is easy to see that $M_{k-1}\in\Lambda^{\circ}_{\mu}$ but $M_{k-1}$ doesn’t stabilize any flag of subspaces of type $\nu$. This proves the parts (a) and (b) of the proposition.

The first part of the statement (c) was proved above. To prove the rest of (c), note that

[TABLE]

where $\nu$, $\nu^{\prime}$ are the types of $W$, $W^{\prime}$ respectively. We deduce that we have $\Lambda_{\nu}\subseteq\Lambda_{{\trianglelefteqslant}\nu}^{\circ}.$ Part (d) of the proposition is obvious. ∎

3.7. Quiver varieties

This section is a reminder on the Nakajima quiver varieties associated with $Q$. We assume that the reader is familiar with the formalism of quiver varieties and we refer to [8], [9], [30], [31], [32], [33], [34] for the proofs of the facts recalled below.

3.7.1. Basics

The space of representations of dimension vectors $v,$ $w$ of the framed quiver associated with $\bar{Q}$ is

[TABLE]

where $\operatorname{Hom}\nolimits_{I}(V,W)$ is the set of $I$-graded ${\operatorname{k}\nolimits}$-linear homomorphisms and

[TABLE]

Since $\operatorname{R}\nolimits(v,w)$ is canonically identified with the cotangent of a vector space, it admits a canonical symplectic structure.

The algebraic group $T\times G(v)\times G(w)$ acts on $\operatorname{R}\nolimits(v,w)$ so that the element $(z_{h},z)\in T$ takes $(a,a^{*})$ to $(a,za^{*})$. Consider the cocharacters $\xi$, $\xi^{*}$ of $T\times G(v)$ given by

[TABLE]

We can view them as cocharacters of $T\times G(v)\times G(w)$ under the obvious inclusion

[TABLE]

Then $\xi(t)$, $\xi^{*}(t)$ act by multiplication by $t$ on the summands of $\operatorname{R}\nolimits(v,w)$ given by

[TABLE]

Equivalently, we may write

[TABLE]

The $G(v)$-action on $\operatorname{R}\nolimits(v,w)$ preserves the symplectic form and admits the moment map

[TABLE]

where we write

[TABLE]

The categorical quotient of the zero set

[TABLE]

by $G(v)$ is the variety

[TABLE]

It is affine, reduced, irreducible, singular in general and $T\times G(w)$-equivariant. Let

[TABLE]

denote the canonical map. We may write $0=\rho_{0}(0,0)$. The set of ${\operatorname{k}\nolimits}$-points of $\mathfrak{M}_{0}(v,w)$ is in bijection with the set of closed $G(v)$-orbits in ${\operatorname{M}\nolimits}(v,w)$ so that $\rho_{0}(\bar{x},\bar{a})$ is identified with the unique closed orbit in the closure of the $G(v)$-orbit of $(\bar{x},\bar{a})$. Equivalently, the set of ${\operatorname{k}\nolimits}$-points of $\mathfrak{M}_{0}(v,w)$ is in bijection with the set of isomorphism classes of semisimple representations in ${\operatorname{M}\nolimits}(v,w)$ and $\rho_{0}$ maps a representation to the sum of its constituents.

Given a character $\theta$ of $G(v)$ we consider the space of semi-invariants of weight $\theta$

[TABLE]

We have the $T\times G(w)$-equivariant projective morphism

[TABLE]

The Hilbert-Mumford criterion implies that $\mathfrak{M}_{\theta}(v,w)$ is the geometric quotient by $G(v)$ of an open subset ${\operatorname{M}\nolimits}_{\theta}(v,w)$ of ${\operatorname{M}\nolimits}(v,w)$ consisting of the $\theta$-semistable representations. Let

[TABLE]

be the open subsets of semistable points. Replacing everywhere ${\operatorname{M}\nolimits}(v,w)$ by $\operatorname{R}\nolimits(v,w)$ we define an open subset $\operatorname{R}\nolimits_{\theta}(v,w)$ of $\operatorname{R}\nolimits(v,w)$ such that ${\operatorname{M}\nolimits}_{\theta}(v,w)=\operatorname{R}\nolimits_{\theta}(v,w)\cap{\operatorname{M}\nolimits}(v,w)$. Let $\rho_{\theta}(\bar{x},\bar{a})$ be the image in $\mathfrak{M}_{\theta}(v,w)$ of the tuple $(\bar{x},\bar{a})\in{\operatorname{M}\nolimits}_{\theta}(v,w)$. We have

[TABLE]

We say that the character $\theta$ given by $\theta(g)=\prod_{i\in I}\det(g_{i})^{-\theta_{i}}$ is generic if neither equations

[TABLE]

with $\theta_{\infty}=-\sum_{i\in I}\theta_{i}v_{i}$ have integer solutions $(u_{i})$ satisfying $0\leqslant u_{i}\leqslant v_{i}$ other than the trivial solutions $(u_{i})=0$ or $(u_{i})=(v_{i}).$ If $\theta$ is generic then any semistable pair $(\bar{x},\bar{a})$ is stable and in that case the map

[TABLE]

is a $G(v)$-torsor. In particular, the variety $\mathfrak{M}_{\theta}(v,w)$ is smooth, symplectic, of dimension

[TABLE]

The character $s$ given by $s(g)=\prod_{i\in I}\det(g_{i})^{-1}$ is generic. We abbreviate

[TABLE]

and we write (semi)stable for $s$-(semi)stable. A representation $(\bar{x},\bar{a})$ in $\operatorname{R}\nolimits(v,w)$ is semistable if and only if it does not admit any nonzero subrepresentation whose dimension vector belongs to $\mathbb{N}^{I}\times\{0\}$.

Remark 3.10*.*

Assume that ${\operatorname{k}\nolimits}=\mathbb{C}$. By [30], the $G(v)$-action on ${\operatorname{M}\nolimits}_{s}(v,w)$ is set-theoretically free and ${\operatorname{M}\nolimits}_{s}(v,w)$ is a smooth scheme. Hence, the natural morphism of smooth schemes

[TABLE]

is a bijection on closed points and, therefore, it is an isomorphism. In other words, the $G(v)$-action on ${\operatorname{M}\nolimits}_{s}(v,w)$ is free. Hence the $G(v)$-scheme ${\operatorname{M}\nolimits}_{s}(v,w)$ is a $G(v)$-torsor over $\mathfrak{M}(v,w)$ and the quotient stack ${\operatorname{M}\nolimits}_{s}(v,w)/G(v)$ is represented by the scheme $\mathfrak{M}(v,w)={\operatorname{M}\nolimits}_{s}(v,w)/\!\!/G(v)$.

3.7.2. Crawley-Boevey’s trick

It may be useful to realize the quiver varieties as moduli spaces of representations of some preprojective algebra. More precisely, consider the quiver $\tilde{Q}=(\tilde{I},\tilde{\Omega})$ obtained from $Q$ by adding one new vertex $\infty$ and $w_{i}$ arrows from $\infty$ to the vertex $i$ for all $i\in I$. For each $v\in\mathbb{N}^{I}$ set

[TABLE]

If $w\neq 0$ we identify $G(v)$ with $PGL(\tilde{v})$. Then, there is a canonical $G(v)$-equivariant isomorphism ${\operatorname{M}\nolimits}(v,w)\simeq{\operatorname{M}\nolimits}(\tilde{v})$ and $\mathfrak{M}(v,w)$ or $\mathfrak{M}_{0}(v,w)$ may be viewed as moduli spaces of (stable, resp. semisimple) modules of dimension $\tilde{v}$ over the preprojective algebra $\tilde{\Pi}$. The symmetric bilinear forms for $Q$ and $\tilde{Q}$ are related as follows

[TABLE]

Let $\tilde{\Pi}$ be the preprojective algebra of $\tilde{Q}$.

3.7.3. Representation types

If the representation $z\in{\operatorname{M}\nolimits}(v,w)$ is semisimple then we can decompose it into its simple constituents $z=z_{1}^{\oplus d_{1}}\oplus z_{2}^{\oplus d_{2}}\oplus\cdots z_{s}^{\oplus d_{s}}$ where the $z_{r}$’s are non-isomorphic simples. If $u_{r}=(v_{r},w_{r})$ is the dimension vector of $z_{r}$, i.e., if $z_{r}$ is a simple representation in ${\operatorname{M}\nolimits}(v_{r},w_{r})$, then we say that $z$ has the representation type

[TABLE]

If $w\neq 0$ and $z$ is stable, then there is a unique integer $r$ such that $w_{r}=w$ and $w_{r^{\prime}}=0$ for all $r^{\prime}\neq r$, hence we may assume that the representation type of $z$ has the following form

[TABLE]

where $(v_{1},v_{2},\dots,v_{s})\vDash v$ and the tuples $(d_{2},v_{2}),\dots(d_{s},v_{s})$ are only defined up to a permutation. Let $RT(v,w)$ be the set of all representation types of dimension $(v,w)$. Let

[TABLE]

be the set of semisimple representations with representation type equal to $\tau$. We have the following stratification by smooth irreducible locally closed subsets

[TABLE]

Defining $\tilde{u}_{1},\dots,\tilde{u}_{s}$ as in §3.7.2 we get

[TABLE]

Since $\mathfrak{M}_{0}(v,w)$ is irreducible, there is a unique representation type $\kappa_{v,w}$ such that $\mathfrak{M}_{0}(\kappa_{v,w})$ is a dense open subset of $\mathfrak{M}_{0}(v,w)$. We call it the generic representation type of $RT(v,w)$.

The stabilizer in $G(v)$ of an arbitrary element of ${\operatorname{M}\nolimits}(\tau)$ is a reductive group which is conjugate by an element of $G(v)$ to the group $G_{\tau}=\prod_{r=1}^{s}GL(d_{r})$. Write $\tau^{\prime}\leqslant\tau$ if and only if $G_{\tau}$ is conjugate to a subgroup of $G_{\tau^{\prime}}$. Then, we have

[TABLE]

Given two representation types

[TABLE]

we define their sum by

[TABLE]

Whenever $\sum_{t=1}^{r}v_{t}\leqslant v$ and $\sum_{t=1}^{r}w_{t}\leqslant w$, the direct sum yields a closed embedding

[TABLE]

The relation with representation types is given by the following relation

[TABLE]

For each representation type $\tau$ we write

[TABLE]

If $\mathfrak{M}(\tau)\neq\emptyset$ then the map $\pi$ restricts to a locally trivial fibration $\mathfrak{M}(\tau)\to\mathfrak{M}_{0}(\tau)$ such that, see [8] and [9, cor. 6.4],

[TABLE]

In particular, if the map $\pi:\mathfrak{M}(v,w)\to\mathfrak{M}_{0}(v,w)$ is birational, then it is semismall.

The map $\pi$ may not be surjective. Its image is an irreducible closed subvariety of $\mathfrak{M}_{0}(v,w)$ which is a union of strata. For later purposes, we will need the following stronger statement. Let us call dimension type of a semisimple representation $z$ the sequence $d=(d_{u})$ where $d_{u}$ is the total number of simple representations (counted with multiplicity) of dimension $u$ occuring in $z$. The representation type determines the dimension type but the converse is false since we lose the information of the multiplicity of each individual simple representation of a given dimension vector. Consider the locally closed subvariety

[TABLE]

parametrizing semisimple representations of dimension type $d$.

Proposition 3.11**.**

For any $v,$ $w$, the image of $\pi$ is a union of strata $\mathfrak{M}_{0}(\!(d)\!)$.

Proof.

See the appendix. ∎

3.7.4. The Bialynicki-Birula decomposition

Let $\gamma$ be a cocharacter of $T\times G(w)$. Composing it with the $T\times G(w)$-action we get a $\mathbb{G}_{m}$-action $\bullet$ on $\mathfrak{M}(v,w)$. We want to describe the fixed point locus $\mathfrak{M}(v,w)^{\bullet}$ and the Bialynicki-Birula attracting variety

[TABLE]

To do this, fix a cocharacter $\rho$ of $G(v)$. Since the $G(v)$-action on ${\operatorname{M}\nolimits}_{s}(v,w)$ commutes with the $T\times G(w)$-action, we can view the product $\gamma\rho$ as a cocharacter of $T\times G(v)\times G(w)$. Let $L$ be the centralizer of $\rho$ in $G(v)$ and set

[TABLE]

Proposition 3.12**.**

• $\mathrm{(a)}$

The maps $G(v)\times_{L}{\operatorname{M}\nolimits}[\rho]\to G(v)\cdot{\operatorname{M}\nolimits}[\rho]$ and $G(v)\times_{P}\operatorname{L}\nolimits[\rho]\to G(v)\cdot\operatorname{L}\nolimits[\rho]$ are invertible,

• $\mathrm{(b)}$

$\mathfrak{M}[\rho]$* is a sum of connected components of $\mathfrak{M}(v,w)^{\bullet}$ such that $\mathfrak{M}(v,w)^{\bullet}=\bigsqcup_{\rho}\mathfrak{M}[\rho],$*

• $\mathrm{(c)}$

$\mathfrak{L}[\rho]=\{z\in\mathfrak{M}(v,w)\,;\,\lim_{t\to 0}t\bullet z\in\mathfrak{M}[\rho]\}$* and $\mathfrak{L}(v,w)^{\bullet}=\bigsqcup_{\rho}\mathfrak{L}[\rho].$*

Proof.

To prove the first claim of (a), observe that for two distinct cocharacters $\rho$ and $\rho^{\prime}$ the freeness of the $G(v)$-action on ${\operatorname{M}\nolimits}_{s}(v,w)$ implies that ${\operatorname{M}\nolimits}[\rho]\cap{\operatorname{M}\nolimits}[\rho^{\prime}]=\emptyset$. For the second claim we must check that for each $g\in G(v)$ we have

[TABLE]

If $z$ and $g\cdot z$ belong to $\operatorname{L}\nolimits[\rho]$, then the following limits exist in ${\operatorname{M}\nolimits}_{s}(v,w)$

[TABLE]

Since the $G(v)$-action on ${\operatorname{M}\nolimits}_{s}(v,w)$ is free, this implies that $g\in P$. Part (b) is well-known, see e.g. [32, §4]. Let us sketch a proof of (c). Put

[TABLE]

We must check that $\mathfrak{L}[\rho]=\mathfrak{L}^{\prime}[\rho]$. We have

[TABLE]

and the Bialynicki-Birula theorem implies that $\mathfrak{L}^{\prime}[\rho]$ is the disjoint union of affine space bundles $\mathfrak{L}^{\prime}_{1},\dots,\mathfrak{L}^{\prime}_{r}$ over the connected components $\mathfrak{M}_{1},\dots,\mathfrak{M}_{r}$ of $\mathfrak{M}[\rho]$ with a contracting $\mathbb{G}_{m}$-action on each $\mathfrak{L}^{\prime}_{i}$ given by $\bullet$. Set $\mathfrak{L}_{i}=\mathfrak{L}^{\prime}_{i}\cap\mathfrak{L}[\rho]$ for some $i=1,\dots r$. It is not difficult to prove that $\mathfrak{L}_{i}$ and $\mathfrak{L}^{\prime}_{i}$ have the same dimension. Hence, since $\mathfrak{L}_{i}$ is locally closed in $\mathfrak{L}^{\prime}_{i}$, it is a dense open subset. Finally, since $\mathfrak{L}_{i}$ contains $\mathfrak{M}_{i}$ and is preserved by the contracting $\mathbb{G}_{m}$-action given by $\bullet$, it equals $\mathfrak{L}^{\prime}_{i}$.

∎

Remark 3.13*.*

We will actually prove later, see Corollary 4.9, that $\mathfrak{M}[\rho]$ is connected for any $\rho$. In particular, the map $\mathfrak{L}[\rho]\to\mathfrak{M}[\rho]$ is an affine fibration.

3.8. Hecke correspondences

3.8.1. Basics

Let $v_{1}$, $v_{2}$ and $w$ be dimension vectors. Set $v=v_{1}+v_{2}$. Fix an $I$-graded subspace $V_{1}\subseteq V$ with dimension vector $v_{1}$ and let $P$ be the corresponding parabolic subgroup of $G(v)$. Fix isomorphisms ${\operatorname{k}\nolimits}^{v_{1}}\simeq V_{1}$ and ${\operatorname{k}\nolimits}^{v_{2}}\simeq V/V_{1}$. Let us write

[TABLE]

Note that $\mathfrak{h}[v_{1},v_{2}\,;\,w]$ is indeed a geometric quotient. It is called a Hecke correspondence. For each $(\bar{x},\bar{a})\in\operatorname{H}\nolimits[v_{1},v_{2}\,;\,w]$ we write

[TABLE]

Proposition 3.14**.**

• $\mathrm{(a)}$

There is a closed embedding $\mathfrak{h}[v_{1},v_{2}\,;\,w]\subseteq\mathfrak{M}(v,w)\times\mathfrak{M}(v_{1},w)$,

• $\mathrm{(b)}$

$\mathfrak{h}[v_{1},v_{2}\,;\,w]$* is projective over $\mathfrak{M}(v,w)$.*

Proof.

We have

[TABLE]

where

[TABLE]

Thus, the assignment

[TABLE]

gives rise to a map $\mathfrak{h}[v_{1},v_{2}\,;\,w]\to\mathfrak{M}(v,w)\times\mathfrak{M}(v_{1},w).$ For each elements $(\bar{x},\bar{a})\in{\operatorname{M}\nolimits}_{s}(v,w)$ and $(\bar{x}_{1},\bar{a}_{1})\in{\operatorname{M}\nolimits}_{s}(v_{1},w)$ there is at most one embedding $\phi_{1}:V_{1}\to V$ such that

[TABLE]

because if there are two of them, say $\phi_{1}$ and $\phi^{\prime}_{1}$, then the $I$-graded subspace ${\operatorname{Im}\nolimits}(\phi_{1}-\phi^{\prime}_{1})$ of $V$ is destabilizing for $(\bar{x},\bar{a})$, hence it is $\{0\}$. This proves the part (a). Part (b) follows from the projectivity of Grassmanians. ∎

For each $G(v_{2})$-invariant locally closed subset $\SS\subseteq{\operatorname{M}\nolimits}(v_{2})$ we define

[TABLE]

We may abbreviate

[TABLE]

Recall that a subvariety $V$ of a symplectic manifold $X$ is isotropic if the restriction of the symplectic form to the smooth locus of $V$ vanishes. Let $X^{{\operatorname{op}\nolimits}}$ denote the manifold $X$ with the opposit symplectic form.

Lemma 3.15**.**

Let $\SS$ be an isotropic subvariety of the symplectic vector space $\operatorname{R}\nolimits(v_{2})$. Then $\mathfrak{h}[v_{1},\SS\,;\,w]$ is an isotropic subvariety of the symplectic manifold $\mathfrak{M}(v,w)^{{\operatorname{op}\nolimits}}\times\mathfrak{M}(v_{1},w)$.

Proof.

Consider the closed embedding, see (3.21),

[TABLE]

The variety $\mathfrak{M}(v,w)^{\operatorname{op}\nolimits}\times\mathfrak{M}(v_{1},w)$ is the symplectic reduction of the symplectic manifold

[TABLE]

relative to the $G(v)\times G(v_{1})$-action which is Hamiltonian. The variety $\mathfrak{h}[v_{1},S]$ is the image in $\mathfrak{M}(v,w)\times\mathfrak{M}(v_{1},w)$ of the $G(v)\times G(v_{1})$-saturation of the set

[TABLE]

under the $G(v)\times G(v_{1})$-action on $\operatorname{R}\nolimits(v,w)^{\operatorname{op}\nolimits}\times\operatorname{R}\nolimits(v_{1},w)$. The symplectic form on $\mathfrak{M}(v,w)^{\operatorname{op}\nolimits}\times\mathfrak{M}(v_{1},w)$ is the unique symplectic form whose pullback to ${\operatorname{M}\nolimits}_{s}(v,w)\times{\operatorname{M}\nolimits}_{s}(v_{1},w)$ equals the restriction of the symplectic form of $\operatorname{R}\nolimits(v,w)^{\operatorname{op}\nolimits}\times\operatorname{R}\nolimits(v_{1},w)$. Therefore, to prove that $\mathfrak{h}[v_{1},\SS]$ is isotropic it is enough to check that (3.24) is an isotropic subvariety of $\operatorname{R}\nolimits(v,w)^{\operatorname{op}\nolimits}\times\operatorname{R}\nolimits(v_{1},w)$.

According to §3.7.1, the symplectic form on $\operatorname{R}\nolimits(v,w)$ is given by

[TABLE]

Thus, if we have

[TABLE]

we deduce that

[TABLE]

where $X_{1}\in\operatorname{End}\nolimits(V_{1})$ is the restriction of $X$, $X_{2}\in\operatorname{End}\nolimits(V/V_{1})$ is the induced operator, etc. Since $\SS$ is an isotropic subvariety of $\operatorname{R}\nolimits(v_{2})$, we deduce that

[TABLE]

∎

Proposition 3.16**.**

The Hecke correspondence $\mathfrak{h}[v_{1},\Lambda_{(v_{2})}\,;\,w]$ is a closed Lagrangian local complete intersection of the symplectic manifold $\mathfrak{M}(v,w)^{{\operatorname{op}\nolimits}}\times\mathfrak{M}(v_{1},w)$.

