Quantization of the minimal nilpotent orbits and the quantum Hikita conjecture

TL;DR

This paper proves a quantum version of Hikita's conjecture by establishing an isomorphism between specialized quantum D-modules of equivariant quantum cohomology and D-modules of graded traces on minimal nilpotent orbits for ADE singularities, extending to BCFG types.

Contribution

It generalizes recent results to verify the quantum Hikita conjecture for ADE and BCFG singularities, linking quantum cohomology and nilpotent orbit structures.

Findings

Isomorphism between quantum D-modules and graded trace D-modules for ADE singularities

Verification of the quantum Hikita conjecture in this setting

Extension of results to BCFG singularities

Abstract

We show that the specialized quantum D-module of the equivariant quantum cohomology ring of the minimal resolution of an ADE singularity is isomorphic to the D-module of graded traces on the minimal nilpotent orbit in the Lie algebra of the same type. This generalizes a recent result of Shlykov [Hikita conjecture for the minimal nilpotent orbit, to appear in Proc. AMS, https://doi.org/10.1090/proc/15281] and hence verifies in this case the quantum version of Hikita's conjecture, proposed by Kamnitzer, McBreen and Proudfoot [The quantum Hikita conjecture, Advances in Mathematics 390 (2021) 107947]. We also show analogous isomorphisms for singularities of BCFG type.

Equations

$${\mathrm{H}}^{\bullet }({\tilde{X}}^{!},\mathbb{C})\cong \mathbb{C}[X^T].$$

$${\mathrm{H}}^{\bullet }(\widetilde{{\mathbb{C}}^2/\Gamma })\cong \mathbb{C}[{\overline{\mathcal{O}}}_{min}^{{\mathbb{C}}^{\times }}]$$

$${\mathrm{QH}}_{({\mathbb{C}}^{\times }{)}^2}^{\bullet }(\widetilde{{\mathbb{C}}^2/\Gamma })\cong Q(\mathcal{A}[{\overline{\mathcal{O}}}_{min}])$$

$${\mathrm{QH}}_{{\mathbb{C}}^{\times }}^{\bullet }(\widetilde{{\mathbb{C}}^2/\Gamma })\cong Q(\mathcal{A}[{\overline{\mathcal{O}}}_{min}]),$$

$$\begin{array}[]{ll}\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(\widetilde{{\mathbb{C}^{2}}/\Gamma})\cong B(\mathscr{A}[\overline{\mathcal{O}}_{min}]),&\mbox{if $\mathbb{C}^{2}/\Gamma$ is an $A_{n}$ singularity,}\\[5.69054pt] \mathrm{H}^{\bullet}_{\mathbb{C}^{\times}}(\widetilde{{\mathbb{C}^{2}}/\Gamma})\cong B(\mathscr{A}[\overline{\mathcal{O}}_{min}]),&\mbox{otherwise,}\end{array}$$

$${\mathrm{QH}}_{{\mathbb{C}}^{\times }}^{\bullet }(\widetilde{{\mathbb{C}}^2/\Gamma })\cong Q(\mathcal{A}[{\overline{\mathcal{O}}}_{min}])\text{mbox}and{\mathrm{H}}_{{\mathbb{C}}^{\times }}^{\bullet }(\widetilde{{\mathbb{C}}^2/\Gamma })\cong B(\mathcal{A}[{\overline{\mathcal{O}}}_{min}])$$

$$\begin{array}[]{ll}\mathrm{QH}^{\bullet}_{R}(\mathcal{B}_{n})\cong Q(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(C_{n})]),&\mathrm{QH}^{\bullet}_{R}(\mathcal{C}_{n})\cong Q(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(B_{n})]),\\[5.69054pt] \mathrm{QH}^{\bullet}_{R}(\mathcal{F}_{4})\cong Q(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(F_{4})]),&\mathrm{QH}^{\bullet}_{R}(\mathcal{G}_{2})\cong Q(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(G_{2})])\end{array}$$

$$\begin{array}[]{ll}\mathrm{H}^{\bullet}_{\mathbb{Z}_{2}\times\mathbb{C}^{\times}}(\mathcal{B}_{n})\cong B(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(C_{n})]),&\mathrm{H}^{\bullet}_{\mathbb{Z}_{2}\times\mathbb{C}^{\times}}(\mathcal{C}_{n})\cong B(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(B_{n})]),\\[5.69054pt] \mathrm{H}^{\bullet}_{\mathbb{Z}_{2}\times\mathbb{C}^{\times}}(\mathcal{F}_{4})\cong B(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(F_{4})]),&\mathrm{H}^{\bullet}_{\mathfrak{S}_{3}\times\mathbb{C}^{\times}}(\mathcal{G}_{2})\cong B(\mathscr{A}[\tilde{\mathcal{O}}_{ms}(G_{2})])\end{array}$$

$$\mathcal{N}:=\left\{x\in\mathfrak{g}:(\mathrm{ad}_{x})^{n}=0\,\,\mbox{for some $n\in\mathbb{N}$}\right\}.$$

$$(e_{\alpha }\star e_{{\alpha }^{\prime }},e_{\gamma })=(e_{\alpha }\cup e_{{\alpha }^{\prime }},e_{\gamma })+\sum _{\beta \in H_2(X,\mathbb{Z})-\{0\}}⟨e_{\alpha },e_{{\alpha }^{\prime }},e_{\gamma }{⟩}_{0,\beta }^{X,T}\cdot q^{\beta }.$$

$$E_i\cap E_j=(e_i,e_j)=-⟨{\alpha }_i,{\alpha }_j⟩.$$

$$e_{\alpha }\star e_{{\alpha }^{\prime }}=-t^2|\Gamma |⟨\alpha ,{\alpha }^{\prime }⟩+\sum _{\gamma \in {\Delta }^{+}}t⟨\alpha ,\gamma ⟩⟨{\alpha }^{\prime },\gamma ⟩\frac{1+q^{\gamma }}{1-q^{\gamma }}{e}_{\gamma }$$

$$\xi \cdot (z_1,z_2)=\left(e^{\frac{2\pi i}{n+1}}{z}_1,e^{-\frac{2\pi i}{n+1}}{z}_2\right).$$

$$v_0=(1,0),v_1=(0,1),v_2=(-1,2),\cdots ,v_{n+1}=(-n,n+1).$$

$$R_{\Sigma }:=\Lambda [X_0,X_1,\cdots ,X_{n+1}]/(I_{\Sigma }+J_{\Sigma }),$$

$$u_1-\sum ⟨u_1,v_i{⟩}_{{\mathbb{R}}^2}{X}_i,u_2-\sum ⟨u_2,v_i{⟩}_{{\mathbb{R}}^2}{X}_i.$$

$$R_{\Sigma }\to {\mathrm{H}}_{({\mathbb{C}}^{\times }{)}^2}^{\bullet }({\mathcal{A}}_n)$$

$${\mathcal{A}}_n\cong ({\mathbb{C}}^{n+2}-Z(\Sigma ))/({\mathbb{C}}^{\times }{)}^n,$$

$$({\lambda }_1,\cdots ,{\lambda }_n)\cdot (z_0,\cdots ,z_{n+1})=({\lambda }_1{z}_0,{\lambda }_1^{-2}{\lambda }_2{z}_1,{\lambda }_1{\lambda }_2^{-2}{\lambda }_3{z}_2,\cdots ,{\lambda }_{n-1}{\lambda }_n^{-2}{z}_{n-1},{\lambda }_n{z}_{n+1}).$$

$$\eta \cdot [z_0:z_1:\cdots z_n:z_{n+1}]=[{\eta }^{t_2}{z}_0:z_1:\cdots z_n:{\eta }^{t_1}{z}_{n+1}].$$

$$\begin{array}{rl}{u}_1& =⟨u_1,v_{n+1}{⟩}_{{\mathbb{R}}^2}{t}_1+⟨u_1,v_0{⟩}_{{\mathbb{R}}^2}{t}_2=-n{t}_1+t_2,\\ u_2& =⟨u_2,v_{n+1}{⟩}_{{\mathbb{R}}^2}{t}_1+⟨u_2,v_0{⟩}_{{\mathbb{R}}^2}{t}_2=(n+1)t_1.\end{array}$$

$$⟨{\epsilon }_i,{\epsilon }_j⟩={\delta }_{ij}$$

$$(\alpha ,{\alpha }^{\prime }{)}_c:=\sum _{i=1}^{n+1}{x}_i{y}_i({\omega }_{i-1}-{\omega }_i),$$

$$e_{\alpha }\cup e_{{\alpha }^{\prime }}=-(n+1)⟨\alpha ,{\alpha }^{\prime }⟩t_1{t}_2-(n+1)\frac{t_1-t_2}{2}(\alpha ,{\alpha }^{\prime }{)}_c+\sum _{\gamma \in {\Delta }^{+}}\frac{t_1+t_2}{2}⟨\alpha ,\gamma ⟩⟨{\alpha }^{\prime },\gamma ⟩e_{\gamma },$$

$$e_{n+1}=t_1-\frac{1}{n+1}\sum _{i=1}^n(i{e}_i),e_0=t_2-\frac{1}{n+1}\sum _{i=1}^n(i{e}_{n+1-i}).$$

$$\mathrm{LHS}=t_1{t}_2-t_1{e}_0-t_2{e}_{n+1}=-t_1{t}_2+t_1{\omega }_1+t_2{\omega }_2.$$

$$\text{mbox}RHS=-t_1{t}_2+\frac{t_1-t_2}{2}({\omega }_1-{\omega }_n)+\frac{t_1+t_2}{2}({\epsilon }_1-{\epsilon }_{n+1})=-t_1{t}_2+t_1{\omega }_1+t_2{\omega }_2.$$

$$⟨e_{\alpha },e_{{\alpha }^{\prime }},e_{\gamma }{⟩}_{0,\beta }^{{\mathcal{A}}_n,({\mathbb{C}}^{\times }{)}^2}=(t_1+t_2)(e_{\alpha },e_{\eta })(e_{{\alpha }^{\prime }},e_{\eta })(e_{\gamma },e_{\eta }),$$

$$e_{\alpha }\star e_{{\alpha }^{\prime }}=-(n+1)⟨\alpha ,{\alpha }^{\prime }⟩t_1{t}_2-(n+1)\frac{t_1-t_2}{2}(\alpha ,{\alpha }^{\prime }{)}_c+\sum _{\gamma \in {\Delta }^{+}}\frac{(t_1+t_2)}{2}⟨\alpha ,\gamma ⟩⟨{\alpha }^{\prime },\gamma ⟩\frac{1+q^{\gamma }}{1-q^{\gamma }}{e}_{\gamma }.$$

$$e_{\alpha }\star e_{\alpha }^{\prime }$$

Full text

Keywords: Symplectic duality, Kleinian singularity, nilpotent orbit, quantum cohomology, quantization

MSC2020: 14B05, 17B08, 53D55, 55N91

1. 1 Introduction

2. 2 Equivariant quantum cohomology of ADE resolutions

3. 2.1 Kleinian singularities

4. 2.2 The $\mathbb{C}^{\times}$-Equivariant quantum cohomology

5. 2.3 The $(\mathbb{C}^{\times})^{2}$-equivariant quantum cohomology of $A_{n}$ resolutions

6. 3 Quantization of the minimal nilpotent orbits

7. 3.1 The coordinate ring of minimal orbits

8. 3.2 Quantization of the minimal nilpotent orbits

9. 3.3 Garfinkle’s construction of the Joseph ideals

10. 3.4 The $B$-algebras

11. 4 The quantum Hikita conjecture

12. 4.1 Specialized quantum D-module

13. 4.2 D-module of graded traces

14. 4.3 Proof of Theorems 1.2 and 1.3

15. 5 Generalization to BCFG type singularities

16. 5.1 Quantization of the minimal special nilpotent orbits

17. 5.2 Equivariant cohomology of the minimal resolutions

18. 5.3 Proof of Theorems 1.4 and 1.5

1 Introduction

Over the past two decades, 3d $\mathcal{N}=4$ mirror symmetry has attracted a lot of attentions from both physicists and mathematicians (see, for example, [7, 13, 25, 40] and references therein). It is also equivalent to the theory of symplectic duality of Braden et al. [5, 6] (see also [30] for a survey). For two (possibly singular) manifolds that are symplectic dual to each other, there are some highly nontrivial identities between the geometry and topology of them. One of the properties predicted by 3d $\mathcal{N}=4$ mirror symmetry and symplectic duality is Hikita’s conjecture. Suppose we are given a pair of symplectic dual conical symplectic singularities, then Hikita’s conjecture relates the coordinate ring of one symplectic variety to the cohomology ring of the symplectic resolution of the other, which is stated as follows.

Conjecture 1.1** (Hikita [22, Conjecture 1.3]).**

Let $X$ and $X^{!}$ be a pair of symplectic dual conical symplectic singularities over $\mathbb{C}$. Suppose $X^{!}$ admits a conical symplectic resolution $\tilde{X}^{!}\rightarrow X^{!}$, and suppose $T$ is a maximal torus of the Hamiltonian action on $X$. Then there is an isomorphism of graded algebras

[TABLE]

In loc. cit. Hikita proved this conjecture in several cases, such as hypertoric varieties, Spaltenstein varieties and the Hilbert schemes of points in the plane. He then asked whether this phenomenon holds for other examples of symplectic duality. In [32], Kamnitzer et. al. proved Hikita’s conjecture for the case of Nakajima quiver varieties of type A and affine Grassmannian slices, which are symplectic dual to each other. In loc. cit., they also stated a conjecture of Nakajima, which generalized Hikita’s conjecture to the equivariant case (see [32, §8]). In his Ph.D. thesis [49], Weekes proved this conjecture for symplectic dual pairs studied in [32]. In a recent paper [33], Krylov and Shlykov called this conjecture the Hikita-Nakajima conjecture and proved it for Gieseker varieties (the ADHM spaces).

In [31], Kamnitzer, McBreen and Proudfoot further generalized the Nakajima-Hikita conjecture to the quantum case, and proved it for nilpotent cones in ADE type Lie algebras and hypertoric varieties. A bit more precisely, in loc. cit., they introduced, for a symplectic dual pair $X$ and $X^{!}$, two concepts: one is the so-called specialized quantum D-module, which is basically the equivariant quantum cohomology of $X^{!}$, and the other is the D-module of graded traces on $X$, which may be understood as the “graded functions” (the universal source of graded traces) of the quantized coordinate ring of $X$. Kamnitzer et. al. conjectured that these two types of D-modules are isomorphic, and called it the quantum Hikita conjecture.

According to 3d mirror symmetry, the minimal nilpotent orbit closure $\overline{\mathcal{O}}_{min}$ in a simple Lie algebra $\mathfrak{g}$ of ADE type is mirror to (or equivalently symplectic dual to) the intersection of a Slodowy slice to the subregular nilpotent orbit with the nilpotent cone in the same Lie algebra; see, for example, [48] and [6, Remark 10.6]. This is highly related to the duality discovered by Spaltenstein [46] and Lusztig [38] (see also [15] for more details). Recall that by Brieskorn [9] and Slodowy [44], the latter is isomorphic to the Kleinian singularity $\mathbb{C}^{2}/\Gamma$ of the same type. If we denote by $\widetilde{\mathbb{C}^{2}/\Gamma}$ the minimal resolution of $\mathbb{C}^{2}/\Gamma$, then in a recent paper [45], Shlykov showed that

[TABLE]

as graded algebras, and hence verified Hikita’s conjecture in these cases. The purpose of this paper is to generalize his work to the quantum case.

Theorem 1.2**.**

Let $\mathfrak{g}$ be a complex semisimple Lie algebra of ADE type, and let $\overline{\mathcal{O}}_{min}$ be the closure of the minimal nilpotent orbit in $\mathfrak{g}$. Let $\widetilde{{\mathbb{C}^{2}}/\Gamma}$ be the minimal resolution of the singularity of the same type. Then the quantum Hikita conjecture holds for the pair $\widetilde{{\mathbb{C}^{2}}/\Gamma}$ and $\overline{\mathcal{O}}_{min}$; more precisely,

1. $(1)$

for $\mathbb{C}^{2}/\Gamma$ being an $A_{n}$ singularity, there is an isomorphism

[TABLE]

of graded modules over $F_{\mathrm{reg}}\otimes\mathrm{Sym}\mathrm{H}^{2}_{(\mathbb{C}^{\times})^{2}}(\widetilde{{\mathbb{C}^{2}}/\Gamma})$; 2. $(2)$

for other types of singularity, there is an isomorphism

[TABLE]

of graded modules over $F_{\mathrm{reg}}\otimes\mathrm{Sym}\mathrm{H}^{2}_{\mathbb{C}^{\times}}(\widetilde{{\mathbb{C}^{2}}/\Gamma})$,

where $\mathrm{QH}^{\bullet}(-)$ is the equivariant quantum cohomology, also called the specialized quantum D-module, and $Q(\mathscr{A}(-))$ is the D-module of graded traces (see §4.1 and 4.2 for the definitions of these two concepts as well as the base rings).

In the above theorem, if we let the quantum parameter $q=0$, then both the specialized quantum D-module and the D-module of graded traces are algebras, and therefore we get the corresponding Hikita-Nakajima conjecture:

Theorem 1.3**.**

With the notations in Theorem 1.2, the Hikita-Nakajima conjecture holds for the pair $\widetilde{{\mathbb{C}^{2}}/\Gamma}$ and $\overline{\mathcal{O}}_{min}$; that is, there are isomorphisms of graded algebras:

[TABLE]

where $B(-)$ is the associated $B$-algebra (see §3.4 for the definition).

In the above two theorems, the $A_{n}$ singularities are toric varieties, and hence we naturally consider the $(\mathbb{C}^{\times})^{2}$-equivariant cohomology for them. We also expect that the isomorphism in Theorem 1.3 in $A_{n}$ case should be identical to the one proved by Weekes [49] by a different method. For singularities of DE type, there is only a natural $\mathbb{C}^{\times}$-action on them, and we can only consider their $\mathbb{C}^{\times}$-equivariant cohomology, which has been studied by Bryan and Gholampour in [11].

On the other side, Joseph gave in [28] the quantizations of the minimal orbit closures in simple Lie algebras. They are the quotients of the corresponding universal enveloping algebras by some two-sided ideals, which are nowadays called the Joseph ideals. Later, Garfinkle in her thesis [20] constructed explicitly the Josephs ideals. Interestingly enough, the Joseph ideals in the type A case are not unique, but are parameterized by the complex numbers $\mathbb{C}$. Thus in the type A case, if we view the number that parameterizes the Joseph ideals as a formal parameter, then the quantizations of the minimal orbits in this case are over the ring of polynomials of two variables, which exactly matches the base ring of the $(\mathbb{C}^{\times})^{2}$-equivariant cohomology of the dual side. For the other types of Lie algebras, the Joseph ideals are uniquely determined, and hence the quantizations are over the polynomials of one variable.

If we take the usual $\mathbb{C}^{\times}$-action on an $A_{n}$ singularity similar to that on the DE singularities and take a specific Joseph ideal in its symplectic dual side (see Remark 4.25), then all the isomorphisms in the above two theorems become

[TABLE]

respectively.

Moreover, Theorem 1.2 can be generalized to the BCFG type singularities as follows. Recall that the Lusztig-Spaltenstein duality says the subregular nilpotent orbit in a Lie algebra of BCFG type is dual to the minimal special nilpotent orbit in its Langlands dual. A theorem of Brylinski and Kostant [12] says that the minimal special nilpotent orbits in these cases are covered by the minimal nilpotent orbits of $D_{n+1}$, $A_{2n-1}$, $E_{6}$ and $D_{4}$ respectively, with the deck transformations $\mathbb{Z}_{2}$ or $\mathfrak{S}_{3}$. On the other hand, Slodowy showed in [44] that the intersections of Slodowy slices to the subregular nilpotent orbit with the nilpotent cone in these Lie algebras, which are also called the simple singularities of BCFG type, can be obtained by those in $A_{2n-1}$, $D_{n+1}$, $E_{6}$ and $D_{4}$, with some extra symmetry also given by either $\mathbb{Z}_{2}$ or $\mathfrak{S}_{3}$. For these types of singularities, Bryan and Gholampour constructed a version of equivariant quantum algebra, denoted by $\mathrm{QH}_{R}^{\bullet}(-)$, according to the associated root systems/Dynkin diagrams, which they also showed to admit a Frobenius algebra structure (see [11]).

Considering the associated D-modules of these varieties, we obtain the following result, which is a corollary of Theorem 1.2:

Theorem 1.4**.**

Let $\mathcal{B}_{n}$, $\mathcal{C}_{n}$, $\mathcal{F}_{4}$ and $\mathcal{G}_{2}$ be the minimal resolutions of singularities of $B_{n}$, $C_{n}$, $F_{4}$ and $G_{2}$ respectively, and let $\tilde{\mathcal{O}}_{ms}[B_{n}]$, $\tilde{\mathcal{O}}_{ms}[C_{n}]$, $\tilde{\mathcal{O}}_{ms}[F_{4}]$ and $\tilde{\mathcal{O}}_{ms}[G_{2}]$ be the normalizations of the closures of the minimal special nilpotent orbits in Lie algebras of BCFG type respectively. Then

[TABLE]

as D-modules over the corresponding base rings.

Again, let the quantum parameter $q=0$, then we get the following (see §5 for more details about the notions):

Theorem 1.5**.**

Let $\mathcal{B}_{n}$, $\mathcal{C}_{n}$, $\mathcal{F}_{4}$ and $\mathcal{G}_{2}$ be the minimal resolutions of singularities of $B_{n}$, $C_{n}$, $F_{4}$ and $G_{2}$ respectively, and let $\tilde{\mathcal{O}}_{ms}[B_{n}]$, $\tilde{\mathcal{O}}_{ms}[C_{n}]$, $\tilde{\mathcal{O}}_{ms}[F_{4}]$ and $\tilde{\mathcal{O}}_{ms}[G_{2}]$ be the normalizations of the closures of the minimal special nilpotent orbits in Lie algebras of BCFG type respectively. Then

[TABLE]

as algebras over $\mathbb{C}[\hbar]$.

The rest of this paper is devoted to the proofs of the above two theorems. It is organized as follows. In §2 we first recall some basic facts on Kleinian singularities, and then compute the equivariant cohomology of the minimal resolutions of these singularities. In §3 we go over the quantizations of the minimal nilpotent orbit closures in Lie algebras of ADE type, which is due to Joseph [28] and Garfinkle [20]. After that, in §3.4 we study with some details the corresponding $B$-algebra of these quantizations. In §4, we first recall Kamnitzer-McBreen-Proudfoot’s version of the quantum Hikita conjecture, and then prove Theorems 1.2 and 1.3. In §5, we study the quantizations of the minimal special orbits and the equivariant cohomologies of the minimal resolutions of BCFG type singularities, and prove Theorems 1.4 and 1.5.

Acknowledgements**.**

In the spring of 2021, Professor Yongbin Ruan gave a series of lectures at Zhejiang University on his project on the mirror symmetry of nilpotent orbits of semi-simple Lie algebras. This paper is also motivated by our study of his lectures. We are extremely grateful to him as well as IASM, Zhejiang University for inviting us to attend the lectures and for offering excellent working conditions. We also thank Xiaowen Hu, Huazhong Ke, Yaoxiong Wen and Farkhod Eshmatov for some very valuable discussions. This work is supported by NSFC Nos. 11890663, 12271377 and 12261131498.

2 Equivariant quantum cohomology of ADE resolutions

In this section, we study the equivariant cohomology of the minimal resolutions of Kleinian singularities. In §2.1 we briefly recall the definition of ADE singularities. In §2.2 we go over Bryan and Gholampour’s result in [11] on the $\mathbb{C}^{\times}$-equivariant cohomology of resolution of ADE singularities, and then §2.3 we further study the $(\mathbb{C}^{\times})^{2}$-equivariant cohomology of the minimal resolution of $A_{n}$ singularities. For $A_{n}$ singularities, we shall use both of them in later sections.

2.1 Kleinian singularities

Let $\Gamma$ be a finite subgroup of $\mathrm{SL}_{2}(\mathbb{C})$. It naturally acts on $\mathbb{C}^{2}$ via the canonical action of $\mathrm{SL}_{2}(\mathbb{C})$. The singularity $\mathbb{C}^{2}/\Gamma$ is called a Kleinian singularity, and has been widely studied. The following table summarizes the classification of Kleinian singularities:

[TABLE]

The singularity $\mathbb{C}^{2}/\Gamma$ has a unique minimal resolution, denoted by $\widetilde{\mathbb{C}^{2}/\Gamma}$, whose exceptional fiber is given by a tree of $\mathbb{CP}^{1}$’s. The corresponding tree, whose vertices are the $\mathbb{CP}^{1}$’s and whose edges between two given vertices are identified with the intersection points of the corresponding $\mathbb{CP}^{1}$’s. It turns out that the trees such constructed are exactly the Dynkin diagrams of the Lie algebra of the same type.

There is another direct relationship between the Kleinian singularities and the Lie algebras; namely, the Kleinian singularities are exactly the Slodowy slices to the subregular nilpotent orbits in the Lie algebra of the same type (see Slodowy [44] for more details).

Let $\mathfrak{g}$ be a Lie algebra. Recall that the nilpotent cone of $\mathfrak{g}$, usually denoted by $\mathcal{N}$, is the set

[TABLE]

Definition 2.1** (Slodowy slice [44]).**

Let $x\in\mathfrak{g}$ be a nilpotent element, and extend it to be an $\mathfrak{sl}_{2}(\mathbb{C})$ triple $\{x,h,y\}\subseteq\mathfrak{g}$. The Slodowy slice associated to $(\mathfrak{g},x)$ is the affine sub-variety $S=x+\ker[y,-]\subseteq\mathfrak{g}$.

It is a transverse slice to the nilpotent orbit of the point $x$.

Theorem 2.2** (Brieskorn [9] and Slodowy [44]).**

Let $\mathfrak{g}$ be simply-laced, $\mathcal{N}\subseteq\mathfrak{g}$ denote the nilpotent cone, and $S_{x}$ be a Slodowy slice to a subregular nilpotent element $x\in\mathcal{O}_{sub}$. The intersection $S_{x}\cap\mathcal{N}$ is a Kleinian surface singularity with the same Dynkin diagram as $\mathfrak{g}$. Moreover, the symplectic resolution $\widetilde{S_{x}\cap\mathcal{N}}\rightarrow S_{x}\cap\mathcal{N}$ is the same as the minimal resolution of the Kleinian singularity $\widetilde{\mathbb{C}^{2}/\Gamma}\rightarrow\mathbb{C}^{2}/\Gamma$.

2.2 The $\mathbb{C}^{\times}$-Equivariant

quantum cohomology

Suppose $T$ is a torus, $X$ is a variety with a $T$-action on it. The $T$-equaivariant quantum cohomology ring $(\mathrm{QH}^{\bullet}_{T}(X),\star)$ of $X$ is a deformation of $(\mathrm{H}^{\bullet}_{T}(X),\cup)$, which is induced by the genus 0 $T$-equivariant Gromov-Witten invariants of $X$. More explicitly, the quantum product $\star$ is defined as follows (see e.g. [34]). For any $e_{\alpha},e_{\alpha^{\prime}},e_{\gamma}\in\mathrm{H}^{\bullet}_{T}(X)$

[TABLE]

Here $(-,-)$ is the Poincare pairing, $q$ is the quantum parameter and $\langle e_{\alpha},e_{\alpha^{\prime}},e_{\gamma}\rangle^{X,T}_{0,\beta}$ is some genus 0 $T$-equivariant Gromov-Witten invariant of $X$.

Now let $\widetilde{\mathbb{C}^{2}/\Gamma}$ be the minimal resolution of an ADE singularity. Observe that the scalar $\mathbb{C}^{\times}$-action on $\mathbb{C}^{2}$ commutes with the action of $\Gamma$, and thus $\mathbb{C}^{\times}$ acts on $\mathbb{C}^{2}/\Gamma$. It lifts to an action on $\widetilde{\mathbb{C}^{2}/\Gamma}$. Let $\{E_{1},E_{2},\cdots,E_{n}\}$ be the set of irreducible components in the exceptional fiber in $\widetilde{\mathbb{C}^{2}/\Gamma}$, which gives a basis of $\mathrm{H}_{2}(\widetilde{\mathbb{C}^{2}/\Gamma},\mathbb{Z})$. It is direct to see that they are invariant under the $\mathbb{C}^{\times}$-action, and hence lifts to a basis $e_{1},e_{2},\cdots,e_{n}$ of the $\mathbb{C}^{\times}$-equivariant cohomology. The intersection matrix $E_{i}\cap E_{j}$ defines a perfect pairing on $\mathrm{H}_{2}(\widetilde{\mathbb{C}^{2}/\Gamma},\mathbb{Z})$, which coincides with the $\mathbb{C}^{\times}$-equivariant Poincare pairing $(e_{i},e_{j})$.