Proof.

By Lemma 3.15, the variety $\mathfrak{h}[v_{1},\Lambda_{(v_{2})}\,;\,w]$ is isotropic. Since the $P$-action on $\operatorname{H}\nolimits[v_{1},\Lambda_{(v_{2})}\,;\,w]$ is free, we deduce that

[TABLE]

Fix $I$-graded vector spaces $V_{1}$, $V_{2}$, $W$ with dimension vectors $v_{1}$, $v_{2}$, $w$. For each $i$ we abbreviate $V_{1,i}=(V_{1})_{i}$ and $V_{2,i}=(V_{2})_{i}$. By construction $\operatorname{H}\nolimits[v_{1},\Lambda_{(v_{2})}\,;\,w]$ is an open subset of the zero fiber of the map

[TABLE]

The domain of the above map being irreducible, using (3.15) we deduce that every irreducible component of $\operatorname{H}\nolimits[v_{1},\Lambda_{(v_{2})}]$ is of dimension at least

[TABLE]

It follows that the variety $\mathfrak{h}[v_{1},\Lambda_{(v_{2})}\,;\,w]$ is a Lagrangian local complete intersection. ∎

3.8.2. One vertex Hecke correspondences.

In this section we study the Hecke correspondences in the particular case where $v_{2}$ is concentrated at a single vertex. Fix $i\in I$ and $l\in\mathbb{N}$ such that $v_{2}=l\delta_{i}$. Write $q=q_{i}$. We concentrate on the case $q>1$. Recall that

[TABLE]

Since the set $\Lambda_{\nu}$ is $G(v_{2})$-invariant, the set $\mathfrak{h}[v_{1},\Lambda_{\nu}\,;\,w]$ is well-defined.

Proposition 3.17**.**

If $q>1$ and $\nu\vDash v_{2}$ then the Hecke correspondence $\mathfrak{h}[v_{1},\Lambda_{\nu}\,;\,w]$ in $\mathfrak{M}(v,w)^{{\operatorname{op}\nolimits}}\times\mathfrak{M}(v_{1},w)$ is either Lagrangian and irreducible or empty.

Proof.

By Lemma 3.15, the variety $\mathfrak{h}[v_{1},\Lambda_{\nu}\,;\,w]$ is isotropic, hence we have

[TABLE]

We will show at the same time that $\mathfrak{h}[v_{1},\Lambda_{\nu}\,;\,w]$ is irreducible and that

[TABLE]

Given $\nu=(\nu_{1},\ldots,\nu_{s})$, we fix a flag

[TABLE]

of $I$-graded subspaces in $V$ of dimension

[TABLE]

Then, we have

[TABLE]

Let $P\subset G(v)$ be the stabilizer of $V_{1}$ and $P_{\nu}\subset P$ be the stabilizer of the flag $F$. By (3.8), the canonical map

[TABLE]

is surjective and proper. It is an isomorphism over the open subset $\operatorname{H}\nolimits[v_{1},\Lambda_{\nu}^{\circ}\,;\,w],$ since for any $z$ in this open set there exists a unique flag of type $\nu$ in $V/V_{1}$ satisfying the semi-nilpotency condition with respect to $z$. Proposition 3.17 is a consequence of the following claims

• (i)

$\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]$ is irreducible of dimension $\geqslant d_{v,w}/2+d_{v_{1},w}/2+\dim P_{\nu}$,

• (ii)

$\operatorname{H}\nolimits[v_{1},\Lambda_{\nu}^{\circ}\,;\,w]$ is non-empty.

We begin with the dimension estimate in (i). Fixing isomorphisms $F_{j}\simeq{\operatorname{k}\nolimits}^{v_{1}+\nu_{\leqslant j}}$ for $j\geqslant 0$ we obtain a closed embedding

[TABLE]

For each $u\in\mathbb{N}^{I}$ and $n\in\mathbb{N}$ we set

[TABLE]

It is an open subset of the zero locus of $(u+n\delta_{i})\cdot u$ equations in a vector space of dimension

[TABLE]

By (3.15), each irreducible component of $\operatorname{H}\nolimits[u,n]$ is of dimension

[TABLE]

where $P_{u,n}$ is the parabolic subgroup in $G(u+n\delta_{i})$ of type $(u,n\delta_{i})$. Since the set ${\operatorname{M}\nolimits}_{s}(u,w)\times{\operatorname{M}\nolimits}_{s}(u+n\delta_{i},w)$ is irreducible, every irreducible component of $\operatorname{H}\nolimits[u,n]$ is of codimension

[TABLE]

in ${\operatorname{M}\nolimits}_{s}(u,w)\times{\operatorname{M}\nolimits}_{s}(u+n\delta_{i},w)$. We have

[TABLE]

Hence every irreducible component of $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]$ is of codimension

[TABLE]

in $\prod_{j}{\operatorname{M}\nolimits}_{s}(v_{1}+\nu_{\leqslant j},w)$ and finally of dimension

[TABLE]

Let us turn to the irreducibility statement in (i). For this, we stratify $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]$. We use Crawley-Boevey’s trick and view elements of ${\operatorname{M}\nolimits}(v,w)$ as representations of the preprojective algebra $\tilde{\Pi}$ in §3.7.2. The stratification we want is obtained in three steps.

$\mathbf{Step\ 1:}$ we stratify ${\operatorname{M}\nolimits}_{s}(v_{1},w)$. Given $z\in{\operatorname{M}\nolimits}_{s}(v_{1},w)$, let $V^{z}_{1}\oplus W\subseteq V_{1}\oplus W$ be the smallest $I$-graded $z$-stable subspace containing $\bigoplus_{j\neq i}V_{1,j}\oplus W$. We have an exact sequence of $\tilde{\Pi}$-modules

[TABLE]

Note that $V_{1}/V^{z}_{1}$ is a representation of $\tilde{\Pi}$ which is supported at the vertex $i$. Set

[TABLE]

We have ${\operatorname{M}\nolimits}_{s}(v_{1},w)=\bigsqcup_{n\geqslant 0}{\operatorname{M}\nolimits}_{s}(v_{1},w)^{n,i}$ with ${\operatorname{M}\nolimits}_{s}(v_{1},w)^{0,i}$ being open (and possibly empty).

Lemma 3.18**.**

Let $z\in{\operatorname{M}\nolimits}_{s}(v_{1},w)$ and let $z^{\prime}$ be any $\tilde{\Pi}$-module supported at the vertex $i$. Then, we have $\operatorname{Hom}\nolimits_{\tilde{\Pi}}(z,z^{\prime})=\operatorname{Hom}\nolimits_{\tilde{\Pi}}(z|_{V_{1}/V^{z}_{1}}\,,\,z^{\prime}).$

Proof.

Any $\tilde{\Pi}$-morphism $z\to z^{\prime}$ must vanish on $\bigoplus_{j\neq i}V_{1,j}\oplus W$, hence on $V^{z}_{1}\oplus W$. ∎

$\mathbf{Step\ 2:}$ we stratify ${\operatorname{M}\nolimits}(v_{2})$. We begin with a general construction. Let $Q$ be any quiver and $M$ a $\Pi$-module. Then, there exists a unique minimal submodule $N\subset M$ such that $M/N$ is semi-nilpotent. Indeed, if $M/N_{1}$ and $M/N_{2}$ are semi-nilpotent, then from the exact sequence

[TABLE]

and the fact that the subcategory of semi-nilpotent representations of $\Pi$ is closed under subobjects, quotients and extensions, it follows that $M/(N_{1}\cap N_{2})$ is also semi-nilpotent. We denote the minimal submodule $N$ above by ${{\operatorname{corad}}}(M)$ and call it the coradical of $M$. The next statement is obvious.

Lemma 3.19**.**

Let $M,$ $L$ be $\Pi$-modules, with $L$ being semi-nilpotent. Then

• $\mathrm{(a)}$

${{\operatorname{corad}}}({{\operatorname{corad}}}(M))={{\operatorname{corad}}}(M)$,

• $\mathrm{(b)}$

$\operatorname{Hom}\nolimits_{\Pi}(M,L)=\operatorname{Hom}\nolimits_{\Pi}(M/{{\operatorname{corad}}}(M),L)$.

In particular, we have $\operatorname{Hom}\nolimits_{\Pi}({{\operatorname{corad}}}(M),L)=\{0\}$. ∎

Let us now consider the special case of the one vertex quiver with $q$ loops with $q>1$. Recall that $v_{2}=l\delta_{i}$. For any integer $n\in\mathbb{N}$ with $n\leqslant l$ we set

[TABLE]

We have a stratification

[TABLE]

with ${\operatorname{M}\nolimits}(v_{2})^{0}$ being a non-empty open subset. Because ${\operatorname{M}\nolimits}(v_{2})$ is irreducible by Lemma 3.9 and of dimension $(2q-1)l^{2}+1$, the same holds for ${\operatorname{M}\nolimits}(v_{2})^{0}$.

$\mathbf{Step\ 3:}$ we stratify $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]$. For each $n_{1}\leqslant n_{2}$ we put

[TABLE]

Under the morphism which assigns to $z$ its associated graded

[TABLE]

the stratification ${\operatorname{M}\nolimits}_{s}(v_{1},w)=\bigsqcup_{\,n_{1}\leqslant n_{2}}{\operatorname{M}\nolimits}_{s}(v_{1},w)^{n_{1},n_{2}}$ pulls back to a stratification

[TABLE]

The next step is to compute the dimension of the strata in (3.27). We begin by the following general result.

Lemma 3.20**.**

Let $v_{1},$ $v_{2}$ be dimension vectors of a quiver $Q$ with $v=v_{1}+v_{2}$. Let $\rho$ be the obvious map $\{\bar{x}\in{\operatorname{M}\nolimits}(v)\;;\;\bar{x}(V_{1})\subseteq V_{1}\}\to{\operatorname{M}\nolimits}(v_{1})\times{\operatorname{M}\nolimits}(v_{2}).$

• $\mathrm{(a)}$

The fiber of $\rho$ over $(\bar{x}_{1},\bar{x}_{2})$ is an affine space of dimension

[TABLE]

• $\mathrm{(b)}$

The restriction of $\rho$ to the preimage of the open set

[TABLE]

is an affine space bundle.

Proof.

The first statement is [9, lem 5.1] combined with (3.3) and (3.4). Let us give more details. Fix a vector space splitting $V=V_{1}\oplus V_{2}$. The fiber of $\rho$ over $(\bar{x}_{1},\bar{x}_{2})$ is then identified with the kernel of the linear map

[TABLE]

By [7, §1], see also §A.1, any finite-dimensional representation $N$ of $\Pi$ of dimension $v$ has a canonical projective resolution

[TABLE]

with

[TABLE]

Considering the above resolution for $N=\bar{x}_{2}$, we have

[TABLE]

Hence $\gamma$ is identified with the map $\operatorname{Hom}\nolimits_{\Pi}(P^{\prime},\bar{x}_{1})\to\operatorname{Hom}\nolimits_{\Pi}(P^{\prime\prime},\bar{x}_{1})$ induced by $f$. So there is a short exact sequence

[TABLE]

Since $\dim\operatorname{Hom}\nolimits_{\Pi}(P,\bar{x}_{1})=v_{1}\cdot v_{2}$ we deduce that

[TABLE]

The first part of the proposition is now a consequence of (3.3) and (3.4). The second statement follows from the fact that $U$ is precisely the set of points of ${\operatorname{M}\nolimits}(v_{1})\times{\operatorname{M}\nolimits}(v_{2})$ for which the vector bundle morphism $\gamma$ is of maximal rank. ∎

Now, we can prove the following lemma.

Lemma 3.21**.**

The stratum $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{n_{1},n_{2}}$ is either empty or of dimension equal to

[TABLE]

Proof.

Let us fix $n_{1}\leqslant n_{2}$. We refine the flag

[TABLE]

by adding a pair of subspaces $F_{-2}\subseteq F_{-1}\subseteq V_{1}$ such that ${\operatorname{codim}\nolimits}_{V_{1}}(F_{-j})=n_{j}\delta_{i}$ for $j=1,2$. Then, we set

[TABLE]

where $z_{1}=z|_{V_{1}\oplus W}$. A pair $z=(\bar{x},\bar{a})$ in ${\operatorname{M}\nolimits}_{s}(v,w)$ belongs to $\operatorname{H}\nolimits_{F}$ if and only if it stabilizes the (refined) flag $W$ and the following conditions are satisfied

[TABLE]

Let $P_{F}\subset G(v_{1})$ be the stabilizer of the flag $F_{-2}\subseteq F_{-1}\subseteq V_{1}$. We have

[TABLE]

Let $\operatorname{H}\nolimits^{\prime}_{F}$ be defined as $\operatorname{H}\nolimits_{F}$ but without the stability condition for $z$ : it is the set of elements $z=(\bar{x},\bar{a})$ in ${\operatorname{M}\nolimits}(v,w)$ satisfying (3.28). Finally, consider the flag

[TABLE]

and set

[TABLE]

By Lemmas 3.18 and 3.19, for all $z=(\bar{x},\bar{a})$ in $\operatorname{H}\nolimits^{\prime}_{W}$ we have

[TABLE]

Therefore by Lemma 3.20 the map

[TABLE]

is an affine fibration of rank equal to

[TABLE]

see (3.16). On the other hand, we have

[TABLE]

Lemma 3.22**.**

$\Lambda^{\prime}_{F/F_{-1}}$* is pure of dimension $q(n_{1}+l)^{2}-\sum_{a<b}\nu_{a}\nu_{b}-n_{1}l$.*

Proof.

There is a natural projection map $\Lambda^{\prime}_{F/F_{-1}}\to\Lambda(n_{1}\delta_{i})$. Thus, the partition $\Lambda(n_{1}\delta_{i})=\bigsqcup_{\mu}\Lambda_{\mu}^{\circ}$ yields a partition

[TABLE]

Fix a composition $\mu$ and a flag of subspaces $F_{0}/F_{-1}=F^{\prime}_{0}\supseteq F^{\prime}_{-1}\supseteq\cdots$ of type $\mu$ in $F_{0}/F_{-1}$. We have

[TABLE]

where $P_{\mu}$ is the stabilizer of $F^{\prime}_{\leqslant 0}$ in $GL(n_{1})\subset GL(n_{1}+l)$ and $F^{\prime\prime}$ is the flag

[TABLE]

It follows that

[TABLE]

Recall that $F^{\prime\prime}$ is a flag of type $\gamma=(\mu,\nu)$, the concatenation of $\mu$ and $\nu$. From the equality

[TABLE]

we deduce that

[TABLE]

The lemma is proved. ∎

We are finally ready to compute the dimension of the stratum $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{n_{1},n_{2}}$. Since $\operatorname{H}\nolimits^{\prime}_{F}$ is of pure dimension by Lemma 3.22, the subset $\operatorname{H}\nolimits_{F}$ is either empty or of the same dimension. Thus $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{n_{1},n_{2}}$ is either empty or of dimension equal to

[TABLE]

So a straightforward computation using (3.29), (3.30) and Lemma 3.22 shows that the stratum $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{n_{1},n_{2}}$ is either empty or of dimension equal to $d(n_{1},n_{2}).$ ∎

Now, we can finish the proof of Proposition 3.17. For all $n_{1}$, $n_{2}$ we have

[TABLE]

with equality if and only if $n_{1}=n_{2}=0$. Indeed, we have $(n_{2}\,,\,v_{1}-n_{2})<0$ as soon as $0<n_{2}<v_{1,i}$. Further, if $n_{2}=v_{1,i}$ then $w\cdot n_{2}+(n_{2}\,,\,v_{1}-n_{2})<0$ as soon as $v_{1,j}>0$ for some $j$ adjacent to $i$ or $w_{i}>0$. Finally, if $v_{1,j}=0$ for all adjacent $j$ and $w_{i}=0$ then ${\operatorname{M}\nolimits}_{s}(v_{1},w)$ is empty.

Hence the set $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{0,0}$ is the only stratum of dimension

[TABLE]

Combining this with the lower dimension estimates for $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]$ in (3.26), we deduce that $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{0,0}$ is non-empty if $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]$ is non-empty. Observe also that $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{0,0}$ is irreducible. These facts imply that $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]$ is irreducible of dimension equal to

[TABLE]

The fact that $\operatorname{H}\nolimits[v_{1},\Lambda^{\circ}_{\nu}\,;\,w]$ is non-empty follows from the definition of $\operatorname{H}\nolimits[v_{1},\Lambda_{F/V_{1}}\,;\,w]^{0,0}$. Proposition 3.17 is proved. ∎

3.8.3. Convolution of one vertex Hecke correspondences

Let us now state a useful corollary of the previous proposition concerning compositions of Hecke correspondences. Let $\star$ denote the convolution product of correspondences as in [5]. Set $v=v_{1}+v_{2}$ with $v_{2}=l\delta_{i}$ and consider a fixed composition $\nu=(\nu_{1},\nu_{2},\dots,\nu_{r})$ of $v_{2}$.

Proposition 3.23**.**

Let $q>1$. The following relation holds in H_{d}^{T}\big{(}\mathfrak{M}(v,w)\times\mathfrak{M}(v_{1},w)\big{)}

[TABLE]

Proof.

To simplify we assume that $r=2$. We use the same notation as in the proof of Proposition 3.17. In particular, we have the parabolic subgroups $P_{\nu}\subseteq P$ in $G(v)$, where $P_{\nu}$ fixes a flag $F=(V_{1}\subsetneq F_{1}\subsetneq V)$ of type $(v_{1},v_{1}+\nu_{2},v)$. From Proposition 3.16 we deduce that the varieties

[TABLE]

are local complete intersections and are dimensionally transverse in

[TABLE]

with intersection equal to the categorical quotient $\operatorname{H}\nolimits(v_{1},\Lambda_{\nu}\,;\,w)/\!\!/P_{\nu}$. We deduce that

[TABLE]

Since $\mathfrak{h}[v_{1},\Lambda_{\nu}\,;\,w]=\operatorname{H}\nolimits[v_{1},\Lambda_{\nu}\,;\,w]/\!\!/P$, the varieties $\operatorname{H}\nolimits[v_{1},\Lambda_{\nu}\,;\,w]/\!\!/P_{\nu}$ and $\operatorname{H}\nolimits[v_{1},\Lambda_{\nu}\,;\,w]/\!\!/P$ are both irreducible by Proposition 3.17. Further, they have the same dimension by (3.31) and Proposition 3.17. Since for any $z$ in the open subset $\operatorname{H}\nolimits[v_{1},\Lambda_{\nu}^{\circ}\,;\,w],$ there exists a unique flag of type $\nu$ in $V/V_{1}$ satisfying the semi-nilpotency condition with respect to $z$, the obvious map

[TABLE]

is generically one to one because the set $\Lambda_{\nu}^{\circ}$ is dense in $\Lambda_{\nu}$ by Proposition 3.8. So the proposition follows from definition of the convolution product which yields

[TABLE]

∎

4. The homology of the semi-nilpotent variety

In this section ${\operatorname{k}\nolimits}$ is any algebraically closed field, except in subsections §§4.3, 4.4 where we assume that ${\operatorname{k}\nolimits}=\mathbb{C}$.

4.1. The Lagrangian quiver variety

4.1.1. Definition

For $\flat=0$ or $1$, and for any dimension vectors $v,$ $w$, we set

[TABLE]

The Lagrangian quiver variety is the geometric quotient

[TABLE]

The closed embedding $\Lambda^{1}(v)\subseteq\Lambda^{0}(v)$ yields closed embeddings

[TABLE]

We use another description of $\mathfrak{L}^{\flat}(v,w)$. Consider the $\mathbb{G}_{m}$-action $\circ$ on $\mathfrak{M}(v,w)$ given by

[TABLE]

Let $\mathfrak{M}(v,w)^{\circ}$ be the fixed points locus and $\mathfrak{L}(v,w)^{\circ}$ be the attracting variety. The following is proved in [3, prop. 3.1, 3.2].

Proposition 4.1**.**

• $\mathrm{(a)}$

$\mathfrak{L}^{0}(v,w)=\mathfrak{L}(v,w)^{\circ}$,

• $\mathrm{(b)}$

there is a $\mathbb{G}_{m}$-action $\bullet$ on $\mathfrak{M}(v,w)$ such that $\mathfrak{L}^{1}(v,w)=\mathfrak{L}^{1}(v,w)^{\bullet}$,

• $\mathrm{(c)}$

$\mathfrak{L}^{\flat}(v,w)$* is a closed Lagrangian subvariety of $\mathfrak{M}(v,w)$ for $\flat=0,1$.*

∎

The $\bullet$-action can be made explicit as in [3, §3.3.2], but we won’t need it here. Observe that all the irreducible components of $\mathfrak{L}^{1}(v,w)$ are also irreducible components of $\mathfrak{L}^{0}(v,w)$.

The $\circ$-action is associated with the cocharacter $\gamma=\xi^{-1}$ of $T\times G(w)$ in (3.13). Given a cocharacter $\rho$ of $G(v)$, we define the subsets $\mathfrak{M}[\rho]$ and $\mathfrak{L}[\rho]$ of $\mathfrak{M}(v,w)$ as in §3.7.4. Hence,

[TABLE]

and the closure of $\mathfrak{L}[\rho]$ is a sum of irreducible components of $\mathfrak{L}^{0}(v,w)$ by [3, prop. 3.5].