Let $\Delta$ be the root system associated to the Dynkin diagram given by this pairing. Following [11], we can identify both $E_{1},\cdots,E_{n}$ and $e_{1},\cdots,e_{n}$ with the simple roots $\alpha_{1},\cdots,\alpha_{n}$ of ADE Lie algebra, and the intersection matrix with minus of the Cartan matrix

[TABLE]

where $\langle\alpha,\alpha^{\prime}\rangle:=K(h_{\alpha},h_{\alpha^{\prime}})$ for roots $\alpha,\alpha^{\prime}\in\Delta$ and corresponding Cartan elements $h_{\alpha},h_{\alpha^{\prime}}$ in $\mathfrak{h}$, $K(-,-)$ is the Killing form. Bryan and Gholampour computed the $\mathbb{C}^{\times}$-equivariant quantum cohomology ring of $\widetilde{\mathbb{C}^{2}/\Gamma}$, which is given as follows.

Theorem 2.3** ([11, Theorem 1]).**

For $e_{\alpha}$, $e_{\alpha^{\prime}}\in\mathrm{QH}^{\bullet}(\widetilde{\mathbb{C}^{2}/\Gamma})$, the quantum product is given by

[TABLE]

where $\Gamma$ is the subgroup of $\mathrm{SL}_{2}(\mathbb{C})$, $e_{\alpha}=c_{1}e_{1}+\cdots+c_{n}e_{n}$, if $\alpha=c_{1}\alpha_{1}+\cdots c_{n}\alpha_{n}$, $c_{1},\cdots,c_{n}\in\mathbb{N}$.

By the root data of Lie algebras of ADE type (see, for example, Bourbaki [4, PLATE I-VII]), we may explicitly write down the cup product in all the cases.

2.3 The $(\mathbb{C}^{\times})^{2}$-equivariant quantum cohomology of $A_{n}$ resolutions

In this subsection we calculate the equivariant cohomology of the minimal resolution of the $A_{n}$ singularity. The main reference we use here is [1, Chapter 8].

2.3.1 Equivariant cohomology

Let $\Gamma=\mathbb{Z}_{n+1}$ with generator $\xi$. The finite group $\Gamma$ acts on $\mathbb{C}^{2}$ as

[TABLE]

The $A_{n}$ singularity is given by $\mathbb{C}^{2}/\Gamma$.

Let $\mathcal{A}_{n}$ be the minimal resolution of $A_{n}$, which is also called the Hirzebruch-Jung resolution. By [42], $\mathcal{A}_{n}$ is a toric variety, corresponding to a 2-dimensional fan $\Sigma$. $\Sigma$ contains $n+1$ cones generated by the following $n+2$ rays $\{\rho_{i}=\mathbb{C}v_{i}|0\leq i\leq n+1\}$ in ${\mathbb{R}}^{2}$, where $v_{i}$ is as follows:

[TABLE]

Denote by $\langle\ ,\ \rangle_{{\mathbb{R}}^{2}}$ the canonical paring of ${\mathbb{R}}^{2}$. Let $M\cong{\mathbb{Z}}^{2}$ be the character group of $(\mathbb{C}^{\times})^{2}$. Suppose $u_{1}=(1,0)$ and $u_{2}=(0,1)$ are basis of $M$. By [1], $u_{1}$ and $u_{2}$ can be treated as the equivariant parameters of the torus action, which corresponds to the regular $(\mathbb{C}^{\times})^{2}$-embedding in $\mathcal{A}_{n}$ given by the fan (4).

Set $\Lambda:=\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(pt)=\mathbb{C}[u_{1},u_{2}]$. Let $X_{0},X_{1},\cdots,X_{n+1}$ be formal variables, one for each ray of $\Sigma$. The following $R_{\Sigma}$ is called the Stanley-Reisner ring:

[TABLE]

where

• $-$

the ideal $I_{\Sigma}$ is generated by all monomials $X_{i}\cdot X_{j}$ such that the corresponding rays $\rho_{i},\rho_{j}$ do not span a cone, that is, $|i-j|\geq 2$;

• $-$

the ideal $J_{\Sigma}$ is generated by the following two elements:

[TABLE]

Denote the equivariant cohomology of $\mathcal{A}_{n}$ by $\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$. There is a ring structure on it induced by the cup product $\cup$. Define a homomorphism

[TABLE]

by $X_{i}\mapsto e_{i}$, where $e_{i}$ is the equivariant class of the $(\mathbb{C}^{\times})^{2}$-invariant divisor corresponding to the ray $\rho_{i}$. The following proposition is proved in [3, Theorem 8] (see also [1, Theorem 3.1]).

Proposition 2.4**.**

The homomorphism $(R_{\Sigma},\cdot)\rightarrow\mathrm{(}\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n}),\cup)$ is a ring isomorphism.

According to [14], $\mathcal{A}_{n}$ can be viewed as a GIT quotient, namely,

[TABLE]

where $Z(\Sigma)=\bigcap_{1\leq i\leq n+1}\{z_{0}\cdots\hat{z_{i}}\cdots z_{n+1}=0\}$, and the $(\mathbb{C}^{\times})^{n}$-action on $\mathbb{C}^{n+2}$ is as follows: for any $(\lambda_{1},\lambda_{2},\cdots,\lambda_{n})\in(\mathbb{C}^{\times})^{n}$,

[TABLE]

Here we use the homogeneous coordinate $[z_{0}:z_{1}:\cdots:z_{n+1}]$ to parametrize the $(\mathbb{C}^{\times})^{n}$-orbit of $(z_{0},\cdots,z_{n+1})$. There is another natural $(\mathbb{C}^{\times})^{2}$-action on $\mathcal{A}_{n}$: for $\eta\in\mathbb{C}^{\times}$,

[TABLE]

This torus action is used to calculate equivariant Gromov-Witten invariants of $\mathcal{A}_{n}$ in [39]. We treat $t_{1}$, $t_{2}$ as equivariant parameters of the above torus action, then $(u_{1},u_{2})$ is determined by $(t_{1},t_{2})$ as following (see e.g [35, Section 4.4]):

[TABLE]

By Proposition 2.4, $\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$ is a free $\Lambda$-module generated by $1,e_{1},e_{2},\cdots,e_{n}$.

Remark 2.5**.**

Recall that, for complex semisimple Lie algebra, there is an isomorphism $\mathfrak{h}\rightarrow\mathfrak{h}^{*}$. In this section, for convenience to further discussion, we identify $e_{i}$ with the $i$-th simple root $\alpha_{i}$ of $\mathfrak{sl}(n+1,\mathbb{C})$ and the cartan element $h_{\alpha_{i}}$, and identify the fundamental weights $\omega_{i}\in\mathfrak{h}^{*}$ with its linear dual $\omega_{i}^{*}\in\mathfrak{h}$.

Then the Cartan subalgebra $\mathfrak{h}={\rm Span}_{\mathbb{C}}\{e_{1},e_{2},\cdots,e_{n}\}$. It is well know that $\mathfrak{h}$ can be embedded into $\mathbb{C}^{n+1}$. And $e_{i}=\varepsilon_{i}-\varepsilon_{i+1}$, where $\{\varepsilon_{j}\}_{j=1,\cdots,n+1}$ is the canonical basis of $\mathbb{C}^{n+1}$. Denote the euclidean pairing on $\mathbb{C}^{n+1}$ by $\langle-,-\rangle$, of which restriction on $\mathfrak{h}$ is the Killing form.

[TABLE]

Definition 2.6**.**

Define a bilinear map $(-,-)_{c}:\mathfrak{h}\otimes\mathfrak{h}\rightarrow\mathfrak{h}$ as follows: for $\alpha=\sum_{i=1}^{n+1}x_{i}\varepsilon_{i}$, $\alpha^{\prime}=\sum_{i=1}^{n+1}y_{i}\varepsilon_{i}$,

[TABLE]

where $\omega_{j}$ is the $j$-th fundamental weight satisfying $\langle\omega_{j},e_{i}\rangle=\delta_{ij}$, and we set $\omega_{0}=\omega_{n+1}=0$.

From the isomorphism of linear space: $\mathfrak{h}\cong\mathrm{H}^{2}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$, $(\alpha,\alpha^{\prime})_{c}$ is in $H^{2}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$, we can rephrase the ring structure of $\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$ as follows.

Theorem 2.7**.**

$(\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n}),\cup)\cong(\Lambda\otimes\mathrm{Sym}^{\bullet}\mathfrak{h})/\widetilde{I}$, where $\widetilde{I}$ is generated by the following relation: for any $e_{\alpha},e_{\alpha^{\prime}}\in\mathfrak{h}$,

[TABLE]

where $\Delta^{+}=\{\varepsilon_{i}-\varepsilon_{j}|1\leq i<j\leq n+1\}$ is the set of positive roots.

Proof.

Plugging (9) into (6), we have in $\mathrm{H}^{\bullet}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$,

[TABLE]

Notice that the two handsides of (12) are both commutative $\Lambda$-bilinear map on $\mathfrak{h}$. It suffice to verify (12) on a basis of $\mathrm{Sym}^{2}\mathfrak{h}$. And we choose a basis of $\mathrm{Sym}^{2}\mathfrak{h}$ which consists of the following four types of vectors:

1. (1)

$e_{i}e_{j},\quad 1\leq i<j\leq n,j-i\geq 2$; 2. (2)

$(t_{2}-e_{0})e_{k},\quad 2\leq k\leq n$; 3. (3)

$(t_{1}-e_{n+1})e_{l},\quad 1\leq l\leq n-1$; 4. (4)

$(t_{2}-e_{0})(t_{1}-e_{n+1})$.

Now we check (12) holds on these vectors. For type (1), when $e_{\alpha}=e_{i}=\varepsilon_{i}-\varepsilon_{i+1}$, $e_{\alpha^{\prime}}=e_{j}=\varepsilon_{j}-\varepsilon_{j+1}$ in (12), LHS vanishes by Proposition 2.4 and (5). Notice that $\langle\alpha,\alpha^{\prime}\rangle$ and $(\alpha,\alpha^{\prime})_{c}$ vanish by $j-i>2$, so the first two terms in RHS vanish. By (10), $\langle\alpha,\gamma\rangle\langle\alpha^{\prime},\gamma\rangle$ does not vanish only when $\gamma=\varepsilon_{i}-\varepsilon_{j}$, $\varepsilon_{i}-\varepsilon_{j+1}$, $\varepsilon_{i+1}-\varepsilon_{j}$ or $\varepsilon_{i+1}-\varepsilon_{j+1}$, one can easily check the last term in RHS vanishes. For type (4), set $\alpha=t_{2}-e_{0}=\frac{1}{n+1}(n\varepsilon_{1}-\varepsilon_{2}-\cdots-\varepsilon_{n+1})=\omega_{1}$, $\alpha^{\prime}=t_{1}-e_{n+1}=\frac{1}{n+1}(\varepsilon_{1}+\cdots+\varepsilon_{n}-n\varepsilon_{n+1})=\omega_{n}$, then by Proposition 2.4 and (5),

[TABLE]

By (10) and (11), we have

[TABLE]

The verification for types (2) and (3) is left to readers. ∎

2.3.2 Equivariant quantum cohomology

In this section, we calculate the quantum cohomology of $\mathcal{A}_{n}$. The exceptional locus of $\mathcal{A}_{n}$ consists of a chain of $n$ rational curves $E_{1},\cdots,E_{n}$ with intersection matrix given by the minus Cartan matrix for the $A_{n}$ root lattice. Notice that $E_{i}$ is $(\mathbb{C}^{\times})^{2}$-invariant, and $e_{i}$ is its equivariant lift in $\mathrm{QH}^{2}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$. In [39], Maulik calculated all genus $(\mathbb{C}^{\times})^{2}$-equivariant Gromov-Witten invariant of $\mathcal{A}_{n}$. The following lemma is a special case of [39, Theorem 1.1].

Lemma 2.8**.**

The Gromov-Witten invariant does not vanish only when $\beta=d(E_{i}+E_{i+1}+\cdots+E_{j-1})$, for some $1\leq i<j\leq n+1$, $d\geq 0$. Furthermore, if $\beta=d(E_{i}+E_{i+1}+\cdots+E_{j-1})$,

[TABLE]

where $e_{\eta}=e_{i}+e_{i+1}+\cdots+e_{j-1}$, $(-,-)$ is the Poincaré pairing satisfying $(e_{k},e_{l})=-\langle\alpha_{k},\alpha_{l}\rangle$.

Remark 2.9**.**

In [39, Theorem 1.1], Maulik calculated the reduced Gromov-Witten invariants of $\mathcal{A}_{n}$. For $\beta\neq 0$, the reduced Gromov-Witten invariants coincides with $(\mathbb{C}^{\times})^{2}$-equivariant Gromov-Witten invariant up to a factor $(t_{1}+t_{2})$; see [39, Section 2.2].

Following [11], we identify both $e_{i}$ and $E_{i}$ with the $i$-th simple root $\alpha_{i}$ of $A_{n}$, then the Gromov-Witten invariant $\langle e_{\alpha},e_{\alpha^{\prime}},e_{\gamma}\rangle^{\mathcal{A}_{n},(\mathbb{C}^{\times})^{2}}_{0,\beta}$ does not vanish only when $\beta$ is a multiple of some positive root.

Theorem 2.10**.**

For any $e_{\alpha},e_{\alpha^{\prime}}\in\mathrm{QH}^{2}_{(\mathbb{C}^{\times})^{2}}(\mathcal{A}_{n})$, we have

[TABLE]

Proof.

By (2) and (14), we have

[TABLE]

Now combine (12) with (2.3.2), we obtain (15). ∎

Remark 2.11**.**

In Theorem 2.3, we consider the conical action on $\widetilde{\mathbb{C}^{2}/\Gamma}$ induced by scalar $\mathbb{C}^{\times}$-action on $\mathbb{C}^{2}$. And in Theorem 2.10, $t_{1},t_{2}$ corresponds to the following $(\mathbb{C}^{\times})^{2}$-action on $\mathbb{C}^{2}$ (see (8))

[TABLE]

So the transformation between two equivariant parameter is $t_{1}=t,t_{2}=t$. It is straightforward to check that in (15) if we set $t_{1}=t_{2}$, then the quantum products for $A_{n}$ singularities are identical to the ones given by (3) .

Remark 2.12**.**

In both Theorems 2.3 and 2.10, if we let $q=0$, then the equivariant quantum cohomology reduces to the usual equivariant cohomology.

3 Quantization of the minimal nilpotent orbits

In this section, we study the quantization of the minimal nilpotent orbits of Lie algebras of ADE type. In [28], Joseph studied the quantizations of these orbits, which are given by the quotients of the universal enveloping algebras by the two-sided ideals called the Joseph ideals. In her Ph.D. thesis [20], Garfinkle gave a new construction of the Joseph ideals, by explicitly writing down the generators and relations.

In §3.1 we briefly go over Shlykov’s result on the minimal nilpotent orbits. In §3.2 we recall Joseph’s result on the quantization of the minimal nilpotent orbits and then in §3.3 we go over Garfinkle’s construction of Joseph’s ideals. In §3.4 we study the $B$-algebra of the quantization of the minimal nilpotent orbits.

3.1 The coordinate ring of minimal orbits

In this subsection, we assume $\mathfrak{g}$ is a complex semisimple Lie algebra, and $\mathcal{O}_{min}$ is the minimal nilpotent orbit of $\mathfrak{g}$. Let us first recall the following.

Proposition 3.1** (c.f. [26, §8.3]).**

Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $\mathcal{O}$ be a nilpotent orbit of $\mathfrak{g}$. Then

[TABLE]

if and only if $\overline{\mathcal{O}}$ is normal.

In particular, $\overline{\mathcal{O}}_{min}$ is normal with isolated singularity (see [47]), and hence

[TABLE]

Due to this proposition, in what follows we shall not distinguish $\mathbb{C}[\mathcal{O}_{min}]$ and $\mathbb{C}[\overline{\mathcal{O}}_{min}]$. The following result is proved by Shlykov in [45].

Theorem 3.2** ([45, Theorem 2.2]).**

Let $I$ be the defining ideal of $\overline{\mathcal{O}}_{min}$ in $\mathrm{Sym}(\mathfrak{g})$, i.e.,

[TABLE]

then its image of the projection

[TABLE]

induced by the inclusion $\mathfrak{h}^{*}\hookrightarrow\mathfrak{g}^{*}$ is given by $\mathrm{Sym}^{\geq 2}\mathfrak{h}$.

Let $G$ be the corresponding Lie group of $\mathfrak{g}$. Then the adjoint action $Ad:G\times\overline{\mathcal{O}}_{min}\to\overline{\mathcal{O}}_{min}$ is Hamiltonian. Let $T$ be the maximal torus of $G$. If we choose a generic action of $\mathbb{C}^{\times}$ on $\overline{\mathcal{O}}_{min}$ such that the fixed point schemes for it and for $T$ are the same, that is $\overline{\mathcal{O}}_{min}^{\mathbb{C}^{\times}}\cong\mathfrak{h}\cap\overline{\mathcal{O}}_{min}$ as a scheme, then the main result of Shlykov [45] says

[TABLE]

is isomorphic to $\mathrm{H}^{\bullet}(\widetilde{\mathbb{C}^{2}/\Gamma})$, where $\mathbb{C}^{2}/\Gamma$ is the Kleinian singularity with the same type of $\mathfrak{g}$.

3.2 Quantization of the minimal nilpotent orbits

We now study the quantization of the minimal nilpotent orbits in Lie algebra of ADE type. We start with some basic concepts on the quantization of Poisson algebras; see, for example, Losev [36] for more details.

Definition 3.3** (Filtered and graded quantizations).**

Suppose $A$ is a commutative $\mathbb{Z}_{\geq 0}$-graded $k$-algebra, equipped with a Poisson bracket whose degree is $-1$, where $k$ is a field of characteristic zero.

1. $(1)$

A filtered quantization of $A$ is a filtered $k$-algebra $\mathcal{A}=\bigcup_{i\geq 0}\mathcal{A}_{i}$ such that the associated graded algebra $\mathrm{gr}\,\mathcal{A}$ is isomorphic to $A$ as graded Poisson algebras. 2. $(2)$

A graded quantization of $A$ is a graded $k[\hbar]$-algebra $A_{\hbar}$ ($\mathrm{deg}\,\hbar=1$) which is free as a $k[\hbar]$-module, equipped with an isomorphism of $k$-algebras: $f:A_{\hbar}/\hbar\cdot A_{\hbar}\to A$ such that for any $a,b\in A_{\hbar}$, if we denote their images in $A_{\hbar}/\hbar\cdot A_{\hbar}$ by $\overline{a},\overline{b}$ respectively, then

[TABLE]

Let $A$ be an filtered associative algebra. Recall that the Rees algebra of $A$ is the graded algebra $Rees(A):=\bigoplus_{i\in\mathbb{Z}}A_{\leq i}\cdot\hbar^{i}$, equipped with the multiplication $(a\hbar^{i})(b\hbar^{j})=ab\hbar^{i+j}$ for $a,b\in A$. Now, suppose $\mathcal{A}$ is a filtered quantization of $A$, then the associated Rees algebra $Rees(\mathcal{A})$ is a graded quantization of $A$.

Example 3.4**.**

The universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ is the filtered quantization of $\mathbb{C}[\mathfrak{g}^{*}]=\mathrm{Sym}(\mathfrak{g})$, and the Rees algebra of $\mathcal{U}(\mathfrak{g})$, $Rees(\mathcal{U}(\mathfrak{g})):=\bigoplus_{i\in\mathbb{Z}}\mathcal{U}(\mathfrak{g})_{\leq i}\cdot\hbar^{i}$ is the graded quantization of $\mathrm{Sym}(\mathfrak{g})$. On the other hand, there is an isomorphism of $\mathfrak{g}$-modules:

[TABLE]

which is called symmetrization.

Since the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ is the quantization of the symmetric algebra $\mathrm{Sym}(\mathfrak{g})$, we need to study the quantization of the ideal $I$ of $\mathrm{Sym}(\mathfrak{g})$. Joseph in [28] found a two-sided ideal of $\mathcal{U}(\mathfrak{g})$ which plays the role of the quantization of $I$.

3.2.1 Joseph’s quantization of the minimal nilpotent orbits

Let us first recall the result of Joseph [28], which is stated as follows.

Theorem 3.5** (Joseph [28] and Garfinkle [20]).**

Let $\mathfrak{g}$ be a complex semisimple Lie algebra.

1. $(1)$

If $\mathfrak{g}$ is the type A Lie algebra, then there exists a family of completely prime two-sided primitive ideals $J^{z}$, parametrized by $z\in\mathbb{C}$, such that

[TABLE] 2. $(2)$

If $\mathfrak{g}$ is not of type A, then there exists a unique completely prime two-sided primitive ideal $J$ such that

[TABLE]

In the above theorem, a two-sided ideal $J$ of $\mathcal{U}(\mathfrak{g})$ is called primitive if it is the kernel of an irreducible representation $(\pi,V)$ of $\mathcal{U}(\mathfrak{g})$, i.e., $J$ is the annihilator of $V$,

[TABLE]

An ideal $J$ of $\mathcal{U}(\mathfrak{g})$ is called completely prime if for all $u,v\in\mathcal{U}(\mathfrak{g})$, $uv\in J$ implies $u\in J$ or $v\in J$. In literature, the ideals $J^{z}$ and $J$ are usually called the Joseph ideals.

In fact, in the original paper [28], Joseph proved that the Joseph ideals in type A Lie algebras are not unique. It is Garfinkle who gave the explicit constructions of the Joseph ideals in Lie algebras of all types, and in particular, formulated the Joseph ideals in type A Lie algebras in the form given in the above theorem.

Since

[TABLE]

we have that, for the symplectic singularity $\overline{\mathcal{O}}_{min}$, the algebra $\mathcal{U}(\mathfrak{g})/J$ is its filtered quantization.

By the above theorem, $Rees(\mathcal{U}(\mathfrak{g})/J)$ is the graded quantization of $\overline{\mathcal{O}}_{min}$, and we sometimes write it as $\mathscr{A}[\overline{\mathcal{O}}_{min}]$; that is, \mathscr{A}[\overline{\mathcal{O}}_{min}]=Rees\big{(}\mathcal{U}(\mathfrak{g})/J\big{)}.

3.3 Garfinkle’s construction of the Joseph ideals

Garfinkle in her thesis [20] gave an explicit construction of the Joseph ideals. In this subsection, we go over her results with some details.

Notation 3.6**.**

Let us fix some notations in representation theory of Lie algebras.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{g}$, $\Delta$ be the set of roots of $\mathfrak{h}$ in $\mathfrak{g}$ and $\Delta^{+}$ be a fixed choice of positive roots. Let $\Pi$ be the set of the simple roots of $\mathfrak{g}$. The Lie algebra $\mathfrak{g}$ has the root space decomposition $\mathfrak{g}=\oplus_{\alpha\in\Delta}\mathfrak{g}^{\alpha}$, and let

[TABLE]

denote the associated subalgebras of $\mathfrak{g}$.

Let $(\pi,V)$ be a representation of $\mathfrak{g}$; for any weight $\lambda\in\mathfrak{h}^{*}$, let $V^{\lambda}=\{v\in V|\pi(h)(v)=\lambda(h)v\;\mbox{for any}\;h\in\mathfrak{h}\}$. Let $V^{\mathfrak{n}^{-}}:=\{v\in V|\pi(x)v=0\;\mbox{for any}\;x\in\mathfrak{n}^{-}\}$.

Denote by $X_{\alpha_{i}}$ and $Y_{\alpha_{i}}$ the root vectors in $\mathfrak{g}^{\alpha_{i}}$ and $\mathfrak{g}^{-\alpha_{i}}$ respectively, and denote by $h_{i}$ the element in $\mathfrak{h}$ corresponding to $\alpha_{i}$ such that $\alpha_{i}(H)=(H,h_{i})$ for all $H\in\mathfrak{h}$, where $(-,-)$ denotes the Killing form. By the construction of the Chevalley basis, $h_{i}=[X_{\alpha_{i}},Y_{\alpha_{i}}]$ and $\{h_{i}\}\cup\{X_{\alpha_{i}}\}\cup\{Y_{\alpha_{i}}\}$ form a basis of $\mathfrak{g}$. Denote by $h_{i}^{\vee}$ the dual element of $h_{i}$ via the Killing form, i.e., $K(h_{i}^{\vee},h_{j})=\delta_{ij}$. $C=\sum_{\alpha\in\Delta^{+}}(X_{\alpha}Y_{\alpha}+Y_{\alpha}X_{\alpha})+\sum_{i=1}^{n}h_{i}h_{i}^{\vee}$ is the casimir element of $\mathcal{U}({\mathfrak{g}})$, $n=\mathrm{rank}(\mathfrak{h})$.

Let $\alpha_{1},\cdots,\alpha_{n}\in\Pi$ with the subscripts the same as [4, PLATE I-VII]). Denote by $\theta$ the highest root in $\Delta$.

Let $I_{\mathfrak{p},\lambda}$ be the left ideal of the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ generated by $\{x-\lambda(x)|x\in\mathfrak{p}\}$.

3.3.1 Joseph ideal for type A Lie algebras

As we stated before, it was Garfinkle who gave the explicit construction of the Joseph’s ideals. Let us recall her results.

Proposition 3.7** **([20, Proposition 3.2] and

[8, §4.4]).

For type A Lie algebras $\mathfrak{g}$, we have the following decomposition of irreducible representations:

[TABLE]

The ideal $I(\overline{\mathcal{O}}_{min})$ is generated by the lowest weight vectors in $V(\theta+\alpha_{2}+\cdots+\alpha_{n-1})$, $V(\theta)$ and $V(0)$, where $V(0)$ is spanned by the Casimir element $C$ of $\mathcal{U}(\mathfrak{g})$.

Garfinkle showed that the Joseph ideal $J$ in the type A case is generated by elements corresponding to the three types of lowest weight vectors in the above proposition. We examine them one by one.

First, for the subrepresentation $V(\theta+\alpha_{2}+\cdots+\alpha_{n-1})$, we have the following:

Lemma 3.8** ([20] §IV.3 Theorem 2 and §5).**

Let $v_{0}$ be the lowest weight of the representation $V(\theta+\alpha_{2}+\cdots+\alpha_{n-1})$. Then $\beta(v_{0})$ is an element of Joseph ideal $J$ of $\mathcal{U}(\mathfrak{g})$.

Next, we consider the lowest weight vector in $V(\theta)$. For convenience, we denote by $X_{i\cdots j}=E_{i,j+1}$ and $Y_{i\cdots j}=E_{j+1,i}$ the root vectors in $\mathfrak{g}$ corresponding to the root $\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{j}\in\Delta^{+}$ and the root $-(\alpha_{i}+\alpha_{i+1}+\cdots+\alpha_{j})$ respectively, where $E_{ij}\in\mathrm{M}_{(n+1)\times(n+1)}(\mathbb{C})$.

Lemma 3.9**.**

The lowest weight vector of the subrepresentation $V(\theta)$ in Proposition 3.7 is

[TABLE]

Proof.

It is straightforward to verify $[Y_{\alpha_{i}},v]=0$ for all $\alpha_{i}\in\Pi$, and thus $v$ is the lowest weight vector. ∎

We next find the generator of $J$ corresponding to (17). Recall that a subalgebra $\mathfrak{p}\subseteq\mathfrak{g}$ such that $\mathfrak{p}\supseteq\mathfrak{b}$ is called a parabolic subalgebra. Let $\Pi^{\prime}\subset\Pi$, we define a parabolic subalgebra as follows: Let $\Delta_{\mathfrak{l}}=\{\gamma\in\Delta|\gamma=\sum_{\alpha\in\Pi^{\prime}}n_{\alpha}\alpha,n_{\alpha}\in\mathbb{Z}\}$, $\Delta_{\mathfrak{u}^{+}}=\{\alpha\in\Delta^{+}|\alpha\notin\Delta_{l}\}$. Then, let $\mathfrak{l}=\mathfrak{h}\oplus\oplus_{\alpha\in\Delta_{l}}\mathfrak{g}^{\alpha}$, $\mathfrak{u}=\oplus_{\alpha\in\Delta_{\mathfrak{u}^{+}}}\mathfrak{g}^{\alpha}$. We call $\mathfrak{p}=\mathfrak{l}\oplus\mathfrak{u}$ the parabolic subalgebra defined by $\Pi^{\prime}$. The following lemma is straightforward.

Lemma 3.10**.**

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and $\mathfrak{p}$ be a parabolic subalgebra defined by $\Pi-\{\alpha_{n}\}$. Suppose $\lambda\in\mathfrak{h}^{*}$. Then the following two conditions are equivalent:

1. $(1)$

$\lambda$* can be extended to a character on $\mathfrak{p}$, i.e., $\lambda|_{[\mathfrak{p},\mathfrak{p}]}=0,\lambda|_{\mathfrak{h}}=\lambda$;* 2. $(2)$

there exists a complex number $z\in\mathbb{C}$ such that $\lambda(h_{n})=z$, while $\lambda(h_{1})=\cdots=\lambda(h_{n-1})=0.$

Based on this lemma, Garfinkle showed the following.