4.1.2. The case of one vertex quivers

Assume that $Q=\operatorname{J}\nolimits(i,q)$ with $q>1$. Then, we have $\Lambda^{1}=\Lambda^{0}$. We abbreviate $\mathfrak{L}=\mathfrak{L}^{\flat}$. Given a dimension vector $v=l\,\delta_{i}$, we fix a composition $\nu=(\nu_{1},\dots,\nu_{r})$ of $v$ and an increasing flag $W$ of type $\nu$. Let $\rho_{\nu}$ be the cocharacter of $G(v)$ which preserves the flag $W$ and has the weight $1-p$ on $W_{p}/W_{p-1}$ for each $p=1,\dots,r.$ We write

[TABLE]

Assume that $w=v$. By Propositions 3.12 and 4.1 the variety $\mathfrak{L}(v,v)$ is the union of the Zariski closure of the attracting sets for the $\circ$-action. Each of these closed set is a finite union of irreducible Lagrangian subvareties of $\mathfrak{M}(v,v)$.

Lemma 4.2**.**

Assume that $w=v=l\delta_{i}$. For each cocharacter $\rho$ of $G(v)$ we have

[TABLE]

Proof.

Fix any cocharacter $\rho$ of $G(v)$. Up to the conjugation by an element of $G(v)$, we may choose some integers $v^{\prime}_{k}\geqslant 0$ such that

[TABLE]

Set $v^{\prime}=(v^{\prime}_{k})$. Set also $w^{\prime}=(w^{\prime}_{k})$ with $w^{\prime}_{k}=l\,\delta_{k,1}$. The local quiver variety $\mathfrak{M}(v^{\prime},w^{\prime})^{\#}$ associated with the quiver $Q$ and the dimension vectors $v^{\prime}$ and $w^{\prime}$ is the geometric quotient by the action of the group $G(v)\times G(v)$ of the set of semistable tuples of linear maps $(\bar{x},\bar{a})$ such that

• •

$x\in\operatorname{Hom}\nolimits_{\mathbb{Z}}(V,V[-1])$ and $x^{*}\in\operatorname{End}\nolimits_{\mathbb{Z}}(V)$,

• •

$a=a_{1}\in\operatorname{Hom}\nolimits(W_{1},V_{0})$ and $a^{*}=a^{*}_{1}\in\operatorname{Hom}\nolimits(V_{1},W_{1}),$

• •

$V_{k}\subseteq V$ is the kernel of $\rho(t)-t^{1-k}$,

• •

$W_{1}=W.$

It is empty unless there is an integer $r$ and a composition $\nu\vDash v$ as above such that $v^{\prime}_{k}=\nu_{k}$ if $k\in[1,r]$ and $v^{\prime}_{k}=0$ else. Further, we have an isomorphism $\mathfrak{M}(v^{\prime},w^{\prime})^{\#}\simeq\mathfrak{M}[\rho].$ This yields the direct part of the equivalence.

To prove the reverse implication, it remains to check that $\mathfrak{M}[\rho_{\nu}]\neq\emptyset$ for all $\nu$. The assignment $\bar{x}\mapsto[\bar{x},0,1]$ yields an open immersion of $G(v)$-varieties

[TABLE]

We claim that we have

[TABLE]

proving the lemma. In order to prove (4.7), we fix an increasing flag $W$ of $I$-graded vector spaces in $V$ of type $\nu$ and we consider the cocharacter $\rho_{\nu}$ as above. For each element $\bar{x}\in\Lambda(v)$ we have

[TABLE]

Let $(\bar{y},0,b^{*})$ denote the limit above. We must prove that

[TABLE]

It is enough to check that if $\bar{x}\in\Lambda^{\circ}_{W}$ then $(\bar{y},0,b^{*})$ is semistable. Since $\bar{x}\in\Lambda_{W}$, taking the associated graded of the representation $(\bar{x},0,1)$ relative to the flag $W$ yields the representation $(\bar{y},0,b^{*})$. Hence, the later can be viewed as a representation in ${\operatorname{M}\nolimits}^{\#}(v,w)$ where the dimension vectors $v$, $w$ are given by

[TABLE]

Since $\bar{x}\in\Lambda^{\circ}_{W}$, the subspace $W_{p}\subseteq V$ is the annihilator of $A_{\bar{x}}(\geqslant\!p)$. Hence any nonzero subspace in $V\oplus W$ which is stable by $\bar{y}$ must have a nonzero component in $W$. Hence $(\bar{y},0,b^{*})$ is semistable. ∎

4.2. The homology of the Lagrangian quiver variety

For each dimension vector $w$ we fix a closed subgroup $H(w)\subset G(w)$.

Proposition 4.3**.**

Let ${\operatorname{k}\nolimits}=\mathbb{C}$ and $\flat$ be either 0 or 1. If $X=\mathfrak{M}(v,w)$ or $\mathfrak{L}^{\flat}(v,w)$ then

• $\mathrm{(a)}$

$H_{*}^{T\times H(w)}(X)$* is pure and even,*

• $\mathrm{(b)}$

the $T\times H(w)$-variety $X$ is equivariantly formal,

• $\mathrm{(c)}$

${\operatorname{cl}\nolimits}:{A}_{*}^{T\times H(w)}(X)\to H_{2*}^{T\times H(w)}(X)$* is an isomorphism.*

Proof.

The proof is similar to the proof of [32, thm. 7.3.5], replacing equivariant K-theory by equivariant Borel-Moore homology or Chow groups. Since our setting differs slightly from loc. cit., let us recall briefly the main arguments.

First, recall the Bialynicki-Birula decomposition. Let $X$ be a ${\operatorname{k}\nolimits}$-variety with a $\mathbb{G}_{m}$-action which embeds equivariantly in a projective space with a diagonalizable $\mathbb{G}_{m}$-action. Assume that the $\mathbb{G}_{m}$-fixed point locus $X^{\mathbb{G}_{m}}$ is contained in the regular locus of $X$. Then we have a partition $X=\bigsqcup_{s}X_{s}$ into locally closed subsets with affine space bundles $p_{s}\,:\,X_{s}\to F_{s}$ such that the $F_{s}$’s are the connected components of $X^{\mathbb{G}_{m}}$. Further, there is a filtration $(X_{\leqslant s})$ of $X$ and an ordering of the components $F_{s}$ such that $X_{s}=X_{\leqslant s}\setminus X_{<s}$. This filtration depends on the choice of the equivariant embedding of $X$ in a projective space with a diagonalizable $\mathbb{G}_{m}$-action.

First, we prove the proposition for $X=\mathfrak{M}(v,w)$.

Consider the actions of $\mathbb{G}_{m}\times\mathbb{G}_{m}$ and $\mathbb{G}_{m}$ on $\mathfrak{M}(v,w)$ given by

[TABLE]

for some positive integers $a,$ $b$. One checks (for instance using the known generators of $\mathrm{k}[\mathfrak{M}_{0}(v,w)]$) that the fixed points locus $\mathfrak{M}_{0}(v,w)^{\triangledown}$ is $\{0\}$. Thus $\mathfrak{M}(v,w)^{\triangledown}$ is smooth and projective. Let $\mathfrak{M}_{1},\dots,\mathfrak{M}_{N}$ be its connected components. Choose $a$, $b$ such that the fixed points locus $\mathfrak{M}(v,w)^{\blacktriangledown}$ coincides with $\mathfrak{M}(v,w)^{\triangledown}.$ Then, the Bialynicki-Birula decomposition yields a partition of $\mathfrak{M}(v,w)$ by smooth locally-closed stable subvarieties $\mathfrak{L}_{1},\mathfrak{L}_{2},\dots,\mathfrak{L}_{N}$ such that

[TABLE]

and it also yields equivariant affine space bundles $p_{s}:\mathfrak{L}_{s}\to\mathfrak{M}_{s}$ such that $\mathfrak{L}_{1}\sqcup\mathfrak{L}_{2}\sqcup\cdots\sqcup\mathfrak{L}_{s}$ is closed for each $s=1,\dots,N$. We say that $\mathfrak{L}_{1},\dots,\mathfrak{L}_{N}$ form an $\alpha$-partition of $\mathfrak{M}(v,w)$.

By [32, lem. 7.1.3, 7.1.4] we must check that the properties (a), (b), (c) above hold for the $T\times H(w)$-varieties $\mathfrak{M}_{1},\mathfrak{M}_{2},\dots,\mathfrak{M}_{N}$. The proof in loc. cit. is done for equivariant algebraic K-theory and equivariant topological K-theory, but it applies also to equivariant Chow groups and equivariant Borel-Moore homology.

To prove this we apply [32, prop. 7.2.1]. For each $s=1,\dots,N$ the $T\times H(w)$-variety $\mathfrak{M}_{s}$ is smooth and projective. Hence, it is enough to prove that the diagonal $\Delta_{s}$ of $\mathfrak{M}_{s}\times\mathfrak{M}_{s}$ satisfies the Kunneth property. Since $\mathfrak{M}_{s}$ is a particular case of graded quiver varieties, this follows from a decomposition of the diagonal $\Delta_{s}$ as in [32, §7.3].

Now, we prove the proposition for $X=\mathfrak{L}^{0}(v,w)$. By [3, prop. 3.3], the Bialynicki-Birula decomposition yields an $\alpha$-partition $\mathfrak{L}^{0}(v,w)=\bigsqcup_{\kappa}X_{\kappa}$ with affine space bundles $X_{\kappa}\to V_{\kappa}$ and $\alpha$-partitions $V_{\kappa}=\bigsqcup_{s}U_{\kappa,s},$ where $s$ runs over a subset $\pi_{\kappa}$ of $\{1,2,\dots,N\},$ with affine space bundles $U_{\kappa,s}\to F_{s}.$ Applying [32, lem. 7.1.3, 7.1.4] successively to the $\alpha$-partitions $V_{\kappa}=\bigsqcup_{s}U_{\kappa,s}$ and $\mathfrak{L}^{0}(v,w)=\bigsqcup_{\kappa}X_{\kappa}$, the properties (a), (b), (c) in the proposition follows from the analogous properties for the $T\times H(w)$-varieties $\mathfrak{M}_{1},\mathfrak{M}_{2},\dots,\mathfrak{M}_{N}$, that we have already checked.

The proof in the case of $X=\mathfrak{L}^{1}(v,w)$ is exactly the same, using [3, prop. 3.4]. ∎

4.3. Purity

In this section we prove Theorem 3.2. In particular, we assume that ${\operatorname{k}\nolimits}=\mathbb{C}$. Part (a) is proved in [2] in the case $\flat=1$. The case of $\flat=0$ is similar. Now, we concentrate on the other claims. Fix any dimension vectors $v$ and $w$ with $v\leqslant w$, i.e., with $w-v\in\mathbb{N}^{I}$. Then, let $\operatorname{Mon}\nolimits_{I}(v,w)$ be the set of injective maps in $\operatorname{Hom}\nolimits_{I}(v,w)$. Since the stability condition is open, by (4.1) we have open immersions

[TABLE]

such that $\beta(\bar{x},\bar{a})=(\bar{x},a^{*})$ and $\alpha(\bar{x},a^{*})=(\bar{x},0,a^{*})$. To simplify the notation we write $G=G(v)$ and $H=G(w).$

The group $T\times G$ acts on $\Lambda^{\flat}(v)$ as in §3.4, hence $T\times G\times H$ acts via the projection

[TABLE]

The group $G\times H$ acts on $\operatorname{Hom}\nolimits_{I}(v,w)$ by $(g,h)\cdot(a^{*})=h\,a^{*}g^{-1},$ the group $T$ so that the factor $(\mathbb{G}_{m})^{\Omega}$ in (3.6) acts trivially and the factor $\mathbb{G}_{m}$ acts by dilatation. We equip the product $\Lambda^{\flat}(v)\times\operatorname{Hom}\nolimits_{I}(v,w)$ with the diagonal $T\times G\times H$-action. The group $T\times H$ acts on $\mathfrak{L}^{\flat}(v,w)$ as in §3.7.1. We define the following maps :

• •

The functoriality with respect to $\rho_{1}$ is an $M^{*}_{T\times G}$-linear map

[TABLE]

• •

The pullback by the first projection $p_{1}\,:\,\Lambda^{\flat}(v)\times\operatorname{Hom}\nolimits_{I}(v,w)\to\Lambda^{\flat}(v)$ gives an $M^{*}_{T\times G\times H}$-linear map

[TABLE]

• •

The pullback by $\beta$ gives an $M^{*}_{T\times G\times H}$-linear map

[TABLE]

• •

The quotient by the $G$-action is a $T\times H$-equivariant principal $G$-bundle $\operatorname{L}\nolimits^{\flat}_{s}(v,w)\to\mathfrak{L}^{\flat}_{s}(v,w)$. Hence, we have an $M^{*}_{T\times H}$-linear descent isomorphism

[TABLE]

Composing $\rho_{1}^{\sharp}$, $p_{1}^{*}$, $\beta^{*}$ and $f^{\heartsuit}$ we get a map

[TABLE]

Now, we set $v=w$. Hence (4.11) reads

[TABLE]

and the $T\times G\times H$-action on $\Lambda^{\flat}(v)\times\mathfrak{g}(v)$ preserves the open subsets $\Lambda^{\flat}(v)\times G(v)$ and $\operatorname{L}\nolimits_{s}^{\flat}(v,v)$. We define the following maps :

• •

The pullback by $\alpha$ gives a map

[TABLE]

• •

The $T\times H$-equivariant principal $G$-bundle

[TABLE]

where $T\times H$ acts on the right hand side as in §3.4, yields a descent isomorphism

[TABLE]

which intertwines the $M^{*}_{T\times G}$-action on the left hand side with the $M^{*}_{T\times H}$-action on the right hand side under the isomorphism $M^{*}_{T\times G}=M^{*}_{T\times H}$ given by the identification $G=H$.

• •

The first projection is a $T\times G$-equivariant principal $H$-bundle $q_{1}:\Lambda^{\flat}(v)\times G(v)\to\Lambda^{\flat}(v).$ Hence we have an $M^{*}_{T\times G}$-linear isomorphism

[TABLE]

Composing $(f^{\heartsuit})^{-1}$, $\alpha^{*}$ and $(q^{\heartsuit})^{-1}$ we get a map

[TABLE]

which intertwines the $M^{*}_{T\times H}$-action on the left hand side with the $M^{*}_{T\times G}$-action on the right hand side.

Note that the map (4.12) is $M^{*}_{T}$-linear but is not $M^{*}_{T\times G}$-linear in general. More precisely, we consider a new $M^{*}_{G}$-action on $M_{*}^{T\times H}(\mathfrak{L}^{\flat}(v,v))$ defined as follows. Let $\mathcal{V}_{i}$ be the universal $T\times H$-equivariant vector bundle on $\mathfrak{L}^{\flat}(v,v)$, which is obtained by restricting the universal bundle on $\mathfrak{M}(v,v)$, see (5.29) below. The $\mathbb{Q}$-algebra $M^{*}_{G}$ is spanned by the classes ${\operatorname{ch}\nolimits}_{l}(\theta_{i})$ with $i\in I$ and $l\in\mathbb{N}$, of the tautological $G$-equivariant bundles $\theta_{i}$ on $\{\bullet\}$. We define the new action so that the element ${\operatorname{ch}\nolimits}_{l}(\theta_{i})$ acts on $M_{*}^{T\times H}(\mathfrak{L}^{\flat}(v,v))$ by the cap product with the cohomology class ${\operatorname{ch}\nolimits}_{l}(\mathcal{V}_{i})$. Then, the map (4.12) is $M^{*}_{T\times G}$-linear relative to the new $M^{*}_{T\times G}$-action on the right hand side.

Lemma 4.4**.**

If $v=w$ then the map (4.12) is injective while (4.13) is surjective.

Proof.

Assume that $v=w$. We abbreviate $\Lambda=\Lambda^{\flat}(v)$ and $\mathfrak{g}=\mathfrak{g}(v)$. The map $i=\alpha\circ\beta$ is the obvious inclusion $\Lambda\times G\to\Lambda\times\mathfrak{g}$. The composed map

[TABLE]

is equal to the identity. Now, we have $i^{*}=\alpha^{*}\circ\beta^{*}$. We deduce that (4.13) is surjective and (4.12) is injective. ∎

Theorem 3.2 follows from Proposition 4.3 with $X=\mathfrak{L}^{\flat}(v,v)$ and from Lemma 4.4. More precisely, to prove (b) we use the fact that if we have an epimorphism $H\to H^{\prime}$ of MHS such that $H$ is pure, then $H^{\prime}$ is also pure, see, e.g., [37, §3.1], and to prove (c) we use the naturality of the cycle map with respect to pullback by open immersions. To prove part (d) we must check that $M_{*}^{T\times G(v)}(\Lambda^{\flat}(v))$ is free as an $M^{*}_{T}$-module. Note that the composed map (4.14) is the identity and it factorizes through the $M^{*}_{T}$-module $M_{*}^{T\times H}(\mathfrak{L}^{\flat}(v,v))$, which is free by Proposition 4.3. So the theorem is proved.

Remark 4.5*.*

Fix a degree $j$. If $v\leqslant w$ then we consider the composed map

[TABLE]

The excision long exact sequence implies that the restriction $i^{*}$ is invertible for degree reasons if $w$ is large enough. Since $p_{1}^{*}$ is invertible, we deduce that the map

[TABLE]

is also injective if $w$ is larger than a constant depending on $j$ and $v$.

4.4. Torsion-freeness

Set $\Bbbk=H^{*}_{T}$ and $\Bbbk[v]=H^{*}_{T\times G(v)}$. Let $K$ be the fraction field of $\Bbbk$. Recall the one-parameter subgroups $\theta$, $\theta^{*}$ of $G_{\Omega}$ defined in §3.3. For each $z\in\mathbb{C}^{\times}$, the element $\theta(z)$ acts by multiplication by $z$ on the summand $Rep({\operatorname{k}\nolimits}Q,v)$ in $\operatorname{R}\nolimits(v)$ and trivially on $Rep({\operatorname{k}\nolimits}Q^{*},v)$, while $\theta^{*}(z)$ acts by multiplication by $z$ on the summand $Rep({\operatorname{k}\nolimits}Q^{*},v)$ and trivially on $Rep({\operatorname{k}\nolimits}Q,v)$. We consider also the one-parameter subgroup $\theta-\theta^{*}$, such that $(\theta-\theta^{*})(z)=\theta(z)\theta^{*}(z)^{-1}$. In this section we prove the following result.

Proposition 4.6**.**

Let $Q$ be an arbitrary quiver, $\flat\in\{0,1\}$ and assume that the torus $T$ contains the one-parameter subgroups $\theta$ and $\theta^{*}$. Then for any dimension vector $v$ of $Q$, the $\Bbbk[v]$-module $H^{T\times G(v)}_{*}(\Lambda^{\flat}(v))$ is torsion-free.

Proof.

The proof proceeds through several reductions. We’ll abbreviate $\gamma=\theta-\theta^{*}$. First, observe that the $\Bbbk[v]$-module

[TABLE]

is torsion free. Since $T$ contains the one-parameter subgroup $\gamma$, we can consider the ideal

[TABLE]

which is the vanishing ideal of $\text{Lie}(\gamma(\mathbb{G}_{m}))$ in $\text{Lie}(T\times G(v))$. Let $T(v)\subset G(v)$ be a maximal torus and let $W(v)$ be the Weyl group of the pair $(G(v),T(v))$. There are isomorphisms of $\Bbbk[v]$-modules, see, e.g., [14, prop. 6],

[TABLE]

Hence, by [18, thm. 6.2], the pushforward is an isomorphism of localized modules

[TABLE]

It is therefore enough to show that the obvious map below is injective

[TABLE]

Equivalently, we must prove that $H_{*}^{T\times G(v)}(\Lambda^{\flat}(v))$ is torsion free over the set $S_{\gamma}$ given by

[TABLE]

By Theorem 3.2(d) there is an embedding

[TABLE]

while by Proposition 5.2(b) below we have

[TABLE]

if $\theta\subset T$. Thus it is enough to show that $H^{T\times G(v)}_{*}({\operatorname{M}\nolimits}(v))\otimes K$ is $S_{\gamma}$-torsion free.

We will now use the following result of B. Davison. Put

[TABLE]

There is a natural action of $T\times G(v)$ on $E(v)$ defined as follows : a tuple $(z_{h},z)\in T$ acts by $z/z_{h}$ on $x_{h^{*}}$ and by $z^{-1}$ on $u$. In particular, we have

[TABLE]

By [10, eq. (29), (30)], there is an isomorphism of $\Bbbk[v]$-modules 111Note that [10] does not consider $T$-equivariant Borel-Moore homology and deals with stacks of nilpotent representations. As explained to us by B. Davison, the proof works verbatim in our situation since the potential $w$ involved in the dimensional reduction process is $T$-invariant, and the present setting of stacks of (not necessarily nilpotent) representations is actually simpler than that considered in [10].