Lemma 3.11** ([20, §IV.3 Proposition 3, §IV.6 Theorem 1 and §V Theorem 1]).**

Let $v\in V(\theta)^{\mathfrak{n}^{-}}$, $\mathfrak{p}$ be the parabolic subalgebra of $\mathfrak{g}$ defined by $\Pi-\{\alpha_{n}\}$, and $\lambda\in\mathfrak{h}^{*}$ satisfy the conditions in Lemma 3.10. Then there exists an element $y\in\mathcal{U}_{1}(\mathfrak{g})^{\mathfrak{n}^{-}}$ depending on $\lambda$ such that $\beta(v)-y\in I_{\mathfrak{p},\lambda}$. In this case, $\beta(v)-y\in J$.

More explicitly, we have that

[TABLE]

is an element in the Joseph ideal $J$.

Third, we find the generator of the Joseph ideal that corresponds to the Casimir element of $\mathfrak{g}$. Let us denote by $C$ the Casimir element. We have the following.

Lemma 3.12** ([20, §IV.3]).**

Let $\mathfrak{g}$ be the $A_{n}$ Lie algebra. Then

[TABLE]

is a generator of $J$, where $c_{\lambda}=\langle\lambda,\lambda\rangle+\langle\lambda,2\delta\rangle$ and $\delta$ is the half of the sum of positive roots.

Proof.

The Casimir element is $C=\sum_{\alpha\in\Pi}X_{\alpha}Y_{\alpha}+Y_{\alpha}X_{\alpha}+\sum_{i=1}^{n}h_{i}h_{i}^{\vee}$, where $n$ is the rank of the corresponding Lie algebra.

For Lie algebra of $A_{n}$, $2\delta=n\alpha_{1}+2(n-1)\alpha_{2}+\cdots+i(n-i+1)\alpha_{i}+\cdots+n\alpha_{n}$. By Lemma 3.10, we have $\lambda=z\lambda_{n}$. Thus

[TABLE]

By [20, §IV.3 §IV.6 Theorem 1 and §V Theorem 1], $C-c_{\lambda}$ is an element of $J$. ∎

By Garfinkle[20], $J$ depends on an element $z\in\mathbb{C}$; to specify its dependence on $z$, in what follows we shall write it as $J^{z}$. Summarizing the above lemmas, we have the following:

Theorem 3.13** ([20]).**

Let $\mathfrak{g}$ be the type A Lie algebra. $v_{0}$ is the lowest weight vector in Lemma 3.8. For each $z\in\mathbb{C}$, there is a Joseph ideal in $\mathcal{U}(\mathfrak{g})$, denoted by $J^{z}$, which is generated by (3.3.1), (3.12) and $\beta(v_{0})$, where $v_{0}$ is given in Lemma 3.8.

3.3.2 Joseph ideal for type D and E Lie algebras

Now we consider the Lie algebra $\mathfrak{g}$ of DE type. Let $\alpha$ be the simple root not orthogonal to the highest root $\theta$; in the case of type D and $E_{6}$, $E_{7}$, $E_{8}$, such an $\alpha$ is unique.

Proposition 3.14** (see [20], [8, §4.4] and [21]).**

Let $\mathfrak{g}$ be the complex semisimple Lie algebra of DE type. Let $\{\theta_{i}\}_{i}$ be the set of the highest roots of the complex semisimple Lie algebras obtained from $\mathfrak{g}$ by deleting $\alpha$ from the Dynkin diagram of $\mathfrak{g}$. Then we have the following decomposition of irreducible representations:

[TABLE]

For the type D Lie algebras, the unique simple root which is not perpendicular to $\theta$ is precisely the simple root $\alpha_{2}$, and thus we have the following:

Fact 3.15**.**

For the $D_{n}(n>4)$ Lie algebra $\mathfrak{g}$, we have the decomposition of irreducible representations:

[TABLE]

where $\theta{{}^{\prime}}=\alpha_{3}+2\alpha_{4}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}$ is the highest root of the Lie algebra corresponding to the sub-Dynkin diagram $D_{n-2}$ of $D_{n}$, which consists of the roots $\alpha_{3},\cdots,\alpha_{n}$.

For the $D_{4}$ Lie algebra, we have the decomposition of irreducible representations:

[TABLE]

For type E Lie algebras, we have the following.

Fact 3.16**.**

(1) For the $E_{6}$ Lie algebra $\mathfrak{g}$, $\alpha=\alpha_{2}$, and we have the following decomposition of representation:

[TABLE]

where $\theta$ is the highest root of Lie algebra of type $E_{6}$.

(2) For the $E_{7}$ Lie algebra $\mathfrak{g}$, we have the following decomposition of representation:

[TABLE]

where $\theta$ is the highest root of Lie algebra of type $E_{7}$, i.e., $\theta=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}$.

(3) For the $E_{8}$ Lie algebra $\mathfrak{g}$ we have the following decomposition of representation:

[TABLE]

where $\theta$ is the highest root of Lie algebra of type $E_{8}$, i.e., $\theta=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}+2\alpha_{8}$.

By Kostant (See [20] §III.2), the ideal $I(\overline{\mathcal{O}}_{min})$ is generated by the lowest weight vectors $v_{i}$ in each summand of $\oplus_{i}V(\theta+\theta_{i})$ and $C$ in $V(0)$. We have the following:

Theorem 3.17** ([20, §IV.3 Theorem 2, §IV.6 Theorem 1 and §V]).**

Let ${\mathfrak{g}}$ be the Lie algebra of type D or E, let $v_{i}$ be the lowest weight vector of the irreducible representation in $V(\theta+\theta_{i})$ in Proposition 3.14. Then the Joseph ideal $J$ is generated by $\beta(v_{i})$ and $C-c_{\lambda}$, where in the $D_{n}$ case, $c_{\lambda}=n^{2}-2n$, and in the $E_{6}$, $E_{7}$ and $E_{8}$ case, $c_{\lambda}=36,\ 84$ and $240$ respectively.

Remark 3.18**.**

According to [20, §IV.4, §IV.6 Theorem 1 and §V], in the $D_{n}$ case, $\lambda(h_{1})=-(n-2)$, $\lambda(h_{2})=\cdots=\lambda(h_{n})=0$. In the $E_{6}$ case, $\lambda(h_{6})=-3$, $\lambda(h_{1})=\cdots=\lambda(h_{5})=0$. In the $E_{7}$ case, $\lambda(h_{7})=-4$, $\lambda(h_{1})=\cdots=\lambda(h_{6})=0$. And in the $E_{8}$ case, $\lambda(h_{8})=-5$, $\lambda(h_{1})=\cdots=\lambda(h_{7})=0$. Recall that $c_{\lambda}=\langle\lambda,\lambda\rangle+\langle\lambda,2\delta\rangle$, where $\delta$ is the half of the sum of positive roots, we get the values of $c_{\lambda}$ in the above theorem.

3.4 The $B$-algebras

The notion of the $B$-algebra of a graded associative algebra is introduced by Braden et. al. in [6], which is the quantization of the fixed point scheme of a scheme with a torus action. It plays an essential role in the equivariant Hikita conjecture (see [31, 32, 33, 49]). In this section, we study the $B$-algebra of the quantizations of the minimal orbits obtained in the previous section.

Suppose ${\mathfrak{g}}$ is a simple Lie algebra, and $Q$ is the root lattice. Let $\mathcal{U}({\mathfrak{g}})$ be the universal enveloping algebra of ${\mathfrak{g}}$, and $J$ be the corresponding Joseph ideal. Recall that there is the PBW filtration of $\mathcal{U}({\mathfrak{g}})$:

[TABLE]

On the other hand, $\mathcal{U}({\mathfrak{g}})$ can be decomposed as

[TABLE]

Furthermore, the Joseph ideal $J$ satisfies the following

[TABLE]

Denote ${\mathscr{A}}={\mathscr{A}}[\overline{\mathcal{O}}_{min}]:=Rees(\mathcal{U}({\mathfrak{g}})/J)$, and then there is a weight decomposition induced by that of $\mathcal{U}({\mathfrak{g}})$,

[TABLE]

Definition 3.19**.**

The B-algebra of ${\mathscr{A}}[\overline{\mathcal{O}}_{min}]$ is defined to be

[TABLE]

The following lemma is immediate.

Lemma 3.20**.**

Let $\mathcal{I}$ be the ideal of $\mathcal{U}_{0}$ given by $\mathcal{I}:=\sum_{\mu>0}\{ab|a\in\mathcal{U}_{\mu},b\in\mathcal{U}_{-\mu}\}.$ Then B({\mathscr{A}})=Rees\big{(}(\mathcal{U}_{0}/\mathcal{I})/\big{(}J_{0}/(J_{0}\cap\mathcal{I})\big{)}\big{)}.

Now we describe $\mathcal{U}_{0}/\mathcal{I}$.

Lemma 3.21**.**

As a (commutative) algebra,

[TABLE]

Proof.

Notice that we have the following simple and important decomposition of $\mathcal{U}({\mathfrak{g}})$ (cf. [20]),

[TABLE]

For any $a\in\mathcal{U}_{0}$, we denote $a_{\mathfrak{h}}\in{\mathfrak{h}}$ to be the ${\mathfrak{h}}$-summand of $a$. Define a map

[TABLE]

Firstly, we show that $\kappa$ is well-defined. In fact, for any $a\in\mathcal{I}$, by the definition of $\mathcal{I}$, $a=a_{+}\cdot a_{-}$, where $a_{+}\in\mathcal{U}_{>0}$, $a_{-}\in\mathcal{U}_{<0}$. Without loss of generality, we assume $a_{+}\in\mathcal{U}_{>0}^{k}$ for some $k>0$. Now we perform the induction on $k$. For $k=1$, $a_{+}\in\mathfrak{n}^{+}$, so $a\in(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})$. Suppose for $k\leq r$, we still have $a\in(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})$. Now consider $a_{+}\in\mathcal{U}_{>0}^{r+1}$, then there exists $v\in\mathfrak{n}^{+}$, $b,c\in\mathcal{U}$, such that $a_{+}=b\cdot v\cdot c$. Then

[TABLE]

We have $v\cdot b\cdot c\cdot a_{-}\in\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})$. Since $[b,v]\cdot c\in\mathcal{U}_{>0}^{\leq r}$, by the assumption of induction, $[b,v]\cdot c\cdot a_{-}\in(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})$. Thus $a\in(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})$. This means that $\mathcal{I}\subseteq(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})$, $\kappa$ is well-defined.

The surjectivity of $\kappa$ is straightforward to check. The injectivity of $\kappa$ is induced by

[TABLE]

This follows from the fact that $\mathfrak{n}^{+}\subseteq\mathcal{U}_{>0}$, $\mathfrak{n}^{-}\subseteq\mathcal{U}_{<0}$. ∎

By Lemmas 3.20 and 3.21, we have

[TABLE]

where $J_{\mathfrak{h}}=\kappa(J_{0}/(J_{0}\cap\mathcal{I}))$. By the definition of $\kappa$, $J_{\mathfrak{h}}$ is just the projection of $J_{0}$ onto $\mathcal{U}({\mathfrak{h}})$ in (21).

Suppose $h_{1},h_{2},\cdots,h_{n}$ is a basis of ${\mathfrak{h}}$, then $\mathcal{U}({\mathfrak{h}})$ is a polynomial ring generated by $h_{1},\cdots,h_{n}$. The degree of the polynomials gives us a natural filtration of $\mathcal{U}({\mathfrak{h}})$,

[TABLE]

Lemma 3.22**.**

$J_{\mathfrak{h}}$* is an ideal of $\mathcal{U}({\mathfrak{h}})$ generated by the set $S_{J}$, where $S_{J}$ is the projection of $J_{0}\cap\mathcal{U}^{2}$ onto $\mathcal{U}({\mathfrak{h}})$ via (21).*

Proof.

Starting from $a_{\mathfrak{h}}\in J_{\mathfrak{h}}$, by the surjectivity of $\kappa$, there exists $a\in J_{0}$ such that $\kappa(a+\mathcal{I})=a_{\mathfrak{h}}$. By [20, Proposition III.3.2], we know that there exist $w\in J\cap\mathcal{U}^{2}$, $b,c\in\mathcal{U}({\mathfrak{g}})$ such that

[TABLE]

Without loss of generality, we assume $b\in\mathcal{U}^{i}$, $c\in\mathcal{U}^{j}$ for some $i,j\geq 0$. Now we claim that, $a=v+a_{1}\cdot w_{0}\cdot a_{2}$, where $w_{0}\in S_{J}$, $a_{1}$, $a_{2}\in\mathcal{U}({\mathfrak{h}})$ and $v\in(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})$. We prove this claim by induction on $k=i+j$.

1. (1)

For $k=0$, we have $b,c\in{\mathbb{C}}$. Therefore $a\in J_{0}\cap\mathcal{U}^{2}$, and it is easy to see the claim holds by (21). 2. (2)

Suppose for $k\leq r$, the claim holds. Now consider $k=r+1$.

1. (2i)

If $b,c\in\mathcal{U}({\mathfrak{h}})$, then $w\in J_{0}\cap\mathcal{U}^{2}$. By (21), we have $w=v+w_{0}$, where $v\in(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})\cap\mathcal{U}^{0}$ and $w_{0}\in S_{J}$. Recall that in the proof of Lemma 3.21, we showed that $(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})\cap\mathcal{U}_{0}=\mathcal{I}$, which is an ideal. Thus $a=bvc+bw_{0}c$ and $bvc\in(\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})+\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-})\cap\mathcal{U}_{0}$. The claim holds. 2. (2ii)

If $b\notin\mathcal{U}({\mathfrak{h}})$, there exits $x\in\mathfrak{n}^{+}$ or $\mathfrak{n}^{-}$ such that $b=b_{1}xb_{2}$. If $x\in\mathfrak{n}^{+}$, then

[TABLE]

Here $xb_{1}b_{2}wc\in\mathfrak{n}^{+}\mathcal{U}({\mathfrak{g}})$ and $[b_{1},x]b_{2}\in\mathcal{U}^{<i}$.

If $x\in\mathfrak{n}^{-}$, then

[TABLE]

Here $b_{1}b_{2}wcx\in\mathcal{U}({\mathfrak{g}})\mathfrak{n}^{-}$, and the remaining 3 terms in the last expression is of the form $\mathcal{U}^{<i}\cdot(J\cap\mathcal{U}^{2})\cdot\mathcal{U}^{j}$. So the claim holds by the assumption of induction. Notice that we can perform a similar discussion on $c$, and the claim holds in the case $k=r+1$.