[TABLE]

where $o(v)=-2\langle v,v\rangle$. The same argument as in Proposition 5.2(b) shows that

[TABLE]

if $\theta\subset T$ or $\theta^{*}\subset T$, where

[TABLE]

Combining the isomorphisms (4.15)-(4.18), if $\theta,\theta^{*}\subset T$ we get an inclusion

[TABLE]

Proposition 4.6 will thus be proved once we show that $H^{T\times G(v)}_{*}(N(v))$ has no $S_{\gamma}$-torsion.

The advantage of $N(v)$ over $\Lambda^{\flat}(v)$ or ${\operatorname{M}\nolimits}(v)$ is that it carries a natural stratification (by Jordan types) with good properties. More precisely, let $(x,u)\in N(v)$ and let $l$ be the index of nilpotence of $u$. Consider the ${\operatorname{k}\nolimits}Q^{*}$-modules $F_{i}={\operatorname{Im}\nolimits}(u^{i-1})/{\operatorname{Im}\nolimits}(u^{i})$ for $i=1,\ldots,l$ and the chain of epimorphisms of ${\operatorname{k}\nolimits}Q^{*}$-modules

[TABLE]

induced by $u$. Let $v_{k}$ be the dimension vector of the kernel of the map $F_{k}\to F_{k+1}$. Note that $v=\sum_{k=1}^{l}kv_{k}$ and that some of the $v_{k}$ may be zero, but $v_{l}\neq 0$. We define the Jordan type of $(x,u)$ as the tuple of dimension vectors $J(x,u)$ given by

[TABLE]

It only depends on $u$. For any such a tuple $J=(v_{1},v_{2},\dots,v_{l})$ we put

[TABLE]

obtaining in this way a partition into locally closed subsets

[TABLE]

where the union ranges over all tuples $J$ such that $\sum_{k}kv_{k}=v$. First, we consider each stratum $N(J)$ separately of $N(v)$. Following [29, $\S 5$], we consider the morphism of stacks

[TABLE]

where the product of stacks on the right hand side is taken over the classifying stack $\bullet\,/\,T$. By [29, prop. 5.1], the map $\pi_{J}$ is a composition of stack vector bundles. Note that although [29, prop. 5.1] is written for coherent sheaves on smooth projective curves, the argument is valid for (smooth) moduli stacks of objects in arbitrary abelian category of homological dimension at most one. As a consequence, the map $\pi_{J}$ induces an isomorphism of $\Bbbk[v]$-modules

[TABLE]

where the tensor product on the left hand side is taken over $\Bbbk$ and the $\Bbbk[v]$-module structure (on the left hand side) comes from the restriction from $G(v)$ to the stabilizer subgroup of an element $u\in\mathfrak{g}(v)$ of the conjugacy class of nilpotent elements of Jordan type $J$, because

[TABLE]

From (4.20) it follows that

[TABLE]

and in particular $H_{*}^{T\times G(v)}(N(J))$ is pure and even.

The purity implies that the excision long exact sequences associated to the partition of $N(v)$ into Jordan strata $N(J)$ splits into short exact sequences, inducing a filtration on $H_{*}^{T\times G(v)}(N(v))$ whose associated graded is $\bigoplus_{J}H^{T\times G(v)}_{*}(N(J))$. Therefore, to prove that $H^{T\times G(v)}_{*}(N(v))$ has no $S_{\gamma}$-torsion, it is enough to check that each $H^{T\times G(v)}_{*}(N(J))$ has no $S_{\gamma}$-torsion.

To do this, we use (4.21) again. Fix an element $u\in\mathfrak{g}(v)$ of the conjugacy class of nilpotent elements of Jordan type $J$. We have

[TABLE]

where $Stab$ stands for the stabilizer of $u$ in $T\times G(v)$. The argument above implies that

[TABLE]

The $H^{*}_{T\times G(v)}$-module structure on the right hand side is induced by the restriction $H^{*}_{T\times G(v)}\to H^{*}_{Stab}$ relative to the inclusion $Stab\subset T\times G(v)$. It is thus enough to show that no element of $S_{\gamma}$ is mapped to zero by this restriction, equivalently that the kernel of this restriction map is contained in $I_{\gamma}$. But this follows from the fact that $\gamma\subset Stab$. This completes the proof of Proposition 4.6. ∎

Remark 4.7*.*

a) In this paper we only use Proposition 4.6 in the particular case of the Jordan quiver (in Theorem 5.18). In this case, a more direct proof can be given which does not use [10]. The proposition is important for further applications. For instance, coupling it with Theorem 5.18 and the methods of [40] or [43], we get a combinatorial realization of the cohomological Hall algebras $\operatorname{Y}\nolimits^{\flat}$ to be introduced in the next section. Namely, they are subalgebras of shuffle algebras generated by some explicit elements. We will not need this here.

b) A (wrong) previous version of Proposition 4.6 involved only the condition that $\theta\subset T$. In fact, one can prove that under this sole condition the map to the shuffle algebra is not injective, see Appendix A.5.

4.5. Kirwan Surjectivity for fixed point quiver varieties

Fix dimension vectors $v,w\in\mathbb{N}^{I}$. Consider the $\mathbb{G}_{m}$-action on $\mathfrak{M}(v,w)$ associated to a cocharacter $\sigma=\sigma_{T}\times\sigma_{G(w)}$ of $T\times G(w)$. The cocharacter $\sigma_{G(w)}$ yields a weight space decomposition $W=\bigoplus_{l}W^{(l)}$ of $I$-graded vector spaces. The cocharacter $\sigma_{T}$ yields a $\mathbb{G}_{m}$-action on $\operatorname{R}\nolimits(v,w)$ such that

[TABLE]

for some integers $d,d(h),d^{*}(h)$ with $h\in\Omega$ such that $d(h)+d^{*}(h)=d$ for any $h\in\Omega$.

Next, let $\rho$ be a cocharacter of $G(v)$. Let $V=\bigoplus_{l\in\mathbb{Z}}V^{(l)}$ be the decomposition in weight spaces according to $\rho$. We will set

[TABLE]

and use similar notations for $W$.

Define ${\operatorname{M}\nolimits}[\rho],$ $\mathfrak{M}[\rho]$ as in §3.7.4 (with respect to $\sigma$). There is an action of the group $T\times G(v^{\bullet})\times G(w^{\bullet})$ on ${\operatorname{M}\nolimits}[\rho]$ and there is a geometric quotient $\mathfrak{M}[\rho]={\operatorname{M}\nolimits}[\rho]//G(v^{\bullet})$. By construction, $\mathfrak{M}[\rho]$ carries a collection of $T$-equivariant tautological bundles $\mathcal{V}_{i,l}$, $\mathcal{W}_{i,l}$ whose Chern classes generate an action of $H^{*}_{T\times G(v^{\bullet})\times G(w^{\bullet})}$. The version of Kirwan surjectivity which we will use (in the particular case of the $\circ$ action) is the following one.

Theorem 4.8**.**

Assume that $d\neq 0$. Then the following map is surjective

[TABLE]

Proof.

A proof is given in the Appendix for the case of the $\circ$ action. The case of a general cocharacter $\sigma$ with $d\neq 0$ can be proved in a similar way. ∎

Corollary 4.9**.**

For any pair of cocharacters $\sigma,\rho$ of $T\times G(w)$ and $G(v)$ respectively for which $d\neq 0$, the variety $\mathfrak{M}[\rho]$ is connected.

Proof.

By Theorem 4.8, the cohomology group $H^{0}(\mathfrak{M}[\rho])$ is one-dimensional.∎

5. The algebra $\operatorname{Y}\nolimits^{\flat}$

Until the end of the paper we assume that ${\operatorname{k}\nolimits}=\mathbb{C}$. For each dimension vector $v$ we fix a closed subgroup $H(v)\subset G(v)$. Set $M^{*}={A}^{*}$ or $H^{*},$ and

[TABLE]

Let $K$ be the fraction field of $\Bbbk$. We abbreviate

[TABLE]

5.1. Definition

Consider the abelian group $\mathbb{X}=\mathbb{Z}^{I}\times\mathbb{Z}$ and the sub-semigroup $\mathbb{X}_{+}=\mathbb{N}^{I}\times(-\mathbb{N})$. Let $\operatorname{Y}\nolimits^{\flat}$ be the $\mathbb{X}$-graded $\Bbbk$-module given by

[TABLE]

For each dimension vector $v$ the $\Bbbk$-submodule of $\operatorname{Y}\nolimits^{\flat}$ given by

[TABLE]

has a canonical $\Bbbk[v]$-module stucture given by the cap-product $\cap$. We define a $\Bbbk$-algebra structure on $\operatorname{Y}\nolimits^{\flat}$. The multiplication is given by correspondences. The same multiplication was first defined in [39], [40] for the algebra $\operatorname{Y}\nolimits$ in the particular case where $|I|=1$. It differs from the usual convolution product on the Borel-Moore homology of the moduli stack of quivers, because it uses a pullback by a non l.c.i. morphism. This pullback is defined as a a refined pullback. We could as well view it as an ordinary pullback relative to some virtual fundamental classes which compensate the singularities of the spaces. This multiplication was later generalized to $\operatorname{Y}\nolimits$ and arbitrary quivers in [43].

To avoid any confusion we may write $\operatorname{Y}\nolimits^{\flat}_{H}$ or $\operatorname{Y}\nolimits^{\flat}_{A}$ for $\operatorname{Y}\nolimits^{\flat}$ to distinguish between the algebras associated with $M_{*}=H_{*}$ and $M_{*}=A_{*}$.

First, we introduce some notation. We write

[TABLE]

Fix an $I$-graded vector subspace $V_{1}\subseteq V$. Let

[TABLE]

be the stabilizer of $V_{1}$ in $G(v)$, its unipotent radical and the Lie algebra of the unipotent radical. Following (3.20), we set

[TABLE]

A closed point of $G(v)/U$ is identified with a pair of maps $\phi_{1},$ $\phi_{2}$ which give an exact sequence of $I$-graded vector spaces

[TABLE]

Thus, the set of triples $(\bar{x},\phi_{1},\phi_{2})$ where $\bar{x}\in\operatorname{R}\nolimits(v)$ and $\phi_{1},$ $\phi_{2}$ are as above with ${\operatorname{Im}\nolimits}(\phi_{1})$ stable by $\bar{x}$ is identified with the fiber product

[TABLE]

For each such triple $(\bar{x},\phi_{1},\phi_{2})$ the representation $\bar{x}$ in $V$ induces representations $\bar{x}_{1}$ in $V_{1}$ and $\bar{x}_{2}$ in $V_{2}$, yielding the diagram

[TABLE]

where the maps $p,q$ are given by $p(\bar{x},\phi_{1},\phi_{2})=\bar{x}$ and $q(\bar{x},\phi_{1},\phi_{2})=\bar{x}_{1}\oplus\bar{x}_{2}$. The map $q$ is smooth while $p$ is proper. The group $G(v)\times L$ acts in the obvious way on the variety $G\times_{U}\operatorname{R}\nolimits_{P}$. The diagram (5.5) yields the following diagram of Artin stacks

[TABLE]

In the diagram (5.6) the map $q$ is not smooth. So, the pullback homomorphism $q^{*}$ is not well-defined. In [39] this pullback is replaced by a refined pullback. Let $\mathfrak{u}$ be the Lie algebra of $U$. The map $q$ in (5.6) factorizes as follows. Consider the maps $c\,:\,\Lambda^{\flat}_{L}\to\mathfrak{u}\times\operatorname{R}\nolimits_{L}$ and $q_{3}\,:\,\operatorname{R}\nolimits_{P}\to\mathfrak{u}\times\operatorname{R}\nolimits_{L}$ given by

[TABLE]

We have the following fiber diagram of Artin stacks

[TABLE]

The vertical maps are closed immersions. The morphism $q_{3}$ is an l.c.i. because $\mathfrak{u}\times\operatorname{R}\nolimits_{L}$ and $\operatorname{R}\nolimits_{P}$ are smooth. Hence, the refined pullback morphism $(q_{3},q_{2})^{!}$ is well-defined.

Now, we have

[TABLE]

The pullback by $q_{1}$ yields a morphism of $\mathbb{Z}$-graded $\Bbbk$-modules

[TABLE]

The refined pullback by $(q_{3},q_{2})$ yields a morphism of $\mathbb{Z}$-graded $\Bbbk$-modules

[TABLE]

The pushforward yields a morphism of $\mathbb{Z}$-graded $\Bbbk$-modules

[TABLE]

The maps above are $T$-equivariant. Since $\dim(G(v))=d_{v}+(v,v)/2$, composing the map $p_{*}\circ(q_{3},q_{2})^{!}\circ q_{1}^{*}$ with the Kunneth homomorphism

[TABLE]

we get a $\Bbbk$-linear map

[TABLE]

hence a morphism of $\mathbb{Z}$-graded $\Bbbk$-modules

[TABLE]

Proposition 5.1**.**

Let $\flat$ be either [math] or $1$.

• $\mathrm{(a)}$

$(\operatorname{Y}\nolimits^{\flat},\star)$* is an $\mathbb{X}_{+}$-graded associative algebra over $\Bbbk$.*

• $\mathrm{(b)}$

$\operatorname{Y}\nolimits^{\flat}(0)\simeq\Bbbk$* with $[\Lambda^{\flat}(0)]\mapsto 1.$ The unit of $\operatorname{Y}\nolimits^{\flat}$ is the element $[\Lambda^{\flat}(0)]$.*

• $\mathrm{(c)}$

$\operatorname{Y}\nolimits^{\flat}(v,0)$* is spanned over $\mathbb{Q}$ by the set $\{[X]\,;\,X\in{\operatorname{Irr}\nolimits}(\Lambda^{\flat}(v))\}$.*

• $\mathrm{(d)}$

${\operatorname{cl}\nolimits}:\operatorname{Y}\nolimits^{\flat}_{A}\to\operatorname{Y}\nolimits^{\flat}_{H}$* is a surjective $\mathbb{Q}$-algebra homomorphism.*

Proof.

Part (a) is proved in [39], [40] in the particular case where $|I|=1$ and in [43] for arbitrary quivers. Part (d) follows from Theorem 3.2. The other claims are obvious. ∎

5.2. Comparison of $\operatorname{Y}\nolimits^{0}$ and $\operatorname{Y}\nolimits^{1}$

We can also define an $\mathbb{X}_{+}$-graded $\Bbbk$-algebra structure on the $\Bbbk$-module

[TABLE]

As $v$ runs over $\mathbb{N}^{I}$, the pushforward by the canonical closed embeddings

[TABLE]

gives $\mathbb{X}$-graded $\Bbbk$-linear maps

[TABLE]

Proposition 5.2**.**

• $\mathrm{(a)}$

The maps $\operatorname{Y}\nolimits^{1}\to\operatorname{Y}\nolimits^{0}$ and $\operatorname{Y}\nolimits^{0}\to\operatorname{Y}\nolimits$ above are $\Bbbk$-algebra homomorphisms.

• $\mathrm{(b)}$

If the torus $T$ contains a one parameter subgroup which scales all the quiver data by the same scalar, then

• $\bullet$

the maps $\operatorname{Y}\nolimits^{1}\otimes\,K\to\operatorname{Y}\nolimits^{0}\otimes\,K$ and $\operatorname{Y}\nolimits^{0}\otimes\,K\to\operatorname{Y}\nolimits\otimes\,K$ are $K$-algebra isomorphisms,

• $\bullet$

for each $v,$ $w$ the pushforward yields an isomorphism

[TABLE]

Proof.

Part (a) follows from the proper base change property of refined pullbacks applied to the map $(q_{3},q_{2})^{!}$ in §5.1. Now, let us check that the map $\operatorname{Y}\nolimits^{\flat}\otimes\,K\to\operatorname{Y}\nolimits\otimes\,K$ is an isomorphism. Although the set $\Lambda^{\flat}(v)$ is not projective and may be very singular, it admits only finitely many orbit types relative to the $T$-action, which means that only finitely many stabilizers occur since $T$ is abelian. Therefore, the localization theorem applies. We have the following lemma.

Lemma 5.3**.**

Let $T$ be a torus and $G$ be a product of general linear groups over ${\operatorname{k}\nolimits}$. Let $V$ be a finite dimensional rational representation of $T\times G$ and $X\subset V$ be a closed $T\times G$-equivariant subset. Let $I\subset{\operatorname{k}\nolimits}[V]$ be the ideal of polynomials vanishing at 0 and $\Lambda\subset X$ be the 0-set of $I^{G}$. Assume that $I^{T\times G}=\{0\}.$ Then, the pushforward yields an isomorphism

[TABLE]

Proof.

Fix a maximal torus $D\subseteq G$ and let $W=N_{G}(D)/D$ be the Weyl group. Then, the $N_{G}(D)$-action on $X$ gives rise to a $W$-action on $H^{D}_{*}(X)$ such that

[TABLE]

Hence, we may assume in the rest of the proof that $G$ is a torus.

Set $A=T\times G$ and $\mathfrak{a}=\text{Lie}(A)$. The pushforward by the inclusion $\Lambda\subset X$ gives an $M^{*}_{A}$-module homomorphism $i_{*}:M^{A}_{*}(\Lambda)\to M^{A}_{*}(X)$ which is invertible over the (open) complement of $\bigcup_{x\in X\setminus\Lambda}\mathfrak{a}_{x}$ in $\mathfrak{a}$. Here $\mathfrak{a}_{x}$ is the annihilator of $x$ in $\mathfrak{a}$.

The assumptions imply that for each point $x\in X\setminus\Lambda$ there is a non zero $G$-invariant polynomial $f\in I$ which does not vanish at $x$ and a non trivial character $\chi:T\to\mathbb{G}_{m}$ such that $f$ has the weight $\chi$ relative to the $T$-action on $V$. We deduce that $\mathfrak{a}_{x}\subseteq\{0\}\times\mathfrak{g}$. Therefore, the map $i_{*}$ is invertible over the open subset $(\mathfrak{t}\setminus\{0\})\times\mathfrak{g}$ of $\mathfrak{a}$.

∎

Now, set $V=\operatorname{R}\nolimits(v)$, $X={\operatorname{M}\nolimits}(v)$ and $G=G(v)$. The pushforward by the inclusion $\Lambda\subseteq\Lambda^{\flat}(v)$ yields a chain of maps

[TABLE]

Since the torus $T$ contains a one parameter subgroup which scales all the quiver data by the same scalar, we have $I^{T}=\{0\}.$ Then, the first part of (b) follows from Lemma 5.3.

Now, we prove the second part of (b). Set

[TABLE]

Set also $V=\operatorname{R}\nolimits(v,w)$ and $G=G(v)$. By Proposition 4.1, we have

[TABLE]

We deduce that for each point $x\in\mathfrak{M}(v,w)\setminus\mathfrak{L}^{\flat}(v,w)$ there is a non zero $G$-invariant polynomial $f\in I$ which does not vanish at $\pi(x)$ and a non trivial character $\chi:T\to\mathbb{G}_{m}$ such that $f$ has the weight $\chi$ relative to the $T$-action on $V$. Set $A=T\times H(w)$. We deduce that $\mathfrak{a}_{x}\subseteq\{0\}\times\mathfrak{h}(w)$. Therefore, the pushforward map

[TABLE]

is invertible over the open subset $(\mathfrak{t}\setminus\{0\})\times\mathfrak{h}(w)$ of $\mathfrak{a}$. ∎

5.3. Dimension

Given commuting formal variables $z_{i}$ with $i\in I$, we write $z^{v}=\prod_{i}(z_{i})^{v_{i}}$ for each dimension vector $v\in\mathbb{N}^{I}$. Consider the formal series in $\mathbb{Q}[[z_{i}\,;\,i\in I]]$ given by

[TABLE]

Let $A^{\flat}_{v}(t)=\sum_{r}a^{\flat}_{v,r}\,t^{r}$ with $\flat=0,1$ be the nilpotent and the 1-nilpotent Kac polynomials in $\mathbb{Z}[t]$ considered in [3] : for each prime power $q$ the integers $A^{0}_{v}(q)$ and $A^{1}_{v}(q)$ count the number absolutely indecomposable nilpotent and 1-nilpotent representations of the quiver $Q$ of dimension $v$ over $\mathbb{F}_{q}$. Then, by [3], we have

[TABLE]

Hence, the variety $\Lambda^{\flat}(v)$ has polynomial count. Let $\tau$ be the rank of $T$. Assume that $M_{*}=H_{*}$. Then, Corollary 2.2 and Theorem 3.2 yield

[TABLE]

We deduce that $\operatorname{Y}\nolimits^{\flat}(v,k)=0$ if $k$ is odd and that we have

[TABLE]

This yields the following formula.

Theorem 5.4**.**

Assume that $M_{*}=H_{*}$. We have

[TABLE]

∎

Example 5.5*.*

Let $Q$ be the Jordan quiver. Assume that $T$ is the rank 2 torus which acts by dilatation on the variables $x$ and $x^{*}$. Then we have $\operatorname{Y}\nolimits^{0}=\operatorname{Y}\nolimits^{1}$ and

[TABLE]

In this case, the algebra $\operatorname{Y}\nolimits^{\flat}$ is (the positive half of) the spherical degenerate double affine Hecke algebra of type $\operatorname{GL}\nolimits(\infty)$ defined in [40].

5.4. Generators

5.4.1. The stratification of $\Lambda^{1}(v)$

Definition 5.6**.**

Fix a vertex $i$ and an integer $l\geqslant 0$.

• $\mathrm{(a)}$

For each element $\bar{x}$ in $\operatorname{R}\nolimits(v)$ let $\varepsilon_{i}(\bar{x})$ be the codimension of the smallest subspace of $V_{i}$ containing \sum_{j\neq i}\sum_{h\in\bar{\Omega}_{ji}}{\operatorname{Im}\nolimits}(\bar{x}_{h}\big{)} and preserved by $\bar{x}_{h}$ for all $h\in\bar{\Omega}_{ii}$.

• $\mathrm{(b)}$

Let $\operatorname{R}\nolimits(v)^{l,i}$ and $\Lambda(v)^{l,i}$ be the locally closed subsets of $\operatorname{R}\nolimits(v)$ given by

[TABLE]

Fix dimension vectors $v$, $v_{1}$, $v_{2}$ with $v=v_{1}+v_{2}$. We use the same notation as in (5.1), (5.3). We define the following sets

[TABLE]

Now, assume that $v_{2}=l\,\delta_{i}$. From (5.7) we get the following fiber diagram of stacks

[TABLE]

Lemma 5.7**.**

Assume that $v_{2}=l\,\delta_{i}$.

• $\mathrm{(a)}$

$j_{1},j_{2},j_{3}$* and $j_{4}$ are open immersions.*

• $\mathrm{(b)}$

$q_{3,0}$* and $q_{2,0}$ are affine space bundles,*

• $\mathrm{(c)}$

$q_{2,0}^{*}\circ j_{1}^{*}=j_{2}^{*}\circ(q_{3},q_{2})^{!},$**

• $\mathrm{(d)}$

$p_{0}$* is an isomorphism,*

• $\mathrm{(e)}$

$p_{0,*}\circ q_{2,0}^{*}$* is an isomorphism $M_{*}^{T\times P}(\Lambda_{L}^{0,i})\to M_{*}^{T\times G}(\Lambda(v)^{l,i})$ such that $[\Lambda_{L}^{0,i}]\mapsto[\Lambda(v)^{l,i}].$*

Proof.

Part (a) is obvious, because, since $v_{2}=l\,\delta_{i}$, we have $\Lambda_{P}^{\geqslant l,i}=\Lambda^{1}_{P}$ and $R_{P}^{\geqslant l,i}=R_{P}$. Fix $I$-graded vector spaces $V$, $V_{1}$, $V_{2}$ with dimension vectors $v$, $v_{1}$, $v_{2}$. We abbreviate

[TABLE]

The fiber of $q_{3}$ over any tuple $(z,\bar{x}_{1}\oplus\bar{x}_{2})$ in $\mathfrak{u}\times\operatorname{R}\nolimits_{L}$ is

[TABLE]

where $\gamma_{\bar{x}_{1}\oplus\bar{x}_{2}}$ is the linear map given by

[TABLE]

So, this fiber is either empty or a torsor over the vector space ${\operatorname{Ker}\nolimits}(\gamma_{\bar{x}_{1}\oplus\bar{x}_{2}})$. Now, since $v_{2}=l\,\delta_{i}$ we have $\mathfrak{u}=\operatorname{Hom}\nolimits(V_{2,i},V_{1,i})$ and, for any $\bar{y}\in\operatorname{Hom}\nolimits_{\bar{\Omega}}(V_{2},V_{1})$,

[TABLE]

Thus, we have

[TABLE]

We deduce that the orthogonal to ${\operatorname{Im}\nolimits}(\gamma_{\bar{x}_{1}\oplus\bar{x}_{2}})$ relative to the trace is

[TABLE]

In particular, if $z\in{\operatorname{Im}\nolimits}(\gamma_{\bar{x}_{1}\oplus\bar{x}_{2}})^{\perp}$ then

[TABLE]

Now, assume that $\varepsilon_{i}(\bar{x}_{1})=0$. From (5.10) we deduce that

[TABLE]

Hence the map $\gamma_{\bar{x}_{1}\oplus\bar{x}_{2}}$ is onto. Thus the fiber $(q_{3})^{-1}(z,\bar{x}_{1}\oplus\bar{x}_{2})$ is not empty and has a constant dimension as $\bar{x}_{1}\oplus\bar{x}_{2}$ runs over the set $\operatorname{R}\nolimits_{L}^{0,i}$. Hence, the map $q_{3,0}$ is an affine space bundle, so the map $q_{2,0}$ is also an affine space bundle. This proves the claim (b). Compare the proof of [2, prop. 1.14].

Part (c) follows from (a), (b) and standard properties of the refined pullback. More precisely, given a cartesian square as in (2.3) and an open immersion $X_{0}\subset X$ we set $Y_{0}=g^{-1}(X_{0})$ and we consider the fiber diagram

[TABLE]

If the map $f_{0}$ is smooth, then $f^{\prime}_{0}$ is smooth by base change, so the pullback $(f_{0}^{\prime})^{*}$ is well-defined and coincides with $(f_{0},f^{\prime}_{0})^{!}$, so the functoriality of the refined pull-back with respect to the smooth morphisms $i$, $j$ yields $i^{*}\circ(f,f^{\prime})^{!}=(f^{\prime}_{0})^{*}\circ j^{*}.$

To prove (d) note that, since we have $v_{2}=l\,\delta_{i}$, for any tuple $(\bar{x},\phi_{1},\phi_{2})$ in $G\times_{U}\Lambda_{P}^{l,i}$ the degree $i$ part of the $I$-graded vector space ${\operatorname{Im}\nolimits}(\phi_{1})$ is equal to $\sum_{j\neq i}\sum_{h\in\bar{\Omega}_{ji}}{\operatorname{Im}\nolimits}(\bar{x}_{h})$, hence it is uniquely determined by $\bar{x}$. Part (e) follows from (b) and (d). ∎

5.4.2. The reduction to one vertex quivers

For each vertex $i\in I$ we abbreviate

[TABLE]

The first step to compute a system of generators of $\operatorname{Y}\nolimits^{1}$ is the following.

Proposition 5.8**.**

The $\Bbbk$-algebra $\operatorname{Y}\nolimits^{1}$ is generated by the subspace $\bigoplus_{i\in I}\operatorname{Y}\nolimits^{1}(\mathbb{N}\,\delta_{i})$.

Proof.

By Proposition 5.1 it is enough to assume that $M_{*}={A}_{*}$. The proof is by induction. For each $i\in I$ we consider the finite filtration

[TABLE]

such that $\operatorname{Y}\nolimits^{1}(v)^{\geqslant l,i}$ is the pushforward of ${A}_{*}^{T\times G}(\Lambda^{1}(v)^{\geqslant l,i})$ in $\operatorname{Y}\nolimits^{1}(v)$ by the closed embedding

[TABLE]

Lemma 5.9**.**

For each dimension vector $v$ we have $\operatorname{Y}\nolimits^{1}(v)=\sum_{i\in I}\operatorname{Y}\nolimits^{1}(v)^{\geqslant 1,i}.$

Proof.

The Mayer-Vietoris exact sequence for Chow groups implies that

[TABLE]

where $i_{X}$ is the embedding of an irreducible component $X\subseteq\Lambda^{1}(v)$. Therefore, it is enough to check that any irreducible component of $\Lambda^{1}(v)$ is contained into a closed subset of the form $\Lambda(v)^{\geqslant\delta_{i}}$ for some vertex $i$. Since any element of $\Lambda^{1}(v)$ belongs to $\Lambda_{W}$ for some restricted increasing flag of $I$-graded vector spaces $(W_{p})$, we have $\Lambda^{1}(v)=\bigcup_{i}\Lambda(v)^{\geqslant\delta_{i}}$. This proves the lemma. ∎

Let $\operatorname{Y}\nolimits^{\star\,v}$ be the subalgebra of $\operatorname{Y}\nolimits^{1}(v)$ generated by $\bigoplus_{i\in I}\operatorname{Y}\nolimits^{1}(\mathbb{N}\,\delta_{i})$. Since $\operatorname{Y}\nolimits^{1}(v)^{>v_{i},i}=\{0\}$, by descending induction we may assume that for some $l>0$ we have

[TABLE]

and we must prove that $\operatorname{Y}\nolimits^{1}(v)^{\geqslant l,i}\subseteq\operatorname{Y}\nolimits^{\star\,v}.$ The excision yields an exact sequence

[TABLE]

It is enough to prove that

[TABLE]

To do that, set $v_{2}=l\,\delta_{i}$ and $v_{1}=v-v_{2}$ and define $L$, $P$, etc, as in (5.1), (5.3). Then, we have a commutative diagram

[TABLE]

More precisely, note that

• •

the commutativity of the upper square and the invertibility of the pullback morphism $q_{2,0}^{*}$ follow from Lemma 5.7,

• •

the commutativity of the middle square follows from proper base change and the equality $\Lambda_{P}^{\geqslant l,i}=\Lambda^{1}_{P}$,

• •

the commutativity of the lower square follows from proper base change relative to the cartesian square

[TABLE]

• •

the invertibility of the morphism $p_{0,*}$ follows from the invertibility of the map $p_{0}$.

We deduce that for each $x\in\operatorname{Y}\nolimits^{1}(v)^{\geqslant l,i}$ there is an element $y\in{A}^{T\times P}_{*}(\Lambda^{1}_{L})$ such that

[TABLE]

from which we deduce that the following inclusion holds

[TABLE]

Now, since $v_{1}<v$, by an increasing induction on $v$ we can assume that $\operatorname{Y}\nolimits^{1}(v_{1})=\operatorname{Y}\nolimits^{\star\,v_{1}}.$ Therefore, the identity (5.11) follows from (5.12). The proposition is proved. ∎

Remark 5.10*.*

In the particular case where the quiver $Q$ has no 1-loops, the proposition implies that the algebra $\operatorname{Y}\nolimits^{1}$ is indeed generated by the subspace $\bigoplus_{i\in I}\operatorname{Y}\nolimits^{1}(\delta_{i}).$

5.4.3. The case of one vertex quivers

Fix a vertex $i\in I$. Since $\operatorname{Y}\nolimits^{0}(\mathbb{N}\,\delta_{i})=\operatorname{Y}\nolimits^{1}(\mathbb{N}\,\delta_{i})$ and $\Lambda^{0}=\Lambda^{1}$, we drop the exponents 0, 1.

Definition 5.11**.**

Given any integer $l>0$ let ${{\operatorname{x}\nolimits}}_{i,l}$ be the element in $\operatorname{Y}\nolimits$ given by

[TABLE]

Proposition 5.12**.**

$\operatorname{Y}\nolimits^{\flat}(\mathbb{N}\,\delta_{i})$* is generated by the elements $u\cap\,{{\operatorname{x}\nolimits}}_{i,l}$ with $l>0$ and $u\in\Bbbk[l\,\delta_{i}]$.*

Proof.

We assume that $Q=\operatorname{J}\nolimits(i,q)$ and $q>0$, since the case $q=0$ is trivial by Remark 5.10. To avoid cumbersome notation we abbreviate ${{\operatorname{x}\nolimits}}_{l}={{\operatorname{x}\nolimits}}_{i,l}$. By Theorem 3.2 it is enough to assume that $M_{*}={A}_{*}$ is the Chow group.

Let us first concentrate on the hyperbolic case. Fix a composition $\nu=(\nu_{1},\dots,\nu_{r})$ of $v$ and choose an increasing $I$-graded flag $W$ in $V$ of type $\nu$. Let $P$ be the parabolic subgroup of $G(v)$ of block type $\nu$ which fixes the flag $W$. Let $L$, $U$ be the standard Levi subgroup of $P$ and its unipotent radical. Let $\mathfrak{l}$, $\mathfrak{u}$ be their Lie algebras. We define the sets $\operatorname{R}\nolimits_{P}$, $\operatorname{R}\nolimits_{L}$, $\Lambda_{P}$ and $\Lambda_{L}$ as in (5.3), using the parabolic subgroup $P$ instead of a 2 blocks parabolic subgroup as in loc. cit. Consider the closed subset $\Lambda_{(L)}$ of $\Lambda_{L}$ given by

[TABLE]

Then, we have an obvious isomorphism

[TABLE]

Consider the following fiber diagram of stacks

[TABLE]

By proper base change we have

[TABLE]

Further, we claim that

[TABLE]

Since we have

[TABLE]

the proposition will follow by induction using the excision exact sequence

[TABLE]

Now, we prove the identity (5.16). Since the map $p$ is an isomorphism $G(v)\times_{P}\Lambda^{\circ}_{W}\to\Lambda^{\circ}_{\nu}$, by base change relatively to (5.14) we must check the relation

[TABLE]

Lemma 5.13**.**

Assume that $q>1$. Then, we have

• $\mathrm{(a)}$

$A_{*}^{T\times P}(\Lambda^{\circ}_{W})={A}^{*}_{T\times P}\cap\,[\Lambda^{\circ}_{W}]$,

• $\mathrm{(b)}$

${{\operatorname{x}\nolimits}}_{\nu_{1}}\star\dots\star{{\operatorname{x}\nolimits}}_{\nu_{r}}=[\Lambda_{\nu}]$.

By Lemma 5.13(a) and the ${A}_{T\times P}^{*}$-linearity of the pullback we are reduced to prove that

[TABLE]

This relation follows from Lemma 5.13(b).

Now, we consider the isotropic case. In this case, the proof of the proposition is similar to the proof in the hyperbolic case. Since $q=1$, the irreducible components of $\Lambda(v)$ are labelled by the partitions $\nu$ of $v$. Then, we have again the fiber diagram (5.14) with the notations as in §3.6.1. Let $\nu_{1}\geqslant\nu_{2}\geqslant\cdots\geqslant\nu_{r}$ be the parts of $\nu$. The proposition follows from the following lemma.

Lemma 5.14**.**

Assume that $q=1$. Then, we have

• $\mathrm{(a)}$

$A_{*}^{T\times P}(\Lambda^{\circ}_{W})={A}^{*}_{T\times P}\cap\,[\Lambda^{\circ}_{W}]$,

• $\mathrm{(b)}$

${{\operatorname{x}\nolimits}}_{\nu_{1}}\star\dots\star{{\operatorname{x}\nolimits}}_{\nu_{r}}\in[\overline{\Lambda_{\nu}^{\circ}}]+\sum_{\mu{\trianglelefteqslant}\nu}\mathbb{Q}\,[\overline{\Lambda^{\circ}_{\mu}}]$.

∎

Proof of Lemma 5.13.

We use the notation in §3.7.4 with $w=v$ and $\gamma=\xi^{-1}$, $\rho=\rho_{\nu}$ as in (3.13), (4.5). Let $P$ denote the parabolic subgroup associated with the cocharacter $\rho_{\nu}$ in (3.19). It is the standard parabolic of block type $\nu$. Since $Q=\operatorname{J}\nolimits(i,q)$ we can omit the upperscript $\flat$ and we abbreviate $\operatorname{L}\nolimits_{s}(v,v)=\operatorname{L}\nolimits_{s}^{\flat}(v,v)$ and $\mathfrak{L}(v,v)=\mathfrak{L}^{\flat}(v,v)$. By (4.4) we have the locally closed subset $\mathfrak{L}[\rho_{\nu}]\subset\mathfrak{L}(v,v)$, which is the geometric quotient of the $T\times P\times G(v)$-equivariant subset $\operatorname{L}\nolimits[\rho_{\nu}]\subset\operatorname{L}\nolimits_{s}(v,v)$. By (4.7), the open immersion $\alpha:\Lambda(v)\times G(v)\to\operatorname{L}\nolimits_{s}(v,v)$ in (4.11) satisfies

[TABLE]

From descent and excision, this yields a surjective ${A}^{*}_{T\times P}\,$-module homomorphism

[TABLE]

This homomorphism maps $[\,\mathfrak{L}[\rho_{\nu}]\,]$ to $[\Lambda^{\circ}_{W}]$. Therefore, the part (a) of the lemma follows from the next result :

Lemma 5.15**.**

We have ${A}_{*}^{T\times P\times G(v)}(\operatorname{L}\nolimits[\rho_{\nu}])={A}^{*}_{T\times P\times G(v)}\cap[\,\operatorname{L}\nolimits[\rho_{\nu}]\,].$

Proof.

We use the same notation as in the proof of Lemma 5.13. Since $\operatorname{L}\nolimits[\rho_{\nu}]$ is an affine space bundle over ${\operatorname{M}\nolimits}[\rho_{\nu}]$, it is smooth. Thus, the Poincaré duality yields

[TABLE]

Moreover, we have a chain of isomorphisms

[TABLE]

By Theorem 4.8 the map

[TABLE]

is a surjective ring homomorphism. It follows that

[TABLE]

is surjective as well. The lemma is proved. ∎

Now, let us prove the part (b). The fundamental class of the set $\Lambda_{(L)}$ in (5.13) is identified with the monomial ${{\operatorname{x}\nolimits}}_{\nu_{1}}\otimes\dots\otimes{{\operatorname{x}\nolimits}}_{\nu_{r}}$ under the isomorphism (5.17). Thus, from the equality (5.15) we deduce that

[TABLE]

In particular, the element ${{\operatorname{x}\nolimits}}_{\nu_{1}}\star\dots\star{{\operatorname{x}\nolimits}}_{\nu_{r}}$ in $\operatorname{Y}\nolimits^{\flat}(v,0)$ is supported on the closed subset $\Lambda_{\nu}$ of $\Lambda(v)$. Since $\Lambda_{\nu}$ is irreducible by Proposition 3.8, we deduce that this class is a rational multiple of $[\Lambda_{\nu}]$. To prove that it is equal to $[\Lambda_{\nu}]$, it is enough to check that

[TABLE]

To prove this, recall first that $\Lambda_{W}$ and $\Lambda_{(L)}$ are both irreducible. Then, applying recursively the Lemma 3.20, we deduce that the map $q_{4}:\Lambda_{W}\to\Lambda_{(L)}$ restricts to a smooth map from a dense open subset of $\Lambda_{W}$ onto a dense open subset of $\Lambda_{(L)}$. Next, a direct computation yields

[TABLE]

[TABLE]

Hence $q_{3}$ and $q_{4}$ have the same relative dimension. This proves the relation (5.19).

∎

Proof of Lemma 5.14.

Part (a) can be proved as in Lemma 5.13(a). Let us give a more direct argument. Recall that $\Lambda_{W}^{\circ}$ is a vector bundle over the $P$-orbit $\mathfrak{u}^{\circ}$. Hence we have the chain of maps

[TABLE]

where $a$ is given by excision, hence it is surjective, and $b$ is the pullback by the bundle $\Lambda_{W}^{\circ}\to\mathfrak{u}^{\circ}$, hence it is an isomorphism. The part (a) follows.

To prove (b), note that the equality (5.18) implies that ${{\operatorname{x}\nolimits}}_{\nu_{1}}\star\dots\star{{\operatorname{x}\nolimits}}_{\nu_{r}}$ is supported on the set $\Lambda_{{\trianglelefteqslant}\,\nu}^{\circ}$. Hence a degree argument implies that

[TABLE]

We must compute the first coefficient. To do that, we factorize the maps $q_{3}$ and $q_{4}$ as in the following fiber diagram

[TABLE]

The maps $q^{\prime}_{3}$, $q^{\prime}_{4}$ are smooth of relative dimension $2\dim(\mathfrak{u})$. The map $q^{\prime\prime}_{3}$ is a regular embedding of relative dimension $-\dim(\mathfrak{u})$ given by the component in $\mathfrak{u}$ of the commutator of elements of $\mathfrak{p}$. Fix an element $e\in\mathfrak{u}^{\circ}$. Let $P_{e}$ be the stabilizer of $e$ for the adjoint action of $P$ on $\mathfrak{u}$. Let $\mathfrak{p}_{e}$ be its Lie algebra. The map $q^{\circ}_{4}$ is given by the inclusion

[TABLE]

Since $\mathfrak{p}_{e}$ is the kernel of the submersion $\mathfrak{p}\to\mathfrak{u}$ such that $x\mapsto[e,x]$, we deduce that the map $q_{4}^{\circ}$ is also a regular embedding of relative dimension $-\dim(\mathfrak{u})$. Hence, we have

[TABLE]

This finishes the proof of the lemma.

∎

5.5. The algebra $\mathbf{Y}^{\flat}$

In this subsection we introduce an extension of $\operatorname{Y}\nolimits^{\flat}$ by adding a Cartan part.