By the above claim, we have $a_{\mathfrak{h}}=a_{1}w_{0}a_{2}$, and the proof of the lemma is complete. ∎

Lemma 3.22 together with (23) tells us that, to calculate $B({\mathscr{A}})$, we only need to calculate $J_{0}\cap\mathcal{U}^{2}$. Firstly we calculate its dimension.

Suppose $I={\rm gr}J=\bigoplus_{k\in{\mathbb{N}}}I^{k}$, where $I^{k}=(J\cap\mathcal{U}^{k})/(J\cap\mathcal{U}^{k-1})$. Then $I$ is an ideal of the polynomial ring ${\rm gr}(\mathcal{U}({\mathfrak{g}}))=\mathrm{Sym}({\mathfrak{g}})$, and the degree of elements in $I^{k}$ is $k$. For $\mu\in Q$, set $I^{k}_{\mu}$ be the component of $I^{k}$ with weight $\mu$.

Lemma 3.23**.**

With the above notations, we have

[TABLE]

Proof.

Since $J\neq\mathcal{U}({\mathfrak{g}})$, $1\notin J$, $J\cap\mathcal{U}^{0}=\{0\}$. Then as a vector space,

[TABLE]

By a theorem of Kostant (see [20, Theorem III.2.1]), as an ideal of $\mathrm{Sym}({\mathfrak{g}})$, $I$ is generated by $I^{2}$, which contains the homogenous elements with degree 2. Therefore $I^{1}=\{0\}$, and $J\cap\mathcal{U}^{1}=\{0\}$.

Now we consider $J\cap\mathcal{U}^{2}$, as a vector space,

[TABLE]

Since the projection $J\cap\mathcal{U}^{k}\rightarrow(J\cap\mathcal{U}^{k})/(J\cap\mathcal{U}^{k-1})$ is compatible with the decomposition (20), we have $J_{0}\cap\mathcal{U}^{2}\cong I^{2}_{0}$. ∎

3.4.1 Some calculations on ${\mathfrak{g}}$-modules

In this subsection, we calculate $\dim I^{2}_{0}$ for ADE Lie algebras. We have the following decomposition of ${\mathfrak{g}}$-module (see [20] or [45]).

Theorem 3.24** (Kostant).**

Suppose ${\mathfrak{g}}$ is a semisimple Lie algebra, and $\theta$ is the highest weight of the adjoint representation ${\mathfrak{g}}$. Then as a ${\mathfrak{g}}$-module,

[TABLE]

where $V(2\theta)$ is the irreducible representation of highest weight $2\theta$, and $L_{2}$ is a representation with underlying space $I^{2}$.

For a ${\mathfrak{g}}$-module $V$, denote by $V_{0}$ the subspace of $V$ with weight [math], then we have the following.

Lemma 3.25**.**

With the notations as above, we have:

[TABLE]

where

[TABLE]

Proof.

Just notice that all elements in $({\rm Sym}^{2}{\mathfrak{g}})_{0}$ is a linear combination of $x_{\mu}x_{-\mu}$, $x_{\mu}\in{\mathfrak{g}}_{\mu}$ and $h_{i}h_{j}$, $h_{i},h_{j}\in{\mathfrak{h}}$. ∎

The calculation of $\dim V(2\theta)_{0}$ is more difficult. The main tool we use is the following formula (see [24, Theorem 22.3]).

Lemma 3.26** (Freudenthal).**

Let $V=V(\lambda)$ be an irreducible ${\mathfrak{g}}$-module of highest weight $\lambda$. Let $\Lambda$ be the set of weights of $V$. For $\mu\in\Lambda$, set the multiplicity $m(\mu)$ to be the dimension of the weight space $V^{\mu}$, then $m(\mu)$ is given recursively as follows:

[TABLE]

where $\delta=\frac{1}{2}\sum_{\alpha\in\Delta^{+}}\alpha$.

Lemma 3.27**.**

For the ADE type Lie algebra ${\mathfrak{g}}$,

[TABLE]

Proof.

Notice that $\dim V(2\theta)_{0}$ is just the multiplicity $m(0)$ in $V(2\theta)$. We prove the lemma case by case.

The $A_{n}$ case: Firstly we list some data in $A_{n}$ case (see [24] or [29]).

[TABLE]

Since $2\theta=2(\varepsilon_{1}-\varepsilon_{n+1})$ is the highest weight of $V(2\theta)$, $m(2\theta)=1$. Since $2\theta=2(\varepsilon_{1}-\varepsilon_{n+1})$, and $m(\mu)$ is invariant under the $W$-action (see [24, Theorem 21.2]), we have

[TABLE]

Now we consider $m(2\varepsilon_{1}-\varepsilon_{n}-\varepsilon_{n+1})$. By (27), we have

[TABLE]

One can check that

[TABLE]

Therefore

[TABLE]

By the $W$-invariance of $m(\mu)$ and $m(\mu)=m(-\mu)$, we have

[TABLE]

Now we consider $m(\varepsilon_{1}+\varepsilon_{2}-\varepsilon_{n}-\varepsilon_{n+1})$. By (27), we have

[TABLE]

By (30), we have $m(2\varepsilon_{1}-\varepsilon_{n}-\varepsilon_{n+1})=m(\varepsilon_{1}+\varepsilon_{2}-2\varepsilon_{n+1})=1$. Furthermore,

[TABLE]

Thus $m(2\theta-\alpha_{1}-\alpha_{n})=1$ by the $W$-invariance of $m(\mu)$, and we have

[TABLE]

Now we calculate $m(\theta)$. By (27),

[TABLE]

By (29), (30) and (31), $m(\theta+\alpha)=1$. Furthermore, we have

[TABLE]

Then $m(\theta)=n$ and by the $W$-invariance of $m(\mu)$,

[TABLE]

Finally, by (27),

[TABLE]

By (29) and (32), we have $m(\alpha)=n$ and $m(2\alpha)=1$. Furthermore,

[TABLE]

Thus (33) is equivalent to

[TABLE]

which induces

[TABLE]

The $D_{n}$ case: The data of $D_{n}$ is as follows:

[TABLE]

The argument is similar to $A_{n}$, so we just list the result and omit the details:

[TABLE]

The type E case: By [10, §4], we know that for $E_{6}$, $m(0)=36$; for $E_{7}$, $m(0)=63$ and for $E_{8}$, $m(0)=120$. They are exactly $\displaystyle\frac{\dim{\mathfrak{g}}-\dim{\mathfrak{h}}}{2}$ in these cases.

In summary, in all the ADE cases, we have $\displaystyle m(0)=\frac{\dim{\mathfrak{g}}-\dim{\mathfrak{h}}}{2}$. ∎

Combining (25), (26) and (28), we have the following.

Proposition 3.28**.**

If ${\mathfrak{g}}$ is of ADE type, then $\dim I^{2}_{0}=\dim({\rm Sym}^{2}{\mathfrak{h}}).$

In conclusion, we have the following:

Theorem 3.29**.**

If ${\mathfrak{g}}$ is of ADE type, then

[TABLE]

where $S_{J}$ is a vector space defined in Lemma 3.22, with $\dim S_{J}\leq\dim({\rm Sym}^{2}{\mathfrak{h}})$.

Proof.

Follows from a combination of (23), Lemmas 3.22 and 3.23, and Proposition 3.28. ∎

4 The quantum Hikita conjecture

As we have mentioned before, the quantum Hikita conjecture was proposed by Kamnitzer, McBreen and Proudfoot in [31]. Under some conditions, the quantum Hikita conjecture implies the Hikita-Nakajima equivariant conjecture. In this section, we first recall the two main objects in the quantum Hikita conjecture, and then prove Theorems 1.2 and 1.3.

4.1 Specialized quantum D-module

Let $X$ be a conical symplectic variety. Assume that $X$ admits a $T\times\mathbb{G}_{m}$-equivariant projective symplectic resolution $\tilde{X}$. There is a natural short exact sequence (see [31, (6)])

[TABLE]

which is called the cohomology exact sequence. The image of an element $u\in\mathrm{H}^{2}_{T\times\mathbb{C}^{\times}}(\tilde{X};\mathbb{C})$ in $\mathrm{H}^{2}(\tilde{X};\mathbb{C})$ is henceforth denoted by $\bar{u}$.

Now let $\mathrm{H}_{2}(\tilde{X};\mathbb{Z})_{\mathrm{free}}$ be the quotient of $\mathrm{H}_{2}(\tilde{X};\mathbb{Z})$ by its torsion subgroup. Okounkov conjectured that there is a finite set $\Delta_{+}\subset\mathrm{H}_{2}(\tilde{X};\mathbb{Z})_{\mathrm{free}}$ and an element $L_{\alpha}\in\mathrm{H}^{2\dim X}(\tilde{X}\times_{X}\tilde{X};\mathbb{C})$ for each $\alpha\in\Delta_{+}$ such that

[TABLE]

for any $u\in\mathrm{H}^{2}_{T\times\mathbb{C}^{\times}}(\tilde{X};\mathbb{C})$, where $\star$ is the quantum product. The minimal such subset $\Delta_{+}$ is called the set of positive Kähler roots.

Remark 4.1**.**

Acording to (3) and (15), in our case, the set $\Delta_{+}$ above is identified with the set of positive roots (denoted by $\Delta^{+}$) of corresponding Lie algebra, after identifying $E_{i}$ with the simple roots $\alpha_{i}$. And Okounkov’s conjecture holds in our case, by setting $L_{\alpha}=-(pr_{1})^{*}\mathrm{PD}(\alpha)\cup(pr_{2})^{*}\mathrm{PD}(\alpha)$ for each $\alpha\in\Delta_{+}$, where $pr_{i}:\tilde{X}\times_{X}\tilde{X}\rightarrow\tilde{X}$ is the $i$-th projection, $i=1,2$, and $\mathrm{PD}(\alpha)$ is the Poincaré dual of $\alpha$. Also, $\hbar$ is $\dfrac{t_{1}+t_{2}}{2}$ in the type A case, and is $t$ in the type DE case.

We next introduce several algebraic structures. Let $F:=\mathbb{C}\{q^{\alpha}\,|\,\alpha\in\mathbb{Z}\Delta^{+}\}$ and F_{\mathrm{reg}}:=F\left[\frac{1}{1-q^{\alpha}}\Big{|}\,\alpha\in\Delta^{+}\right]. Let $E:=F\otimes\mathrm{Sym}\mathrm{H}^{2}_{T\times\mathbb{C}^{\times}}(\tilde{X};\mathbb{C})$ equipped with the multiplication satisfying $uq^{\alpha}=q^{\alpha}(u+\hbar\langle\alpha,\bar{u}\rangle)$ for all $\alpha\in\mathbb{N}\Delta_{+}$ and $u\in\mathrm{Sym}\mathrm{H}^{2}_{T\times\mathbb{C}^{\times}}(\tilde{X};\mathbb{C})$. Let $E_{\mathrm{reg}}$ be the Ore localization with respect to the multiplication set generated by $1-q^{\alpha}$ for $\alpha\in\Delta_{+}$ (it is argued in [31, §4.2] that the multiplicative set satisfies the Ore condition).

Let $Q_{\mathrm{reg}}(\tilde{X}):=F_{\mathrm{reg}}\otimes\mathrm{H}^{\bullet}_{T\times\mathbb{C}^{\times}}(\tilde{X};\mathbb{C})$. Then $E_{\mathrm{reg}}$ acts on $Q_{\mathrm{reg}}$ as follows: elements in $F_{\mathrm{reg}}\subset E_{\mathrm{reg}}$ acts by multiplication on the first tensor factor, while an element $u\in\mathrm{H}^{2}_{T\times\mathbb{C}^{\times}}(\tilde{X};\mathbb{C})$ acts by the operator $\hbar\partial_{\bar{u}}+u\star$, where $\partial_{\bar{u}}(q^{\alpha})=\langle\alpha,\bar{u}\rangle q^{\alpha}$.

Definition 4.2**.**

The specialized quantum D-module of $\tilde{X}$ is the $E_{\mathrm{reg}}$-module $Q_{\mathrm{reg}}(\tilde{X})$.

Remark 4.3**.**

Now let us move to the case of the minimal resolutions of ADE singularities $\widetilde{{\mathbb{C}^{2}}/\Gamma}$. In this case, the quantum cohomology is generated by the exceptional divisors, whose product contains no terms of degree higher than 2, and therefore $Q_{\mathrm{reg}}(\widetilde{{\mathbb{C}^{2}}/\Gamma})$ is nothing but the equivariant quantum cohomology algebra of $\widetilde{{\mathbb{C}^{2}}/\Gamma}$; see also [31, Remark 4.1]. Let $q=0$, then $Q_{\mathrm{reg}}(\widetilde{{\mathbb{C}^{2}}/\Gamma})$ becomes the equivariant cohomology ring.

4.2 D-module of graded traces

We now introduce the notion of D-module of graded traces. The general construction is quite complicated (see [31, §3] for details), and in this subsection, we only focus on the case that $X$ is the minimal nilpotent orbits in ADE type Lie algebras.

Let $X$ be a conical symplectic variety, suppose $T$ is a maximal torus of the Hamiltonian action on $X$. Then there is an exact sequence (see [31, §2.1]):

[TABLE]

where ${\mathscr{A}}^{1}_{0}$ denote the weight [math] degree $1$ part of ${\mathscr{A}}$. When $X=\mathcal{O}_{min}$ of type $A_{n}$, this exact sequence has the form:

[TABLE]

and when $X=\mathcal{O}_{min}$ of DE type, this exact sequence has the form:

[TABLE]

All the $\mathfrak{h}$ above are the Cartan subalgebra corresponding to the Lie algebra type.

Remark 4.4**.**

Choosing a splitting of the exact sequence above is equivalent to choosing a quantum comoment map $\mathrm{Sym}\mathfrak{h}\rightarrow\mathrm{Sym}{\mathscr{A}}_{0}^{1}$, we also denote by $h_{\alpha}$ the element in ${\mathscr{A}}^{1}_{0}$, which is the image of $h_{\alpha}\in\mathfrak{h}$ under this quantum comoment map.