5.5.1. Definition

Let $\Bbbk(\infty)$ be Macdonald’s ring of symmetric functions with coefficients in $\Bbbk$. We view it as the free commutative $\Bbbk$-algebra generated by the power sum polynomials $p_{l}$ for each integer $l>0$. It carries a comultiplication $\Delta$, a counit $\eta$ and an antipode $S$ given by $\Delta(p_{l})=p_{l}\otimes 1+1\otimes p_{l},$ $\eta(p_{l})=0$ and $S(p_{l})=-p_{l}.$ It is equipped with the $\mathbb{Z}$-grading such that the element $p_{l}$ has the degree $\varepsilon l$. We define

[TABLE]

For each vertex $i\in I$ and each integer $l>0$, let $p_{i,l}$ be the element of $\mathbf{Y}(0)$ given by

[TABLE]

where $p_{l}$ is at the $i$-th spot. For any dimension vector $v$, restricting a representation of $G(\infty^{I})$ to the subgroup $G(v)$, yields a $\Bbbk$-algebra homomorphism

[TABLE]

Let $\mathcal{U}_{i}$ be the tautological $T\times G(v)$-equivariant vector bundle on $\Lambda^{\flat}(v)$, which is given by

[TABLE]

with the obvious $T\times G(v)$-action. Consider the $\mathbf{Y}(0)$-action $\circledast$ on $\operatorname{Y}\nolimits^{\flat}$ such that the element $p_{i,l}$ acts via the cap product with the class ${\operatorname{ch}\nolimits}_{l}(\mathcal{U}_{i})$. We use Sweedler’s notation for the coproduct. The following result is an easy consequence of the definitions and is left to the reader.

Proposition 5.16**.**

For each $x\in\mathbf{Y}(0)$ and each $y,z\in\operatorname{Y}\nolimits^{\flat}$ we have

[TABLE]

We deduce that the $\mathbb{X}$-graded $\Bbbk$-algebra $\operatorname{Y}\nolimits^{\flat}$ is a $\mathbf{Y}(0)$-module $\mathbb{X}$-graded algebra. ∎

Definition 5.17**.**

The COHA is the smash product $\mathbf{Y}^{\flat}$ of $\mathbf{Y}(0)$ and $\operatorname{Y}\nolimits^{\flat}$. It is a free $\mathbb{X}_{+}$-graded $\Bbbk$-algebra, with $\mathbf{Y}^{\flat}(v,k)=\operatorname{Y}\nolimits^{\flat}(v,k)$ for each nonzero dimension vector $v$ and each integer $k$ and $\mathbf{Y}^{\flat}(0,k)$ equal to the degree $k$ part of $\mathbf{Y}(0)$. We set $\mathbf{Y}_{K}=\mathbf{Y}^{1}\otimes K.$

5.5.2. The generators of $\mathbf{Y}^{1}$

Let $Q$ be any quiver. Set

[TABLE]

Theorem 5.18**.**

• $\mathrm{(a)}$

The $\Bbbk$-algebra $\mathbf{Y}^{1}$ is generated by the subset

[TABLE]

• $\mathrm{(b)}$

Assume that $M_{*}=H_{*}$, and that $T$ contains $\theta,\theta^{*}$. Then, the $K$-algebra $\mathbf{Y}_{K}$ is generated by the subset

[TABLE]

Proof.

Part (a) follows from Propositions 5.8 and 5.12. To prove (b) we must check that if $Q$ is the Jordan quiver with vertex $i$ and $T$ is the torus $\mathbb{G}_{m}\times\mathbb{G}_{m}$ generated by the cocharacters $\theta$, $\theta^{*}$, then the $K$-algebra $\mathbf{Y}_{K}$ is generated by $\mathbf{Y}(0)\otimes K$ and the element ${{\operatorname{x}\nolimits}}_{i,1}.$ To do this, let us abbreviate ${{\operatorname{x}\nolimits}}_{l}={{\operatorname{x}\nolimits}}_{i,l}$ for each integer $l>0$ as in the proof of Proposition 5.12. We may also omit the symbol $\delta_{i}$ to unburden the notation. Let $u\in\Bbbk[\delta_{i}]$ be the first equivariant Chern class of the linear character of $G(\delta_{i})$. For each integer $k$, consider the element $D_{k}=u^{k}\cap{{\operatorname{x}\nolimits}}_{1}$ in $\operatorname{Y}\nolimits(\delta_{i})$. To simplify the notation we omit the symbol $\delta_{i}$. We must check that for each $l$ the element ${{\operatorname{x}\nolimits}}_{l}$ belongs to the subalgebra $\operatorname{Y}\nolimits_{\bullet}\subset\operatorname{Y}\nolimits$ generated by $\{D_{k}\,;\,k\in\mathbb{N}\}$. To do that, consider the closed embedding

[TABLE]

By proper base change, the pushforward gives an $\mathbb{X}_{+}$-graded $\Bbbk$-algebra homomorphism

[TABLE]

where $\operatorname{Z}\nolimits(v)=H_{*}^{T\times G(v)}(\operatorname{R}\nolimits(v))$. The map $i_{*}$ is injective because the $\Bbbk[v]$-module $\operatorname{Y}\nolimits(v)$ is torsion-free by Proposition 4.6. We identify $\operatorname{Z}\nolimits(v)$ with $\Bbbk[v]$ under the map

[TABLE]

The multiplication in $\operatorname{Z}\nolimits$ has been computed in [40]. Let $(t,t^{*})$ be the basis of $\mathfrak{t}^{*}$ dual to $(\theta,\theta^{*})$. Modulo the canonical identification

[TABLE]

this product is identified with the shuffle product given by

[TABLE]

where SYM denotes summing over all permutations of the variables $x_{1},\dots,x_{v+w}$ and $\zeta$ is the rational function

[TABLE]

Now, let $\operatorname{Z}\nolimits_{\bullet}\subset\operatorname{Z}\nolimits$ be the image of $\operatorname{Y}\nolimits_{\bullet}$. To finish the proof, it remains to show that $i_{*}({{\operatorname{x}\nolimits}}_{l})\in\operatorname{Z}\nolimits_{\bullet}\otimes K$ for each $l$. To do that, we may use [35, thm. 2.8] which implies that $\operatorname{Z}\nolimits_{\bullet}\otimes K$ consists of the symmetric polynomials over $K$ which satisfy the wheel conditions in [35, def. 2.5]. Indeed, since ${{\operatorname{x}\nolimits}}_{l}$ is the fundamental class of $\Lambda_{(l\delta_{i})}=\{0\}\times\mathfrak{gl}(l)$, the element $i_{*}({{\operatorname{x}\nolimits}}_{l})$ coincides with the polynomial

[TABLE]

hence it satisfies the wheel conditions. ∎

5.6. The representation of $\mathbf{Y}^{\flat}$ in the homology of quiver varieties

Now, we turn to the action of $\mathbf{Y}^{\flat}$ on the homology of quiver varieties. We set $M_{*}=H_{*}$. We need this to compare $\mathbf{Y}^{1}$ with the Yangian introduced in [27], see [41].

5.6.1. The faithful representation of $\operatorname{Y}\nolimits^{\flat}$

Let $F_{w}$ be the $\mathbb{X}$-graded $\Bbbk\langle w\rangle$-module given by

[TABLE]

Set $|w\rangle$ equal to the fundamental class $[\mathfrak{M}(0,w)]$ in $F_{w}.$

Proposition 5.19**.**

Let $\flat$ be either [math] or $1$.

• $\mathrm{(a)}$

The $\mathbb{X}$-graded $\Bbbk$-algebra $\operatorname{Y}\nolimits^{\flat}$ acts on the $\mathbb{X}$-graded $\Bbbk\langle w\rangle$-module $F_{w}$.

• $\mathrm{(b)}$

If $H(v)=G(v)$ then the action on the element $|v\rangle$ yields an injective map $\operatorname{Y}\nolimits^{\flat}(v)\to F_{v}(v).$ In particular, the representation of $\operatorname{Y}\nolimits^{\flat}$ in $\bigoplus_{w}F_{w}$ is faithful.

• $\mathrm{(c)}$

The action of $\operatorname{Y}\nolimits^{\flat}(v)$ on the element $|w\rangle$ yields a $\Bbbk\langle w\rangle$-linear map

[TABLE]

Its image is the pushward of $H_{*}^{T\times H(w)}(\mathfrak{L}^{\flat}(v,w))$ by the inclusion $\mathfrak{L}^{\flat}(v,w)\subset\mathfrak{M}(v,w)$.

Proof.

The proof of (a) is similar to the proof given in [39, §7.6], [40, §6] in the particular case where $|I|=1$. See also [43]. Let $L$, $P$, $U$, $\mathfrak{u}$, $v_{1}$, $v_{2}$ be as in (5.1), (5.2). Let $\operatorname{R}\nolimits_{P}$ and $\operatorname{R}\nolimits_{s,P}$ be as in (3.20). Set

[TABLE]

For each pair $(\bar{x},\bar{a})$ in $\operatorname{R}\nolimits_{P}$ let $\bar{x}_{1}$, $\bar{x}_{2}$, $\phi_{1}$, $\phi_{2}$ be as in (5.4), (5.5) and define

[TABLE]

Then, we have the following fiber diagram of $P$-varieties

[TABLE]

where the maps $c$, $q_{2}$ and $q_{3}$ are given by

[TABLE]

Since the map $q_{3}$ is an l.c.i. morphism, the refined pullback $(q_{3},q_{2})^{!}$ is well-defined. Consider the obvious maps

[TABLE]

where $p$ is as in Proposition 3.14. As in (5.8), the composition $p_{*}\circ(q_{2},q_{3})^{!}\circ q_{1}^{*}$ defines a $\Bbbk$-linear map

[TABLE]

Taking the sum over all $v_{1},v_{2}\in\mathbb{N}^{I}$ yields an $\mathbb{X}$-graded $\Bbbk$-module homomorphism

[TABLE]

The proof that this map gives indeed a representation of $\operatorname{Y}\nolimits^{\flat}$ on $F_{w}$ is the same as in the case of the algebra $\operatorname{Y}\nolimits$ and the Jordan quiver in [39], [40], see, e.g., [43].

Now, we prove part (b). Set $v=v_{2}$ and $v_{1}=0$ in the diagram (5.25). Set

[TABLE]

We get the following fiber diagram of $G(v)$-varieties

[TABLE]

where $p$ and the vertical maps are the obvious closed immersions, while $q_{2}$ and $q_{3}$ are the projections on the first factors. The maps $q_{2}$ and $q_{3}$ are l.c.i. morphisms of the same relative dimension. So, we have $(q_{3},q_{2})^{!}=q_{2}^{*}$ by the excess intersection formula. As in §4.3, we define the maps

[TABLE]

Thus, we have $q_{2}=p_{1}\circ\beta$. Further, the map

[TABLE]

coincides with the composed map

[TABLE]

Now, set $w=v$ and $H(v)=G(v)$. Then, the map $q_{2}^{*}\circ\rho_{1}^{\sharp}$ is injective by Lemma 4.4. Finally, by Proposition 4.3, the pushforward by $p$ yields an injective map

[TABLE]

because the left hand side is torsion free by loc. cit. and $p_{*}$ induces an isomorphism after base change from $\Bbbk[v]$ to its fraction field by the localization theorem.

Finally, to prove (c) note that by Proposition 4.3, the cycle map gives an isomorphism

[TABLE]

Since the $\Bbbk$-algebra $\operatorname{Y}\nolimits^{\flat}_{A}$ acts on the $\Bbbk\langle w\rangle$-module

[TABLE]

by convolution, the map (5.27) fits into the fiber diagram

[TABLE]

Since $q_{2}$ is the composition of the vector bundle $p_{1}$ and the open immersion $\beta$, the lower map is surjective by excision. The right vertical map is surjective by Proposition 4.3. This finishes the proof of the proposition. ∎

Now, for any $i$, $l$ we compute the action of the element ${{\operatorname{x}\nolimits}}_{i,l}$ on $F_{w}$. To do so, assume that $v_{2}=l\,\delta_{i}.$ Then, we consider the subvariety of $\mathfrak{h}[v_{1},v_{2}\,;\,w]$ given by

[TABLE]

Let $\mathfrak{M}(w)$ be the disjoint sum of all $\mathfrak{M}(v,w)$’s and $\mathfrak{C}_{i,l}(w)$ be the disjoint sum of all $\mathfrak{C}(v_{1},v_{2}\,;\,w)$’s where $v,v_{1},v_{2}$ are as above. We view $\mathfrak{C}_{i,l}(w)$ as a closed $T\times G(w)$-invariant subvariety of the symplectic manifold $\mathfrak{M}(w)^{{\operatorname{op}\nolimits}}\times\mathfrak{M}(w)$ which is proper over the first factor. Hence, it acts by convolution on $F_{w}$. We use the same symbol to denote $\mathfrak{C}_{i,l}(w)$ and the convolution by this correspondence, which is a $\Bbbk\langle w\rangle$-linear operator on $F_{w}$.

Proposition 5.20**.**

For each $i$, $l$, $w$ the following hold

• $\mathrm{(a)}$

$\mathfrak{C}_{i,l}(w)$* is either empty or a Lagrangian local complete intersection in $\mathfrak{M}(w)^{{\operatorname{op}\nolimits}}\times\mathfrak{M}(w)$.*

• $\mathrm{(b)}$

If either $q_{i}>1$ or $q_{i}=0$ and $l=1$ then $\mathfrak{C}_{i,l}(w)$ is irreducible.

• $\mathrm{(c)}$

${{\operatorname{x}\nolimits}}_{i,l}$* acts on $F_{w}$ via the operator $\mathfrak{C}_{i,l}(w)$.*

Proof.

If $q_{i}=0$ then $\mathfrak{C}(v_{1},v_{2}\,;\,w)$ coincides with the Hecke correspondence in [31] (for the case of quiver without loops) and all statements are straightforwards. If $q_{i}>0$ then parts (a), (b) follow from Propositions 3.16, 3.17.

Now, we prove (c). Let $\operatorname{R}\nolimits_{s,L}$, $\operatorname{R}\nolimits_{s,P}$ and $q_{2}$, $q_{3}$ be as in (5.24), (5.25). We assume that $q_{i}>0$, the proof in the case $q_{i}=0$ is very similar. We have the Cartesian square

[TABLE]

The map $q_{3}$ is an l.c.i. morphism of relative dimension $v_{1}\cdot v_{2}-(v_{1},v_{2})+w\cdot v_{2}$, because $\mathfrak{u}\times\operatorname{R}\nolimits_{s,L}$ and $\operatorname{R}\nolimits_{s,P}$ are smooth. Since $\mathfrak{C}(v_{1},v_{2}\,;\,w)$ is a local complete intersection, so is also the variety $\operatorname{H}\nolimits[v_{1},\Lambda_{(v_{2})}\,;\,w]$ by (3.23). Hence, the map $q_{2}$ is also an l.c.i. morphism of relative dimension $v_{1}\cdot v_{2}-(v_{1},v_{2})+w\cdot v_{2}$. We deduce that the pullback morphism $q_{2}^{*}$ is well-defined and that $(q_{3},q_{2})^{!}=q_{2}^{*}$ by the excess intersection formula, proving the claim. ∎

Corollary 5.21**.**

If $i\in I^{\operatorname{h}\nolimits}$ and $\nu\vDash v_{2}$, then the element $[\Lambda_{\nu}]$ in $\operatorname{Y}\nolimits^{\flat}(v_{2})$ acts on $F_{w}$ by convolution with the correspondence $\mathfrak{h}(\Lambda_{\nu})$.

Proof.

Set $\nu=(\nu_{1},\dots,\nu_{r})$. By Lemma 5.13 we have $[\Lambda_{\nu}]={{\operatorname{x}\nolimits}}_{\nu_{1}}\star\cdots\star{{\operatorname{x}\nolimits}}_{\nu_{r}}$ in $\operatorname{Y}\nolimits^{\flat}(v_{2})$. Hence, the claim follows from Propositions 3.23, 5.20.

∎

5.6.2. The geometric realization of $\mathbf{Y}^{1}$ via Hecke correspondences

For each dimension vector $w$, let $A_{w}$ be the $\mathbb{X}$-graded $\Bbbk\langle w\rangle$-algebra given by

[TABLE]

where $A_{w}(v_{2},k_{2})$ consists of all $\Bbbk\langle w\rangle$-linear endomorphisms of $F_{w}$ which are homogeneous of degree $(v_{2},k_{2})$. Under the convolution, any cycle in M_{*}^{T\times H(w)}\big{(}\mathfrak{M}(u,w)\times\mathfrak{M}(v,w)\big{)} which is proper over $\mathfrak{M}(u,w)$ gives rise to an operator $F_{w}(v)\to F_{w}(u),$ see [5] for details. Therefore, for each $i\in I$ and $l\in\mathbb{Z}_{>0}$ the familly of correspondences $\mathfrak{C}_{i,l}(w)$ defines an element

[TABLE]

Let $\mathcal{V}_{i}$ be the universal $T\times H(w)$-equivariant vector bundles on $\mathfrak{M}(w)$, given by

[TABLE]

Let $\psi_{i,l}$ be the operators in $\prod_{w}A_{w}$ given by the cap product with the $T$-equivariant cohomology class ${\operatorname{ch}\nolimits}_{l}(\mathcal{V}_{i})$.

Theorem 5.22**.**

Set $H(w)=G(w)$ for all $w$’s. Then, the $\Bbbk$-algebra $\mathbf{Y}^{1}$ is isomorphic to the $\Bbbk$-subalgebra of $\prod_{w}A_{w}$ generated by

[TABLE]

Proof.

We extend the $\operatorname{Y}\nolimits^{1}$-action on $F_{w}$ to a $\mathbf{Y}^{1}$-action taking the element $p_{i,l}\in\mathbf{Y}(0)$ to $\psi_{i,l}$. Then, the theorem follows from Theorem 5.18 and Propositions 5.19, 5.20, because $H(w)=G(w)$.

∎

Appendix A

A.1. Proof of Proposition 3.1

The proposition is proved in the literature under the assumption that $Q$ does not carry oriented cycles (or edge loops). The following argument which works in full generality was explained to us by W. Crawley-Boevey. Let us begin by sketching the proof that $\Pi$ is of homological dimension two. We may and will restrict ourselves to the case of a connected quiver. For any vertex $i\in I$ we denote by $e_{i}$ the corresponding idempotent of $\Pi$. Consider the complex of $(\Pi,\Pi)$-bimodules

[TABLE]

where all the tensor products are taken over $\mathbb{C}$, and where the maps are defined as follows : $k$ is the multiplication map, $g(ae_{h^{\prime\prime}}\otimes e_{h^{\prime}}b)=ahe_{h^{\prime}}\otimes e_{h^{\prime}}b-ae_{h^{\prime\prime}}\otimes e_{h^{\prime\prime}}hb$ and

[TABLE]

where $\epsilon_{h}=1$ if $h\in\Omega$ and $\epsilon_{h}=-1$ if $h\in\Omega^{*}$. The fact that (A.1) is a complex is a direct consequence of the defining relations of $\Pi$. We claim that it is in addition exact. This exactness everywhere except for the leftmost term may be checked using standard arguments. Recall that we have assumed that $Q$ is not of finite Dynkin type. Equipping $\Pi$ with the $\mathbb{N}$-grading obtained by assigning degree $1$ to any edge $h\in\bar{\Omega}$ we then have that $\Pi$ is a Koszul algebra and that its Hilbert series

[TABLE]

is equal to $H_{Q}(t)=(\operatorname{Id}\nolimits+t\,(\mathbf{Q}+{}^{t}\mathbf{Q})+t^{2}\,\operatorname{Id}\nolimits)^{-1}$ where $\mathbf{Q}$ is the adjacency matrix of $Q$, see [26], [16]. Observing that $f,g,k$ are of respective degrees $1,1,0$ we deduce that

[TABLE]

as wanted. This proves that $\Pi$ is of homological dimension two. Next, let $M,$ $N$ be any finite dimensional $\Pi$-modules. Tensoring (A.1) by $M$ yields the projective resolution of $M$

[TABLE]

Applying the functor $\operatorname{Hom}\nolimits(N,\cdot)$ we obtain the following complex computing $\operatorname{Ext}\nolimits^{\bullet}(N,M)$ :

[TABLE]

It remains to observe that this complex is canonically isomorphic to the dual of the same complex in which the roles of $M$ and $N$ are exchanged.

A.2. Proof of Lemma 3.9

Being the fiber of the moment map $\mathfrak{gl}(v)^{2q}\to\mathfrak{sl}(v)$, the set ${\operatorname{M}\nolimits}(v)$ has all its irreducible components of dimension at least $(2q-1)v^{2}+1$. Note also that ${\operatorname{M}\nolimits}(v)$ is defined over $\mathbb{Z}$ and may be therefore reduced to any finite field. By the Lang-Weil theorem the irreducibility of ${\operatorname{M}\nolimits}(v)$ will thus be a consequence of the estimate as $l\to\infty$

[TABLE]

By [28, thm. 5.1] the number of points of ${\operatorname{M}\nolimits}(v)(\mathbb{F}_{l})$ is given by the following generating series

[TABLE]

where Exp is the plethystic exponential and $A_{v}(l)$ is the Kac polynomial, see e.g. [3, §2]. It is well-known that $A_{v}(l)$ is a monic polynomial of degree $1-\langle v,v\rangle=1+(g-1)v^{2}$. The coefficient of $z^{v}$ in (A.4) reads

[TABLE]

where the sum ranges over all $r$-tuples $((n_{1},v_{1}),\ldots,(n_{r},v_{r}))$ satisfying $\sum_{i}n_{i}v_{i}=v$. We claim that only the first term on the right hand side of (A.5) contributes to the leading term. Indeed, $\frac{l}{l-1}A_{v}(l)\sim l^{1+(q-1)v^{2}}$ while for any $r\geqslant 2$ and tuple $(n_{i},v_{i})_{i}$ we have

[TABLE]

and it is elementary to check that $\sum_{i}n_{i}(1+(q-1)v_{i}^{2})<1+(q-1)v^{2}$ whenever $r\geqslant 2$. It now follows by taking the leading term of (A.5) that $\#{\operatorname{M}\nolimits}(v)(\mathbb{F}_{l})\sim l^{(2q-1)v^{2}+1}$ as wanted.