For $X=\overline{\mathcal{O}}_{min}\subset\mathfrak{g}$, let $\Delta^{+}$ be the set of positive roots of $\mathfrak{g}$, which is called the equivariant roots of $X$. Let $S:=\mathbb{C}\{q^{\mu}|\mu\in\mathbb{N}\Delta^{+}\}$ and S_{\mathrm{reg}}:=S\bigg{[}\dfrac{1}{1-q^{\mu}}\bigg{|}\mu\in\Delta^{+}\bigg{]} be the localization of $S$. Let $R:=S\otimes\mathrm{Sym}\mathscr{A}_{0}^{1}$ be the $\mathbb{C}[\hbar]$-algebra with the multiplication satisfying $xq^{\mu}=q^{\mu}(x+\hbar\langle\lambda,\bar{x}\rangle)$, for all $\mu\in\mathbb{N}\Delta_{+}$ and $x\in\mathscr{A}_{0}^{1}$; here $\bar{x}\in\mathfrak{h}$ by the above exact sequence. Let $\mathfrak{S}\subset R$ be the multiplicative set generated by $1-q^{\mu}$ for all $\mu\in\Delta_{+}$; it is shown in [31, Lemma 3.4] that $\mathfrak{S}$ satisfies the Ore condition, and hence we may define the Ore localization $R_{\mathrm{reg}}:=\mathfrak{S}^{-1}R$, which, as a vector space, is isomorphic to $S_{\mathrm{reg}}\otimes\mathrm{Sym}\mathscr{A}_{0}^{1}$.

Now we endow $S\otimes{\mathscr{A}}_{0}$ with the structure of an $\mathbb{N}$-graded left $R$-module by putting

[TABLE]

for all $h_{\alpha}\in{\mathscr{A}}_{0}^{1}$, $a\in{\mathscr{A}}_{0}$ and $\gamma,\mu\in\Delta^{+}$. Let

[TABLE]

which turns out to be an $R$-submodule (see [31, Proposition 3.5]).

Definition 4.5** ([31, §3.3]).**

Let $X=\overline{O}_{min}$ be the closure of the minimal nilpotent orbit in $\mathfrak{g}$. Then the D-module of graded traces of $X$ is $M:=(S\otimes\mathscr{A}_{0})/\mathcal{I}_{q}$. Its localization $M_{\mathrm{reg}}:=R_{\mathrm{reg}}\otimes_{R}M$ is denoted by $Q(\mathscr{A}[\overline{O}_{min}])$ or simply $Q(\mathscr{A})$.

Since $Q(\mathscr{A})$ is the main object that we will be studying, in what follows we shall also call it the D-module of graded traces. Note that in $Q({\mathscr{A}})$, for $h_{\alpha}\in{\mathscr{A}}_{0}^{1},1\otimes a\in Q({\mathscr{A}})$, we have

[TABLE]

The elements in $Q({\mathscr{A}})$ satisfy

[TABLE]

Remark 4.6**.**

In fact, when $q=0$, the submodule $\mathcal{I}_{q}$ reduces to the two-sided ideal $\mathcal{I}$ of ${\mathscr{A}}_{0}$ in Lemma 3.20, and thus $Q(\mathscr{A})|_{q=0}$ is isomorphic to the B-algebra $B({\mathscr{A}})$ in Definition 3.19.

Remark 4.7**.**

Since the Poisson bracket on $\mathbb{C}[\overline{\mathcal{O}}_{min}]$ has degree $-1$, the algebra ${\mathscr{A}}^{1}_{0}$ plays the role of ${\mathscr{A}}^{2}_{0}$ in [31].

Recall that $Q(\mathscr{A})$ is finitely generated:

Lemma 4.8** ([31, Proposition 3.13]).**

If $B({\mathscr{A}})$ is finitely generated as a module over $\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar],$ then $Q(\mathscr{A})$ is finitely generated as a module over $S_{\mathrm{reg}}\otimes\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$.

We shall prove that $B({\mathscr{A}})$ is finitely generated by $\{1,h_{1},\cdots,h_{n}\}$ as a $\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$-module at the end of this subsection (see Corollary 4.22). From this we obtain the following.

Lemma 4.9**.**

$Q({\mathscr{A}})$* is finitely generated by $\{1,h_{1},\cdots,h_{n}\}$ as a free $S_{\mathrm{reg}}\otimes\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$-module, where $n=\mathrm{rank}(\mathfrak{h})$.*

Proof.

The proof is similar to the one of [31, Proposition 3.13]. Since $\{1,h_{1},\cdots,h_{n}\}$ generates $B({\mathscr{A}})$ as a $\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$-module, for any $a\in{\mathscr{A}}_{0}$, we have

[TABLE]

where $a_{j}\in{\mathscr{A}}_{\mu_{j}}$, $b_{j}\in{\mathscr{A}}_{-\mu_{j}}$, $\mu_{1},\cdots,\mu_{e}\in\Delta^{+}$, $r_{i}\in S_{\mathrm{reg}}\otimes\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$.

Since $[a_{j},b_{j}]=a_{j}b_{j}-b_{j}a_{j}=\hbar c$, $a_{j}b_{j}=q^{\mu}_{j}b_{j}a_{j}$, we have $a_{j}b_{j}=\dfrac{-q^{\mu_{j}}}{1-q^{\mu_{j}}}\hbar c$. On the other hand, the degree of $c$ is lower than the degree of $a$, and thus $a=\sum_{i=0}^{n}r_{i}h_{i}$ in $Q({\mathscr{A}})$, that is, $a$ is a combination of $h_{i}$ with coefficients in $S_{\mathrm{reg}}\otimes\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$.

Furthermore, if $h_{i}=\sum_{j=1,j\neq i}^{n}f_{j}h_{j}$ in $Q({\mathscr{A}})$, $h_{i}-\sum_{j=1,j\neq i}^{n}f_{j}h_{j}$ is an element in $S_{reg}\otimes\mathcal{I}_{q}$ or $S_{reg}\otimes J_{0}/(S_{reg}\otimes\mathcal{I}_{q}\cap S_{reg}\otimes J_{0})$, it is impossible. ∎

Thus we have

[TABLE]

where $\mathcal{J}_{\mathrm{reg}}:=S_{\mathrm{reg}}\otimes(J_{0}\cap\mathcal{U}^{2})$, which is a $S_{\mathrm{reg}}\otimes\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$-submodule of $S_{\mathrm{reg}}\otimes{\mathscr{A}}_{0}$.

In the rest of this subsection, we study the D-module of graded traces of the minimal nilpotent orbits in Lie algebras of ADE types. By Lemma 4.9, we only need to study the ${\mathscr{A}}_{0}^{1}$-action on all the generators of $Q({\mathscr{A}})$, i.e., the $n$ linear-independent elements like $1\otimes h_{\alpha^{\prime}}$ in $Q({\mathscr{A}})$.

Note that for $h_{\alpha}\in{\mathscr{A}}_{0}^{1},1\otimes h_{\alpha^{\prime}}\in Q({\mathscr{A}})$, we have

[TABLE]

Once one has fixed the choice of the comoment map as in Remark 4.4, the element $1\otimes h_{\alpha}h_{\alpha^{\prime}}$ in $Q({\mathscr{A}})$ comes from the element $h_{\alpha}h_{\alpha^{\prime}}$ in ${\mathscr{A}}_{0}$.

Theorem 4.10**.**

Let $h_{\alpha}\in{\mathscr{A}}_{0}^{1}$, $1\otimes h_{\alpha^{\prime}}\in Q({\mathscr{A}})$.

1. $(1)$

In the $D_{n}$ and $E_{6},E_{7}$ and $E_{8}$ case,

[TABLE] 2. $(2)$

In the $A_{n}$ case, once one has fixed the choice of $z$, then

[TABLE]

where $(-,-)_{c}$ is defined in Definition 2.6, and $\langle-,-\rangle$ is the positive definite symmetric bilinear form in the root system.

We prove this theorem in the rest of this subsection. We call (36) and ($(2)$) the quantum relations in $Q({\mathscr{A}})$. Notice that, for complex semisimple Lie algebra, there is an isomorphism

[TABLE]

where $K(-,-)$ is the Killing form. For Lie algebra of type ADE, the preimage of $\alpha$ under this map is $h_{\alpha}$, and thus we have $\langle\alpha,\alpha^{\prime}\rangle=K(h_{\alpha},h_{\alpha^{\prime}})$.

4.2.1 D-module of graded traces in the type DE cases

Suppose ${\mathfrak{g}}$ is a DE type Lie algebra, denote by $V(\lambda)$ the irreducible ${\mathfrak{g}}$ representations with highest weight $\lambda$. First let us recall the following well known result:

Lemma 4.11**.**

Let $C$ be the Casimir element of ${\mathfrak{g}}$. Then $C$ acts on $V(\lambda)$ as a scaling $c_{\lambda}=\langle 2\delta,\lambda\rangle+\langle\lambda,\lambda\rangle$, where $\delta$ is the half of the sum of positive roots.

Let us recall a result of Kostant (see [19]), which in fact holds for any Lie algebras:

Lemma 4.12**.**

Suppose $V=V_{1}\oplus V_{2}$ is a spliting of $\mathfrak{g}$-module, $C$ acts on $V_{1}$ as scaling $c_{1}$ and acts on $V_{2}$ as scaling different from $c_{1}$, then as a vector space, $V_{2}=Im(C-c_{1}\mathrm{Id})$.

Next, we have the following lemma.

Lemma 4.13**.**

$\sum_{\gamma\in\Delta^{+}}\langle\alpha,\gamma\rangle h_{\gamma}=\dfrac{1}{2}\langle 2\delta+\theta,\theta\rangle h_{\alpha}$, for any $h_{\alpha}\in\mathfrak{h}$.

Proof.

Consider the $\mathfrak{g}$-module $V(\theta)=\mathfrak{g}$, for any $\alpha\in\mathfrak{h}$, we have

[TABLE]

By Lemma 4.11 Casimir operator $C$ acts on $V(\lambda)$ as a scaling $c_{\lambda}=\langle 2\delta,\lambda\rangle+\langle\lambda,\lambda\rangle$, we have

[TABLE]

Lemma 4.14**.**

Let $(L_{2})_{0}$ be the weight 0 subspace of $L_{2}$ in Theorem 3.24. Then $(L_{2})_{0}$ is spanned by vectors of the following form:

[TABLE]

where $\beta_{1},\beta_{2}\in\mathfrak{h}^{*}$.

Proof.

For any two distinct simple roots $\alpha_{i},\alpha_{j}\in\Pi$, let

[TABLE]

Since these vectors are linearly independent, and $\#\{h_{i}h_{j}\}_{1\leq i\leq j\leq n}=\dim\mathrm{Sym}^{2}\mathfrak{h}=\dim(L_{2})_{0}$, to show the lemma it suffices to show $v_{ij}\in(L_{2})_{0}$.

In fact, we have

[TABLE]

By Lemma 4.13, we have

[TABLE]

Set $V(2\theta)=V_{1}$, $L_{2}=V_{2}$ in Lemma 4.12, we know $v_{ij}\in L_{2}$, and the weight of $v_{ij}$ is [math], thus $v_{ij}\in(L_{2})_{0}$. ∎

By Proposition 3.28 and Lemma 4.14 , we have the following:

Corollary 4.15**.**

There is an isomorphism of vector spaces:

[TABLE]

Considering the Killing form $K(-,-):\mathfrak{h}\otimes\mathfrak{h}\rightarrow\mathbb{C}$, since it is a symmetric bilinear form, it induces a linear functional $K:\mathrm{Sym}^{2}\mathfrak{h}\rightarrow\mathbb{C}$. Recall that the subrepresentation $L_{2}$ has a further $\mathfrak{g}$-module decomposition

[TABLE]

where $W_{2}=\oplus_{i}V(\lambda_{i})$ with $c_{\lambda_{i}}\neq 0$ and $V(\lambda_{i})$ is the highest weight representation. (See Facts 3.15 and Fact 3.16). We then have the following:

Lemma 4.16**.**

Suppose $v\in(W_{2})_{0}$, then $K\circ\Psi^{-1}(v)=0$.

Proof.

Set $V_{1}=V(0)$, $V_{2}=W_{2}$ in Lemma 4.12, we know that $v=C\cdot u$ for some $u\in(L_{2})_{0}$. By Lemma 4.14, without loss of generality, assume $u=h_{\beta_{1}}h_{\beta_{2}}+\sum_{\gamma\in\Delta^{+}}\langle\beta_{1},\gamma\rangle\langle\beta_{2},\gamma\rangle X_{\gamma}Y_{\gamma}$, then

[TABLE]

Now to show the lemma it suffices to check that

[TABLE]

In fact, by Lemma 4.13, we have

[TABLE]

The lemma follows. ∎

Corollary 4.17**.**

$\Psi|_{\ker K}:\ker K\rightarrow(W_{2})_{0}$* is an isomorphism of vector spaces.*

Proof.

Since $K$ is surjective, $\dim\ker K=\dim\mathrm{Sym}^{2}\mathfrak{h}-1$. On the other hand, $(L_{2})_{0}=(W_{2})_{0}\oplus V(0)$, and $\dim V(0)=1$, so by Proposition 3.28, $\dim(W_{2})_{0}=\dim(L_{2})_{0}-1=\dim\ker K$. Then by Lemma 4.16, \Psi^{-1}\big{(}(W_{2})_{0}\big{)}\subseteq\ker K, and thus $\Psi^{-1}|_{\ker K}$ is an isomorphism. ∎

Now, we fix the notation $J_{0}^{2}:=J_{0}\cap\mathcal{U}^{2}$. Recall that $\beta$ is the symmetrization map given in Example 3.4.

Lemma 4.18**.**

If $v\in(W_{2})_{0}$, then $\beta(v)\in J^{2}_{0}$.

Proof.

Suppose $v\in(W_{2})_{0}$. Recall that $W_{2}=\oplus_{i=1}^{k}V(\lambda_{i})$, and thus $v=v_{1}+\cdots+v_{k}$ such that $v_{i}\in V(\lambda_{i})$.

Since $V(\lambda_{i})$ is irreducible, there exists an element $T_{i}\in\mathcal{U}({\mathfrak{g}})$ such that $v_{i}=T_{i}\cdot v_{-\lambda_{i}}$. Thus $v=\sum_{i=1}^{k}T_{i}(v_{-\lambda_{i}})$, Since by Theorem 3.17, $\beta(v_{-\lambda_{i}})\in J_{0}^{2}$, we have

[TABLE]

Proof of Theorem 4.10 (1).

By Corollary 4.15, we can choose a basis of $\mathrm{Sym}^{2}\mathfrak{h}$: $\phi_{1},\cdots,\phi_{l}$ for $l=\dim\mathrm{Sym}^{2}\mathfrak{h}$, such that $\Psi(\phi_{1}),\cdots,\Psi(\phi_{l-1})$ are a basis of $(W_{2})_{0}$ and $\Psi(\phi_{l})$ is a basis of $V(0)$ such that $\beta(\Psi(\phi_{l}))=C$.