Remark A.1*.*

One may alternatively use Beilinson and Drinfeld’s notion of a very good stack, see [1, sec. 1.1.1.] and show that $Rep({\operatorname{k}\nolimits}Q,v)/G(v)$ is very good for any $v$ and any $Q=\mathcal{Q}(i,q)$ with $q>1$.

A.3. Proof of Proposition 3.11

A simple representation $S$ of $\tilde{\Pi}$ will be called rigid if $\operatorname{Ext}\nolimits^{1}(S,S)=\{0\}$. Since

[TABLE]

a simple representation $S$ is rigid if and only if $(\dim S,\dim S)=2$. Alternatively, a representation $S$ is rigid if and only if every other simple representation of the same dimension is isomorphic to $S$. On the other hand, if $S$ is non rigid then there are infinitely many non-isomorphic simple representations of the same dimension (recall that we work over the field ${\operatorname{k}\nolimits}=\mathbb{C}$), and we have $(\dim S,\dim S)\leqslant 0$ (note that $(u,u^{\prime})\in 2\mathbb{Z}$ for any dimension vectors $u,u^{\prime}$). For any dimension type $d$ there exists a unique representation type $\tau=\tau(d)$ of dimension type $d$ in which all non rigid simple representations occur with multiplicity one. The stratum $\mathfrak{M}_{0}(\tau)$ is open and dense in $\mathfrak{M}_{0}(\!(d)\!)$. Since $\pi$ is proper, the set ${\operatorname{Im}\nolimits}(\pi)$ is closed and the proposition will be proved once we show that for any $d$ we have $\mathfrak{M}_{0}(\tau)\subset{\operatorname{Im}\nolimits}(\pi)$ as soon as there exists $\tau^{\prime}$ of dimension type $d$ such that $\mathfrak{M}_{0}(\tau^{\prime})\subset{\operatorname{Im}\nolimits}(\pi)$. By definition, the existence of such a $\tau^{\prime}$ means that there exists a stable representation $M\in{\operatorname{M}\nolimits}_{s}(v,w)$ of semisimple type $\tau^{\prime}$. In other words, there exists a collection of simple $\tilde{\Pi}$-modules $S_{1},S_{2},\ldots,S_{r}$ such that $\#\{i\;;\;\dim S_{i}=u\}=d_{u}$ for any $u$, and an iterated extension

[TABLE]

which is stable. Note that we automatically have $\dim S_{1}\in\mathbb{N}^{I}\times\{w\}$ and $\dim S_{i}\in\mathbb{N}^{I}\times\{0\}$ for any $i>1$. Our aim is then to prove, under the above hypothesis, the existence of a similar iterated extension $N_{1}\subset N_{2}\subset\cdots\subset N_{r}$ which is stable, for which $T_{i}:=N_{i}/N_{i-1}$ is simple, satisfies $\dim T_{i}=\dim S_{i}$ and in which all non-rigid simple factors $T_{i}$’s are non isomorphic. We will use the following simple observation, which might be of independent interest.

Lemma A.2**.**

Let $N_{1}\subset N_{2}\subset\cdots\subset N_{r}$ be a filtration with simple factors $T_{i}:=N_{i}/N_{i-1}$, such that $\dim T_{1}\not\in\mathbb{N}^{I}\times\{0\}$ while $\dim T_{i}\in\mathbb{N}^{I}\times\{0\}$ for any $i>1$. Let $\xi_{i}\in\operatorname{Ext}\nolimits^{1}_{\tilde{\Pi}}(T_{i},N_{i-1})$ be the associated elements. Then $N_{r}$ is stable if and only if $\xi_{i}\neq 0$ for all $i$.

Proof.

The module $N_{r}$ is stable if and only if it contains no submodule whose dimension vector belongs to $\mathbb{N}^{I}\times\{0\}$, and hence if and only if $\operatorname{Hom}\nolimits_{\tilde{\Pi}}(T_{i},N_{j})=\{0\}$ for all $i,j$. This in turn is equivalent to the condition that $\operatorname{Hom}\nolimits_{\tilde{\Pi}}(T_{i},N_{i})=\{0\}$ for all $i$. From the exact sequence

[TABLE]

and the fact that $\xi_{i}=\partial(\operatorname{id}\nolimits)$ we see that $N_{i}$ is stable if and only if $\xi_{i}\neq 0$ and $N_{i-1}$ is stable. The claim follows. ∎

Let us fix some $w$ and prove the Proposition by induction on $v$. Let us also fix a dimension type $d$ such that $\sum_{u}d_{u}=(v,w)$. Let us assume that Proposition 3.11 is proved for $(w,v)$ for any dimension vector $v^{\prime}<v$. Let $\tau^{\prime}$ be a representation type of dimension type $\underline{d}$, and of dimension $(v,w)$. Let $M_{1}\subset M_{2}\subset\cdots\subset M_{r}$ be a filtration as above of representation type $\tau^{\prime}$, and set $S_{i}=M_{i}/M_{i-1}$. If $r=1$ there is nothing to prove, so we assume that $r>1$. There are two cases to consider :

$\mathbf{Case\ 1:}$ $S_{r}$ is not rigid. By our induction hypothesis, there exists a filtration $N_{1}\subset N_{2}\subset\cdots\subset N_{r-1}$ such that $T_{i}:=N_{i}/N_{i-1}$ is a simple $\tilde{\Pi}$-module of dimension $\dim S_{i}$, $N_{r-1}$ is stable and all non-rigid simple factors $T_{j}$ are non isomorphic. Let us choose a simple object $T_{r}$ of the same dimension as $S_{i}$, but non-isomorphic to all the $T_{j}$ for $j<i$. This is possible since there are infinitely many non-isomorphic simples of dimension $\dim S_{r}$. We have to show that $\operatorname{Ext}\nolimits^{1}(T_{r},N_{r-1})\neq\{0\}$. Let $i_{0}\leqslant i$ be the smallest index for which $\dim T_{i_{0}}=\dim S_{r}$. Since $(\dim T_{r},\dim T_{r})\leqslant 0$ and $(\dim T_{r},\dim S)=-\dim\operatorname{Ext}\nolimits^{1}(T_{r},S)\leqslant 0$ for any $S$ which is not isomorphic to $T_{r}$, we have

[TABLE]

therefore $(\dim T_{r},\dim N_{i-1})<0$ and hence $\operatorname{Ext}\nolimits^{1}(T_{r},N_{i-1})\neq\{0\}$. We choose any nonzero $\xi\in\operatorname{Ext}\nolimits^{1}(T_{r},N_{i-1})$ to define $N_{r}$. By Lemma A.2 the representation $N_{r}$ is stable, and by construction it is of representation type $\tau$.

$\mathbf{Case\ 2:}$ $S_{r}$ is rigid. Set $n=\dim\operatorname{Hom}\nolimits(M_{r},S_{r})$. We claim that there exists a filtration $M^{\prime}_{1}\subset M^{\prime}_{2}\subset\cdots\subset M^{\prime}_{r-1}\subset M^{\prime}_{r}=M_{r}$ of $M_{r}$ for which $M^{\prime}_{r-j}/M^{\prime}_{r-j-1}\simeq S_{r}$ for $j=0,\ldots,n-1$ and

[TABLE]

Indeed, let $M$ be any $\tilde{\Pi}$-module satisfying $\operatorname{Hom}\nolimits(S_{r},M)=\{0\}$ and $\operatorname{Hom}\nolimits(M,S_{r})\neq\{0\}$. Applying $\operatorname{Hom}\nolimits(\cdot,S_{r})$ to any exact sequence \textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0} yields a short exact sequence

[TABLE]

from which we get $\dim\operatorname{Hom}\nolimits(K,S_{r})=\dim\operatorname{Hom}\nolimits(M,S_{r})-1$. Similarly, applying $\operatorname{Hom}\nolimits(S_{r},\bullet)$ we get the short exact sequence

[TABLE]

from which we deduce that $\dim\operatorname{Ext}\nolimits^{1}(S_{r},K)=\dim\operatorname{Ext}\nolimits^{1}(S_{r},M)+1.$ Iterating this process $n$ times starting with $M$ we get a filtration of $M_{r}$ with the required properties. The representation type $\nu^{\prime}$ of $M^{\prime}_{r-n}$ is obtained from $\tau^{\prime}$ by decreasing the multiplicity of $S_{r}$ by $n$. By the induction hypothesis, there exists a stable representation $N_{r-n}$ of type $\nu$, obtained from $\tau$ by decreasing the multiplicity of $S_{k}$ by $n$. Now, we have

[TABLE]

hence $\dim\operatorname{Ext}\nolimits^{1}(S_{r},N_{r-n})\geqslant n$. Reversing the process above, we may now consider $n$ successive non split extensions of $N_{r-n}$ by $S_{r}$. The resulting module $N_{r}$ is stable by Lemma A.2 and is of type $\tau$ as wanted.

The induction step is completed, and Proposition 3.11 is proved.

A.4. Kirwan surjectivity for local quiver varieties

In this appendix, we provide an independent proof of the Kirwan surjectivity for the fixed point quiver varieties used in this paper.

A.4.1. Local quiver varieties

We define a new quiver $Q^{\#}=(I^{\#},\Omega^{\#})$ defined as follows :

• •

$I^{\#}=\{(i,l)\}=I\times\mathbb{Z}$,

• •

$\Omega^{\#}=\{(h,l):(h^{\prime},l)\to(h^{\prime\prime},l-1)\,;\,h\in\Omega\}\sqcup\{(h,l):(h^{\prime},l)\to(h^{\prime\prime},l)\,;\,h\in\Omega^{*}\}$.

For each dimension vectors $v$, $w$ of $Q^{\#}$ we define

[TABLE]

where we abbreviate $V={\operatorname{k}\nolimits}^{v}$, $W={\operatorname{k}\nolimits}^{w}$ and $(V[-1])_{i,l}=V_{i,l-1}$. The group $G(v)\times G(w)$ acts in the obvious way on $\operatorname{R}\nolimits(v,w)^{\#}$. The map

[TABLE]

is $G(v)\times G(w)$-equivariant. Let

[TABLE]

The affine variety $\mathfrak{M}_{0}(v,w)^{\#}$ carries an action of the group $T\times G(w)$. Let $s$ be the character of $G(v)$ defined by

[TABLE]

The local quiver variety associated with $Q$, $v$, $w$ is the quasiprojective variety given by

[TABLE]

We may abbreviate $\mathfrak{M}(v,w)^{\#}=\mathfrak{M}_{s}(v,w)^{\#}$. There is a natural projective morphism

[TABLE]

A representation $(\bar{x},\bar{a})$ in ${\operatorname{M}\nolimits}(v,w)^{\#}$ is said to be semistable if and only if it does not admit any nonzero subrepresentation whose dimension vector belongs to $\mathbb{N}^{I^{\#}}\times\{0\}$. Let

[TABLE]

be the open subset consisting of the semistable points. This set is preserved by the $T\times G(v)\times G(w)$-action on $\operatorname{R}\nolimits(v,w)^{\#}$. The same arguments as in [30], [33] implies that $\mathfrak{M}(v,w)^{\#}$ is the geometric quotient of ${\operatorname{M}\nolimits}_{s}(v,w)^{\#}$ by $G(v)$ and is smooth quasi-projective with an action of $T\times G(w)$. If $Q^{\#}$ (or equivalently $Q$) has no oriented cycles then $\mathfrak{M}_{0}(v,w)^{\#}$ is reduced to a single point and thus $\mathfrak{M}(v,w)^{\#}$ is projective. However, this is not the case in general.

The varieties $\mathfrak{M}(v,w)^{\#}$ arise as a union of fixed point connected components for some suitable $\mathbb{G}_{m}$-actions on the Nakajima quiver varieties attached to the quiver $Q$, see §3.7.4. More precisely, fix dimension vectors $u,$ $y$ in $\mathbb{N}^{I}$ and consider the $\circ$-action on $\mathfrak{M}(u,y)$ in (4.3). We have

[TABLE]

where the union ranges over the elements $v,$ $w$ in $\mathbb{N}^{I^{\#}}$ such that

[TABLE]

This decomposition is compatible with the $T$-action.

A.4.2. The homology of local quiver varieties

For any vertex $(i,l)\in I^{\#}$ the universal and the tautological $T\times G(w)$-equivariant vector bundles over $\mathfrak{M}(v,w)^{\#}$ are denoted by

[TABLE]

They are defined as in (5.29). Consider the subspace

[TABLE]

generated by the action of the Chern character components ${\operatorname{ch}\nolimits}_{k}(\mathcal{V}_{i,l})$ and ${\operatorname{ch}\nolimits}_{k}(\mathcal{W}_{i,l})$, for all possible $i,l,k$, on the fundamental class $[\mathfrak{M}(v,w)^{\#}]$.

Theorem A.3**.**

For any $v,$ $w$ we have $M_{v,w}^{\#}=M_{*}^{T\times G(w)}(\mathfrak{M}(v,w)^{\#})$.

When $Q$ has no oriented cycle then $Q^{\#}$ has no oriented cycle either, hence $\mathfrak{M}(v,w)^{\#}$ is projective. In this case, the theorem follows from Nakajima’s resolution of the diagonal as in [32, §7.3]. We will prove the theorem by constructing a suitable compactification of the local quiver variety. A very similar construction has independently appeared in McGerty and Nevins’ recent proof of Kirwan surjectivity for (non-graded, non-local) quiver varieties, see [25].

Corollary A.4**.**

For any $v,w\in\mathbb{N}^{I\#}$, the variety $\mathfrak{M}(v,w)^{\#}$ is connected.

A.4.3. The quivers $Q^{\diamond}$ and $\bar{Q}^{\diamond}$

Consider a new quiver $Q^{\diamond}=(I^{\diamond},\Omega^{\diamond})$ defined as follows :

• •

$I^{\diamond}=\{i_{1}\,;\,i\in I\}\sqcup\{i_{2}\,;\,i\in I\}\simeq I\times\{1,2\},$

• •

$\Omega^{\diamond}=\{h:(h^{\prime})_{1}\longrightarrow(h^{\prime\prime})_{2}\,;\,h\in\Omega\}\sqcup\{i:i_{1}\longrightarrow i_{2}\,;\,i\in I\}.$

Example. We have

[TABLE]

The quiver $Q^{\diamond}$ is strongly bipartite : all arrows go from a vertex in $I\times\{1\}$ to one in $I\times\{2\}$. In particular, it contains no oriented cycles.

The moduli stacks of representations of $Q$ and $Q^{\diamond}$ are related in the following fashion. For each $v\in\mathbb{N}^{I}$ we define $v^{\diamond}\in\mathbb{N}^{{}I^{\diamond}}$ such that $v^{\diamond}_{i_{1}}=v^{\diamond}_{i_{2}}=v_{i}$. Let

[TABLE]

be the open subset consisting of the representations $x^{\diamond}$ such that $x^{\diamond}_{i}$ is invertible for all $i$. Then, the assignment $x^{\diamond}\mapsto(x\,,\,g)$ where

[TABLE]

gives an isomorphism

[TABLE]

hence an isomorphism of Artin stacks

[TABLE]

This allows us to view the stack $Rep({\operatorname{k}\nolimits}Q,v)\,/\,G(v)$ as an open substack of $Rep({\operatorname{k}\nolimits}\,Q^{\diamond},{}v^{\diamond})\,/\,G(v^{\diamond})$.

We will now perform a similar construction for the double quiver. Let $\bar{Q}^{\diamond}$ be the double quiver of $Q^{\diamond}$. For each $v,$ $w$ in $\mathbb{N}^{I}$ we define $v^{\diamond}$, $w^{\diamond}$ in $\mathbb{N}^{I^{\diamond}}$ such that

[TABLE]

The group $G({}v^{\diamond})$ acts in a Hamiltonian fashion on

[TABLE]

and we denote by $\mu_{\diamond}:\operatorname{R}\nolimits({}v^{\diamond},{}w^{\diamond})\to\mathfrak{g}(v^{\diamond})$ the associated moment map. Let $\operatorname{R}\nolimits({}v^{\diamond},{}w^{\diamond})^{\heartsuit}$ be the open subset of $\operatorname{R}\nolimits({}v^{\diamond},{}w^{\diamond})$ consisting of pairs $(\bar{x}^{\diamond},\bar{a}^{\diamond})$ such that $x^{\diamond}_{i}$ is an isomorphism for all $i$. We also put

[TABLE]

Now, let $\operatorname{R}\nolimits(v,w)$ and ${\operatorname{M}\nolimits}(v,w)$ be as in §3.7.1. Consider the map

[TABLE]

defined, for all $h\in\Omega$ and $i\in I$, by

[TABLE]

A direct computation gives

Lemma A.5**.**

Given $v,w$ let $v^{\diamond},w^{\diamond}$ be as in (A.10). Then, the map $\phi$ gives an isomorphism

[TABLE]

∎

We will next consider appropriate stability conditions on $\operatorname{R}\nolimits(v^{\diamond},w^{\diamond})$. Given a tuple of integers $(\theta_{i})$, let $\theta$ be the character of $G(v^{\diamond})$ defined by $\theta(g)=\prod_{i\in I^{\diamond}}\det(g_{i})^{-\theta_{i}}$. We set

[TABLE]

which is a quasi-projective $T\times G(w^{\diamond})$-variety equipped with a projective morphism

[TABLE]

We can describe the set of closed points of $\mathfrak{M}_{\theta}(v^{\diamond},w^{\diamond})$ explicitly. Set

[TABLE]

We say that an element $(\bar{x}^{\diamond}\,,\,\bar{a}^{\diamond})$ is $\theta$-semistable if the following conditions are satisfied :

• •

for any $(\bar{x}^{\diamond},\bar{a}^{\diamond})$-stable $I^{\diamond}$-graded subspace $U\subseteq V^{\diamond}$ we have

[TABLE]

• •

for any $I^{\diamond}$-graded subspace $U\subseteq V^{\diamond}$ such that $U\oplus W^{\diamond}$ is $(\bar{x}^{\diamond},\bar{a}^{\diamond})$-stable we have

[TABLE]

We will further say that $(\bar{x}^{\diamond},\bar{a}^{\diamond})$ is stable if the above inequalities are strict for proper subspaces $U$. Let us denote by $\operatorname{R}\nolimits_{\theta}(v^{\diamond},w^{\diamond})$ the open subset of $\operatorname{R}\nolimits(v^{\diamond},w^{\diamond})$ consisting of $\theta$-semistable elements and by ${\operatorname{M}\nolimits}_{\theta}(v^{\diamond},w^{\diamond})$ its intersection with ${\operatorname{M}\nolimits}(v^{\diamond},w^{\diamond})$. Then [34, prop. 2.9] yields

[TABLE]

We also set

[TABLE]

We say that $\theta$ is generic if neither equations

[TABLE]

have integer solutions $(u_{i})$ satisfying $0\leqslant u_{i}\leqslant v_{i}^{\diamond}$ other than the trivial solutions

[TABLE]

If $\theta$ is generic then any semistable pair $(\bar{x}^{\diamond},\bar{a}^{\diamond})$ is stable and in that case the map

[TABLE]

is a $G(v^{\diamond})$-torsor, hence the variety $\mathfrak{M}_{\theta}(v^{\diamond},w^{\diamond})$ is smooth.

Lemma A.6**.**

Given $v,w$ let $v^{\diamond},w^{\diamond}$ be as in (A.10). Let $\theta\in\mathbb{Z}^{I^{\diamond}}$ be generic and such that

[TABLE]

Then, the map $\phi$ restricts to an isomorphism

[TABLE]

Proof.

Fix a tuple $z=(\bar{x}^{\diamond},\bar{a}^{\diamond})$ in ${\operatorname{M}\nolimits}(v^{\diamond},w^{\diamond})^{\heartsuit}$. We will prove that

[TABLE]

Let $U$ be an $I^{\diamond}$-graded subspace of $V^{\diamond}$. Let us first assume that $U$ is $z$-stable and non-zero. Since $x^{\diamond}_{i}$ is invertible, we have

[TABLE]

If there exists $j\in I$ such that $\dim(U_{j_{1}})<\dim(U_{j_{2}}),$ then

[TABLE]

because $\theta_{j_{2}}\ll 0$. Thus $U$ is not destabilizing. On the other hand, if $x_{i}^{\diamond}$ is an isomorphism $U_{i_{1}}\to U_{i_{2}}$ for all $i$, then we have

[TABLE]

hence $U$ is destabilizing.

Assume now that $U\oplus W^{\diamond}$ is $z$-stable and proper. As above, we have $\dim(U_{i_{1}})\leqslant\dim(U_{i_{2}})$ for all $i$. Thus

[TABLE]

for otherwise $\dim(U_{i_{1}})=\dim(U_{i_{2}})=v_{i}$ for all $i$ and $U$ is not a proper subspace of $V^{\diamond}$. Therefore $U$ is not destabilizing.

By the above, we conclude that $z$ is $\theta$-semistable if and only if there does not exist a nonzero $z$-stable $U\subset V^{\diamond}$ stable under all the maps $x_{i}^{-1}$. This is easily seen to be equivalent to the condition that $\phi^{-1}(z)$ belongs to ${\operatorname{M}\nolimits}_{s}(v,w)\times T\times G(v)$. ∎

Corollary A.7**.**

Given $v,w$ let $v^{\diamond},w^{\diamond}$ be as in (A.10). Let $\theta$ be a generic character as in (A.12). Then, we have an isomorphism $\psi\,:\,\mathfrak{M}(v,w)\,{\to}\,\mathfrak{M}_{\theta}(v^{\diamond},w^{\diamond})^{\heartsuit}$ and an open immersion

[TABLE]

∎

A.4.4. The compactification of local quiver varieties

From now on we fix $u,y,u^{\diamond},y^{\diamond}$ and $\theta$ as above. The existence of such a character $\theta$ is clear. For simplicity, we abbreviate

[TABLE]

It is straightforward to check from (A.11) that the $\bullet$-actions on $\mathfrak{M}$ and $\mathfrak{M}^{\diamond}$ given by

[TABLE]

are compatible with the open embedding $i$ in Corollary A.7. Applying (A.7) to the quivers $Q$ and $Q^{\diamond}$, we get the decompositions into smooth disjoint subvarieties

[TABLE]

where $v,$ $w$, $v^{\diamond}$, $w^{\diamond}$ run over the sets of dimension vectors in $\mathbb{N}^{I^{\#}}$ and $\mathbb{N}^{I^{\diamond,\#}}$ such that

[TABLE]

The variety $\mathfrak{M}(v^{\diamond},w^{\diamond})^{\#}$ is realized in the same way as $\mathfrak{M}(v,w)^{\#}$ by replacing the quiver $Q$ by the quiver $Q^{\diamond}$ throughout. Taking the fixed points under the $\bullet$-actions, the map $i$ yields an open immersion

[TABLE]

where $v^{\diamond},w^{\diamond}$ are defined by

[TABLE]

Since the quiver $Q^{\diamond}$ has no oriented cycle, each $\mathfrak{M}^{\diamond,\#}$ is projective and we have obtained in this way the desired compactification of $\mathfrak{M}^{\#}$.

A.4.5. Proof of Theorem A.3

Now, let us consider the resolution of the diagonal. The smooth variety $\mathfrak{M}^{\diamond}\times\mathfrak{M}^{\diamond}$ carries $T\times G(w^{\diamond})\times\mathbb{G}_{m}$-equivariant universal and tautological bundles given, for all $i\in I^{\diamond}$, by

[TABLE]

Note that $\mathcal{W}_{i}^{\diamond}=\{0\}$ for $i\in I_{2}$ and that $\mathcal{W}_{i}^{\diamond,1}=\mathcal{W}_{i}^{\diamond,2}$ is trivial as a vector bundle. We will simply denote this bundle by $\mathcal{W}_{i}^{\diamond}$. We set also $\mathcal{V}^{\diamond}=\bigoplus_{i\in I^{\diamond}}\mathcal{V}_{i}^{\diamond}$ and $\mathcal{W}^{\diamond}=\bigoplus_{i\in I^{\diamond}}\mathcal{W}_{i}^{\diamond}$. Consider the complex of equivariant bundles on $\mathfrak{M}^{\diamond}\times\mathfrak{M}^{\diamond}$ given by

[TABLE]

where

• •

$C=\bigoplus_{h\in\bar{\Omega}^{\diamond}}\operatorname{Hom}\nolimits(\mathcal{V}_{h^{\prime}}^{\diamond,1},\mathcal{V}_{h^{\prime\prime}}^{\diamond,2})[\varepsilon_{h}]\;\oplus\;\operatorname{Hom}\nolimits_{I^{\diamond}}(\mathcal{V}^{\diamond,1},\mathcal{W}^{\diamond})[1]\;\oplus\;\operatorname{Hom}\nolimits_{I^{\diamond}}(\mathcal{W}^{\diamond},\mathcal{V}^{\diamond,2}),$

• •

$\varepsilon_{h}=1$ if $h\in\Omega^{\diamond,*}$ and [math] otherwise,

• •

$[1]$ denotes a grading shift with respect to the $\mathbb{G}_{m}$-action,

• •

the maps $\mathbf{a},\mathbf{b}$ are defined as

[TABLE]

[TABLE]

Lemma A.8**.**

Over the open subset $\mathfrak{M}^{\diamond,\heartsuit}\times\mathfrak{M}^{\diamond}$ the map $\mathbf{a}$ is injective and the map $\mathbf{b}$ is surjective.

Proof.

Fix tuples $z_{1}\in\mathfrak{M}^{\diamond,\heartsuit}$ and $z_{2}\in\mathfrak{M}^{\diamond}$. Set $z_{1}=(\bar{x}_{1}^{\diamond},\bar{a}_{1}^{\diamond})$ and $z_{2}=(\bar{x}_{2}^{\diamond},\bar{a}_{2}^{\diamond})$. We will first prove that the map $\mathbf{a}$ is injective over the point $(z_{1},z_{2})$. Let $(\xi_{i})\in{\operatorname{Ker}\nolimits}(\mathbf{a})$ and put

[TABLE]

Since $(\xi_{i})\in{\operatorname{Ker}\nolimits}(\mathbf{a})$, the space $K$ is a subrepresentation of $z_{1}$ while $I$ is a subrepresentation of $z_{2}$. Since $z_{1}\in\mathfrak{M}^{\diamond,\heartsuit}$ we have $\dim(K_{i_{1}})\leqslant\dim(K_{i_{2}})$ and hence $\dim(J_{i_{1}})\geqslant\dim(J_{i_{2}})$ for all $i\in I^{\diamond}$. We deduce that

[TABLE]

with a strict inequality if there exists $j$ such that $J_{j_{2}}\neq\{0\}$ or $\dim(J_{j_{1}})>\dim(J_{j_{2}})$, i.e., if $J\neq\{0\}$. By $\theta$-semistability of $z_{2}$ we conclude that $J=\{0\}$ and that $\mathbf{a}$ is injective.

To prove that $\mathbf{b}$ is surjective, we dualize the previous argument. Using the trace pairing, we may view the transpose $\mathbf{b}^{*}$ of $\mathbf{b}$ as a map

[TABLE]

where

[TABLE]

Let $(\xi_{i})\in{\operatorname{Ker}\nolimits}(\mathbf{b}^{*})$ and set

[TABLE]

Because $(\xi_{i})\in{\operatorname{Ker}\nolimits}(\mathbf{b}^{*})$, the spaces $J$ and $K$ are respectively subrepresentations of $z_{1}$ and $z_{2}$. Since $z_{1}\in\mathfrak{M}^{\diamond,\heartsuit}$ it follows that $\dim(J_{i_{1}})\leqslant\dim(J_{i_{2}}),$ hence $\dim(K_{i_{1}})\geqslant\dim(K_{i_{2}})$ for all $i$. If $\dim(K_{j_{1}})>\dim(K_{j_{2}})$ for some $j$, then we have

[TABLE]

which would contradict the semistablity of $z_{2}$. Thus $\dim(J_{i_{1}})=\dim(J_{i_{2}})$ for all $i$. But then

[TABLE]

with strict inequality as soon as $J_{i}\neq\{0\}$ for some $i$. By the semistability of $z_{1}$, this forces $J_{i}=\{0\}$ for all $i$, i.e., we have $(\xi_{i})=0$ as wanted. Lemma A.8 is proved. ∎

By Lemma A.8 the restriction of the complex (A.14) to $\mathfrak{M}^{\diamond,\heartsuit}\times\mathfrak{M}^{\diamond}$ is quasi-isomorphic to an equivariant vector bundle $\mathcal{E}={\operatorname{Ker}\nolimits}(\mathbf{b})/{\operatorname{Im}\nolimits}(\mathbf{a})$. Moreover, the bundle $\mathcal{E}$ carries a section which vanishes precisely on the diagonal $\Delta\mathfrak{M}^{\diamond,\heartsuit}$ in

[TABLE]

Observe that $\Delta\mathfrak{M}^{\diamond,\heartsuit}$ is a closed subset of $\mathfrak{M}^{\diamond,\heartsuit}\times\mathfrak{M}^{\diamond}$. Indeed, see [30], the map

[TABLE]

defines a section of $\mathcal{E}$ which vanishes precisely over the set of pairs $(z_{1},z_{2})$ such that $\operatorname{Hom}\nolimits(z_{1},z_{2})\neq\{0\}$ as ${\operatorname{k}\nolimits}\bar{Q}^{\diamond}$-modules. By the semistability condition, both $z_{1}$ and $z_{2}$ are simple so that $\operatorname{Hom}\nolimits(z_{1},z_{2})\neq\{0\}$ if and only if $z_{1}=z_{2}$. Thus the zero set of $s$ is $\Delta\mathfrak{M}^{\diamond,\heartsuit}.$

As a consequence, we have the following equality in ${A}^{T\times G(w^{\diamond})\times\mathbb{G}_{m}}_{*}(\mathfrak{M}^{\diamond,\heartsuit}\times\mathfrak{M}^{\diamond})$ :

[TABLE]

Let us now fix dimension vectors $v,$ $w$ for $Q^{\#}$ and denote by $v^{\diamond},$ $w^{\diamond}$ the associated dimension vectors for $Q^{\diamond}$. We abbreviate $\mathfrak{M}^{\#}=\mathfrak{M}(v,w)^{\#}$ and $\mathfrak{M}^{\diamond,\#}=\mathfrak{M}(v^{\diamond},w^{\diamond})^{\#}$ as before. Let $\mathcal{E}^{\#}$ be the restriction of $\mathcal{E}$ to $\mathfrak{M}^{\#}\times\mathfrak{M}^{\diamond,\#}$ under the map (A.13). We have

[TABLE]

Consider the following diagram

[TABLE]

in which $\delta~{}:\mathfrak{M}^{\#}\to\mathfrak{M}^{\#}\times\mathfrak{M}^{\#}$ is the diagonal embeddings and $p_{1}$ is the projection onto the first factor. The maps $\delta,$ $\delta^{\prime}$ and $p_{1}$ are proper. Let $\alpha\in{A}^{T\times G(w)}_{*}(\mathfrak{M}^{\#})$ and choose $\alpha^{\diamond}\in{A}^{T\times G(w^{\diamond})}_{*}(\mathfrak{M}^{\diamond,\#})$ such that $i^{*}(\alpha^{\diamond})=\alpha$.

We have, on the one hand

[TABLE]

On the other hand, using the Giambelli formula we may write

[TABLE]

for some classes $a_{k}$ belonging to the subring of ${A}^{*}_{T\times G(w)}(\mathfrak{M}^{\#})$ generated by the classes of the tautological bundles $\mathcal{V}_{i,l}$ and $\mathcal{W}_{i,l}$ as $(i,l)$ runs over $I^{\#}$, and some classes $b_{k}$ belonging to the subring of ${A}^{*}_{T\times G(w^{\diamond})}(\mathfrak{M}^{\diamond,\#})$ generated by the classes of the tautological bundles $\mathcal{V}^{\diamond}_{i,l}$ and $\mathcal{W}^{\diamond}_{i,l}$ as $(l\delta_{i})$ runs over $I^{\diamond,\#}$. We have isomorphisms $i^{*}(\mathcal{V}^{\diamond}_{i_{k},l})\simeq\mathcal{V}_{i,l}$ and $\mathcal{W}^{\diamond}_{i,l}\simeq\mathcal{W}_{i,l}$. Further, we have

[TABLE]

where $p:\mathfrak{M}^{\diamond,\#}\to\bullet$ is the map to a point.

Combining (A.15) and (A.16), we deduce that

[TABLE]

The statement for the equivariant homology groups can be deduced from that for the Chow homology groups. Indeed, by the same argument as in Proposition 4.3 (c), since there is a contracting $\mathbb{G}_{m}$-action on $\mathfrak{M}^{\#}$ whose fixed point subvariety is a projective graded quiver variety, the cycle map

[TABLE]

is surjective. This map preserves the tautological subrings. Hence, we have

[TABLE]

Theorem A.3 is proved.

A.5. Non-injectivity of the shuffle realization map for small tori

In this section we provide an explicit instance in which the map to the shuffle algebra

[TABLE]

is not injective when the assumption $\theta,\theta^{*}\subset T$ is not verified. We will consider the case where $Q$ is the Jordan quiver and $v=2$. For any torus $T$ we have, by Theorem A,

[TABLE]

Note that the dimension of $\Lambda(2)$ is $4$. In addition, by Theorem 5.18,

[TABLE]

We will consider the image of the map $i_{*}$ and compare its graded dimension with that of $H^{T\times G(2)}_{\star}(\Lambda(2))$ for various choices of $T$. For this, we first compute the image of $i_{*}$ for the torus $T=\theta\times\theta^{*}$ and then specialize. Let us write $H^{*}_{T}=\mathbb{Q}[t,t_{*}]$ and $H^{*}_{G(2)}=\mathbb{Q}[z_{1}+z_{2},z_{1}z_{2}]$. Since $\Lambda_{(2)}$ is the zero section of the the map $\operatorname{R}\nolimits(2)\to Rep({\operatorname{k}\nolimits}Q^{*},2)$ we have

[TABLE]

while a direct computation using the shuffle multiplication, see (5.23), yields

[TABLE]

[TABLE]

[TABLE]

We describe the image of $i_{*}$ in small cohomological degree. We have

[TABLE]

[TABLE]

Let $T^{\prime}\subset T$ be a one dimensional subtorus. Let us denote by $t,$ $t_{*}$ the images of $t,$ $t_{*}$ under the restriction map $H^{*}_{T\times G(2)}\to H^{*}_{T^{\prime}\times G(2)}$. Then it is easy to check that the span of the six elements $\{tx_{0},t_{*}x_{0},tx_{0}^{\prime},t_{*}x_{0}^{\prime},x_{1},x_{1}\}$ is of dimension $4$ over $\mathbb{Q}$ if and only if $tt_{*}(t+t_{*})\neq 0$ and of dimension $3$ otherwise (observe that $x^{\prime}_{1}-x_{1}=2tt_{*}(t+t_{*})$). In particular, the map $i_{*}$ is not injective when $T=\theta$ or $T=\theta^{*}$, or $T=\theta-\theta^{*}$.

Index of Notations

1. 2.1.

$H_{*}$, ${A}_{*}$, $M_{*}$, $\varepsilon=1$ or $2$, $H_{*}^{G}$, ${A}_{*}^{G}$, $M_{*}^{G}$, 2. 2.2.

$f^{\heartsuit}$, 3. 2.3.

$(f,f^{\prime})^{!}$, ${\operatorname{cl}\nolimits}:{A}_{*}^{G}(X)\to H^{G}_{2*}(X)$, ${\operatorname{ch}\nolimits}=\sum_{l\geqslant 0}{\operatorname{ch}\nolimits}_{l}$, 4. 3.1.

$Q,$ $Q^{*}$, $\bar{Q}$, $\Omega$, $\Omega^{*}$, $\bar{\Omega}$, $\varepsilon(h)$, $h^{\prime},$ $h^{\prime\prime}$, $q_{ij},$ $\bar{q}_{ij},$ $q_{i},$ $\Omega_{ij},$ $\bar{\Omega}_{ij}$, $\langle\bullet\,,\,\bullet\rangle,$ $(\bullet\,,\,\bullet)$, $d_{v}$, $\delta_{i}$, ${\operatorname{k}\nolimits}Q$, $\bar{x}=(x,x^{*})$, $\operatorname{R}\nolimits(v)$, 5. 3.2.

$G(v)$, $\mathfrak{g}(v)$, $\mu:\operatorname{R}\nolimits(v)\to\mathfrak{g}(v)$, ${\operatorname{M}\nolimits}(v),$ $\Pi$, $\langle\bullet\,,\,\bullet\rangle_{\Pi}$, 6. 3.3.

$\theta(z),$ $\theta^{*}(z),$ $T=T_{\operatorname{sp}}\times T_{\operatorname{dil}}$, 7. 3.4.

$\Lambda_{W},$ $\Lambda_{\nu}$, $\Lambda^{\flat}(v)$, $\flat=0$ or $1$, 8. 3.5.

$\Lambda^{\circ}_{W},$ $\Lambda^{\circ}_{\nu}$ ,${}^{\circ}\!\Lambda_{F},$ ${}^{\circ}\!\Lambda_{\nu}$, $\Lambda_{(v)}$, 9. 3.6.

$\operatorname{J}\nolimits(i,q)$, $\mu\,{\trianglelefteqslant}\,\nu$, 10. 3.7.

$\operatorname{R}\nolimits(v,w),$ $\operatorname{Hom}\nolimits_{I}(V,W)$, $G(w)$, $\bar{x},$ $\bar{a}$, ${\operatorname{M}\nolimits}(v,w)$, $\mathfrak{M}_{0}(v,w),$ $\mathfrak{M}(v,w)$, $\pi_{\theta}:\mathfrak{M}_{\theta}(v,w)\to\mathfrak{M}_{0}(v,w)$, $\rho_{\theta}$, $R_{\theta}(v,w),$ ${\operatorname{M}\nolimits}_{\theta}(v,w),$ ${\operatorname{M}\nolimits}_{s}(v,w)$, $d_{v,w}$, $\mathfrak{M}(v,w)_{\mathbb{A}^{1}},$ $\mathfrak{M}_{0}(v,w)_{\mathbb{A}^{1}},$ $\tilde{Q}=(\tilde{I},$ $\tilde{\Omega}),$ $\tilde{v},$ $\tau$, $RT(v,w)$, $\mathfrak{M}_{0}(\tau)$, $\kappa_{v,w}$, ${\operatorname{M}\nolimits}(\tau),$ ${\operatorname{M}\nolimits}_{s}(\tau)$, $\mathfrak{M}(\tau)$, $\mathfrak{L}(v,w)$, ${\operatorname{M}\nolimits}[\rho],$ $\operatorname{L}\nolimits[\rho],$ $\mathfrak{M}[\rho],$ $\mathfrak{L}[\rho]$, 11. 3.8.

$\operatorname{H}\nolimits[v_{1},v_{2}\,;\,w],$ $\mathfrak{h}[v_{1},v_{2}\,;\,w],$ $\operatorname{H}\nolimits[v_{1},S\,;\,w],$ $\mathfrak{h}[v_{1},S\,;\,w],$ $\operatorname{H}\nolimits[v_{1},S],$ $\mathfrak{h}[v_{1},S]$, 12. 4.1.

$L^{\flat}_{s}(v,w),$ $\mathfrak{L}^{\flat}(v,w)$, $\circ:\mathbb{G}_{m}\times\mathfrak{M}(v,w)\to\mathfrak{M}(v,w)$, $\bullet:\mathbb{G}_{m}\times\mathfrak{M}(v,w)\to\mathfrak{M}(v,w)$, 13. 4.2.

$H(w)\subset G(w),$ 14. 4.4.

$\Bbbk$, $\Bbbk[v]$, 15. 5.

$H(v)\subset G(v)$, $\Bbbk\langle v\rangle$, $K,$ 16. 5.1.

$\mathbb{X},$ $\mathbb{X}_{+}$, $\operatorname{Y}\nolimits^{\flat},$ $\operatorname{Y}\nolimits^{\flat}_{A},$ $\operatorname{Y}\nolimits^{\flat}_{M},$ $\operatorname{Y}\nolimits^{\flat}(v),$ $\operatorname{Y}\nolimits^{\flat}(v,k),$ 17. 5.2.

$\operatorname{Y}\nolimits$ 18. 5.3.

$\lambda^{\flat}(q,z)$, $\tau={{\operatorname{rank}}}(T)$, $A^{\flat}_{v}(t),$ 19. 5.4.

$\varepsilon_{i}(\bar{x})$, $\operatorname{R}\nolimits(v)^{l,i},$ $\Lambda^{1}(v)^{l,i}$, $\operatorname{Y}\nolimits^{1}(v)^{\geqslant l,i},$ $x_{i,l},$ $x_{l}$, 20. 5.5.

$\mathbf{Y}^{\flat}$, $\Bbbk(\infty)$, $\mathbf{Y}(0)$, $\mathbf{Y}_{K}$, $\mathcal{U}_{i}$, $I^{\operatorname{r}\nolimits},$ $I^{\operatorname{e}\nolimits},$ $I^{\operatorname{h}\nolimits}$, 21. 5.6.

$F_{w}$, $F_{w}(v)$, $F_{w}(v,k)$, $A_{w}$, $A_{w}(v)$, $A_{w}(v,k)$, $\mathfrak{C}(v_{1},v_{2}\,;\,w)$, $\mathfrak{C}_{i,l}(w)$, $\mathfrak{C}_{i,l}$, $\mathfrak{M}(w)$, $\mathcal{V}_{i},$ $\psi_{i,l}$.