By Lemma 4.18, \beta\big{(}\Psi(\phi_{i})\big{)}\in J_{0}^{2} for $i=1,\cdots,l-1$. Then for $i=1,\cdots,l-1$, without loss of generality, assume \beta\big{(}\Psi(\phi_{i})\big{)}\in J_{0}^{2} has the form

[TABLE]

Thus by (34) we have the relations

[TABLE]

in the D-module $Q({\mathscr{A}})$. By Lemma 4.16, $K(\phi_{i})=0$ implies $\langle\beta_{1},\beta_{2}\rangle=0$, the above equation coincides with (36), i.e., it satisfies the quantum relation.

Now for $\Psi(\phi_{l})=C$, it gives the element in the Joseph ideal

[TABLE]

By (34) it descends to

[TABLE]

in the D-module $Q({\mathscr{A}})$. Combining the table in §2.1 with Theorem 3.17, we get $\dfrac{c_{\lambda}}{n}=\dfrac{|\Gamma|}{4}$, where $n=rank(\mathfrak{h})$, and then by a direct calculation, the above element gives (36). ∎

4.2.2 D-module of graded traces in the type A case

Considering the bilinear map $(-,-)_{c}:\mathfrak{h}\otimes\mathfrak{h}\rightarrow\mathfrak{h}$, since it is a symmetric, it induces a linear map $K_{c}:\mathrm{Sym}^{2}\mathfrak{h}\rightarrow\mathfrak{h}$. Recall that from §3.3.1, the subrepresentation $L_{2}$ has a further $\mathfrak{g}$-module decomposition

[TABLE]

Then we have the following lemma:

Lemma 4.19**.**

Suppose $v\in V(\theta+\alpha)_{0}\oplus V(0)_{0}$ in Proposition 3.7, then $K_{c}\circ\Psi^{-1}(v)=0$.

Proof.

Set $V(\theta)=V_{1}$, $V(\theta+\alpha)\oplus V(0)=V_{2}$ in Lemma 4.12, we have

[TABLE]

for some $u\in L_{2}$.

By Lemma 4.14, without loss of generality, assume

[TABLE]

then we have:

[TABLE]

We only need to check the following identity on a basis of $\mathrm{Sym}^{2}\mathfrak{h}$:

[TABLE]

In fact, for $h_{\beta_{1}}h_{\beta_{2}}=h_{\alpha_{i}}h_{\alpha_{j}}$ ($j-i\geq 2$), $h_{\beta_{1}}h_{\beta_{2}}=h_{\alpha_{i}}h_{\alpha_{i+1}}$ ($1\leq i\leq n-1$) and $h_{\beta_{1}}h_{\beta_{2}}=h_{\alpha_{i}}^{2}$ ($1\leq i\leq n$) respectively, the above identity is a straightforward calculation, and is left to the readers. ∎

Now consider a subspace $J_{\theta}\subseteq J_{0}^{2}$, which is defined by

[TABLE]

We have:

Lemma 4.20**.**

For $v\in V(\theta)_{0}$, there is a unique $y_{v}\in\mathcal{U}^{1}({\mathfrak{g}})\cap\mathcal{U}_{0}({\mathfrak{g}})\cong\mathfrak{h}$ such that $\beta(v)+y_{v}\hbar$ is in $J_{\theta}.$

Proof.

Since $V(\theta)$ is irreducible, there exists an element $L\in\mathcal{U}({\mathfrak{g}})$ such that $v=L(v_{-\theta})$. Hence $\beta(v)=\beta\circ L(v_{-\theta})=L(\beta(v_{-\theta}))$. By Lemma 3.11, $L(\beta(v_{-\theta})-y\hbar))\in J_{\theta}$. Denote by $y_{v}:=-L(y)$, then $\beta(v)+y_{v}\in J_{\theta}$.

Assume there is another $\tilde{y}_{v}$ such that $\beta(v)+\tilde{y}_{v}\in J_{\theta}$, then $y_{v}-\tilde{y}_{v}\in J_{0}^{1}$. However, from the proof of Lemma 3.23, $J_{0}^{1}=J_{0}\cap\mathcal{U}^{1}\subseteq J\cap\mathcal{U}^{1}=\{0\}$, we get $\tilde{y}_{v}=y_{v}$. ∎

By Lemma 4.20, we get a symmetric bilinear map

[TABLE]

and then we have:

Lemma 4.21**.**

Restricted on \Psi^{-1}\big{(}V(\theta)_{0}\big{)}, $K_{Q}=-(\dfrac{n+1}{2}+z)K_{c}$.

Proof.

Recall that we have obtained $\beta(v_{-\theta})-y$ in (3.3.1). For $k=2,3,\cdots,n$, by applying $ad_{X_{12\cdots k-1}}\circ ad_{X_{k,k+1,\cdots n}}$ to (3.3.1), we obtain

[TABLE]

Then,

[TABLE]

On the other hand,

[TABLE]

Comparing the above two terms, we see that they differ up to a scaler $-(\frac{n+1}{2}+z)$.

Now for $k=1$, denote $ad_{X_{12\cdots k-1}}=\mathrm{Id}$, by applying $ad_{X_{12\cdots n}}$ to (3.3.1), we obtain the same result.

Observe that the following $n$ vectors in $\mathrm{Sym}^{2}\mathfrak{h}$ are linearly independent

[TABLE]

and $\dim\Psi^{-1}(V(\theta)_{0})=\dim(V(\theta)_{0})=n$. Since $V(\theta)_{0}\cong\mathfrak{h}$ and $v_{1},v_{2},\cdots,v_{n}$ is a basis of $\Psi^{-1}(V(\theta)_{0})$, we have $K_{Q}=-(\dfrac{n+1}{2}+z)K_{c}$ on the whole vector space \Psi^{-1}\big{(}V(\theta)_{0}\big{)}. ∎

Proof of Theorem 4.10 (2).

We prove the theorem on a basis of $\mathrm{Sym}^{2}(\mathfrak{h})$. By Corollary 4.15, we can choose a basis of $\mathrm{Sym}^{2}\mathfrak{h}$: $\phi_{1},\cdots,\phi_{l}$, where $l=\dim(n^{2}-n-2)/2$, and $\psi_{1},\cdots,\psi_{n}$, such that $\Psi(\phi_{1}),\cdots,\Psi(\phi_{l-1})$ are a basis of $V(\theta+\alpha)_{0}$, $\Psi(\phi_{l})$ is a basis of $V(0)_{0}$ such that $\beta(\Psi(\phi))=C$, and $\Psi(\psi_{1}),\cdots,\Psi(\psi_{n})$ is a basis of $V(\theta)_{0}$.

First, for $u\in V(\theta+\alpha)_{0}$, $\beta(u)\in J_{0}^{2}$ has the form

[TABLE]

which induces the relation

[TABLE]

in $Q({\mathscr{A}})$. By Lemma 4.16, if $u\in V(\theta+\alpha)_{0}\oplus V(\theta)_{0}$, then $\langle\beta_{1},\beta_{2}\rangle=0$; by Lemma 4.19, if $u\in V(\theta+\alpha)_{0}\oplus V(0)_{0}$, then $(\beta_{1},\beta_{2})_{c}=0$. Thus the above equation is exactly ($(2)$).

Second, by Lemma 4.21, if $\psi_{j}\in V(\theta)_{0}$, then $K_{Q}(h_{\beta_{1}},h_{\beta_{2}})=-(\dfrac{n+1}{2}+z)(\beta_{1},\beta_{2})_{c}$. By Lemma 4.20, we know that $\beta(\psi_{j})+y_{\Psi(\psi_{j})}\in J_{0}^{2}$. The image of $\beta(\psi_{j})+y_{\Psi(\psi_{j})}$ in $Q({\mathscr{A}})$ is

[TABLE]

which also coincides with ($(2)$).

At last, it is straightforward to see $\beta(\Psi(\phi))-c_{\lambda}$ gives the relation ($(2)$). ∎

As a corollary to Theorem 4.10, we have the following.

Corollary 4.22**.**

Let $h_{\alpha},\ h_{\alpha^{\prime}}\in B({\mathscr{A}})$.

1. $(1)$

In the $D_{n}$ and $E_{6},E_{7}$ and $E_{8}$ case,

[TABLE] 2. $(2)$

In the $A_{n}$ case, once one has fixed the choice of $z$, then

[TABLE]

where $(-,-)_{c}$ is defined in Definition 2.6, and $\langle-,-\rangle$ is the positive definite symmetric bilinear form in the root system.

Proposition 4.23**.**

$B({\mathscr{A}})$* is finitely generated by $\{1,h_{1},\cdots,h_{n}\}$ as a $\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$-module. More precisely, when $\mathfrak{g}$ is of type A, $B({\mathscr{A}})$ is finitely generated as a $\mathbb{C}[z,\hbar]$-module, and when $\mathfrak{g}$ is of DE type, $B({\mathscr{A}})$ is finitely generated as a $\mathbb{C}[\hbar]$-module.*

Proof.

By Theorem 3.29, $\dim S_{J}\leq\dim({\rm Sym}^{2}{\mathfrak{h}}),$ and from Corollary 4.22, for $D_{n}$ and $E_{n}$ case

[TABLE]

So $\dim S_{J}=\dim({\rm Sym}^{2}{\mathfrak{h}})$, which is spanned by the above elements. By Theorem 3.29, we see that $B({\mathscr{A}})$ is finitely generated by $\{1,h_{1},\cdots,h_{n}\}$ as a $\mathrm{Sym}\mathrm{H}_{2}(\widehat{X},\mathbb{C})\otimes\mathbb{C}[\hbar]$-module. The argument of the $A_{n}$ case is similar. ∎

4.3 Proof of Theorems 1.2 and 1.3

The quantum Hikita conjecture, proposed by Kamnitzer, McBreen and Proudfoot in [31], is stated as follows.

Conjecture 4.24**.**

Suppose $X$ and $X^{!}$ are symplectic dual to each other. Then there is an isomorphism $M_{\mathrm{reg}}(X)\cong Q_{\mathrm{reg}}(X^{!})$ of graded modules over $R_{\mathrm{reg}}(X)\cong E_{\mathrm{reg}}(X^{!})$ sending $1\in M_{\mathrm{reg}}(X)$ to $1\in Q_{\mathrm{reg}}(X^{!})$.

The Kleinian singularities and the minimal nilpotent orbits are expected to be symplectic dual to each other. Recall that by Remark 4.3, in the ADE singularities case, the quantum D-module is nothing but the equivariant quantum cohomology, and thus the above conjecture is exactly Theorem 1.2 of the current paper.

Proof of Theorem 1.2.

In the $A_{n}$ case, let

[TABLE]

By comparing (15) with ($(2)$), we get the isomorphism. In the DE case, let

[TABLE]

By comparing (3) with (36), we get the isomorphism. ∎

Proof of Theorem 1.3.

Let $q=0$. Then the quantum D-modules reduces to the equivariant cohomology by Remark 2.12, and the D-module of graded traces $Q({\mathscr{A}})$ reduces to $B({\mathscr{A}})$ by Remark 4.6. Thus Theorem 1.2 reduces to Theorem 1.3. ∎

Remark 4.25**.**

In the above two theorems, for type A Lie algebras, we have assumed the Joseph ideals $J^{z}$ are parametrized by the formal parameter $z$, which is also called the Kähler parameter. Now, if we view $z$ as a complex number, then (40) would not give an isomorphism of two algebras unless $t_{1}=t_{2}$ and $z=-(n+1)/2$. That means, if we specify $z=-(n+1)/2$ in the quantization of the minimal nilpotent orbits in the $A_{n}$ Lie algebra, then for all ADE singularities, the isomorphisms in the above theorems have a uniform form

[TABLE]

5 Generalization to BCFG type singularities

In this section, we generalize the isomorphisms in Theorems 1.2 and 1.3 to the cases of BCFG type singularities.

First recall that for the ADE type Lie algebras, the minimal nilpotent orbits are the same as the minimal special nilpotent orbits, but for the other types of Lie algebras, they are different. The Lusztig-Spaltenstein duality says that for BCFG type Lie algebras, the minimal special orbits are dual to the subregular nilpotent orbits. Also, for BC type Lie algebras, they are Langlands dual to each other, and for simple Lie algebras of the rest types, they are Langlands self-dual.

We shall that the normalization of the minimal special nilpotent orbit in one type Lie algebra is symplectic dual to the intersection of a Slodowy slice to the subregular nilpotent orbit with the nilpotent cone in its Langlands dual, where the latter is called a simple singularity of BCFG type by Slodowy [44].

5.1 Quantization of the minimal special nilpotent orbits

In this subsection, we denote by $\mathcal{O}_{ms}$ the minimal special nilpotent orbit in a Lie algebra of BCFG type. A theorem of Panyushev in [43] says that the normalization $\tilde{\mathcal{O}}_{ms}$ of the closure of $\mathcal{O}$ has symplectic singularities, and hence it makes sense to study the symplectic duality for $\tilde{\mathcal{O}}_{ms}$.

Let $\mathfrak{g}$ be a simple Lie algebra. Automorphisms of the Dynkin diagram of $\mathfrak{g}$ induce automorphisms of the root vectors of $\mathfrak{g}$ and hence of $\mathfrak{g}$. Dynkin diagrams with nontrivial automorphisms are only those of $A_{n}(n>1)$, $D_{n}(n>2)$ and $E_{6}$. They are given by (c.f. [29, §7.9]):

1. $-$

For $A_{n}$, the automorphism group is $\mathbb{Z}_{2}$ with the nontrivial automorphism given by $\alpha_{i}\mapsto\alpha_{n+1-i}$; 2. $-$

For $D_{4}$, there are two types of automorphisms, the first automorphism group is $\mathfrak{S}_{3}$ which is generated by the permutations of $\alpha_{1},\alpha_{3}$ and $\alpha_{4}$, while the second automorphism group is $\mathbb{Z}_{2}$ given by permuting $\alpha_{3}$ and $\alpha_{4}$. For $D_{n+1}$, $n\geq 2$ and $n\neq 3$, the automorphism group is $\mathbb{Z}_{2}$, which is given by permuting $\alpha_{n}$ and $\alpha_{n+1}$. 3. $-$

For $E_{6}$, the automorphism is $\mathbb{Z}_{2}$, given by permuting $\alpha_{1}$ and $\alpha_{6}$, $\alpha_{3}$ and $\alpha_{5}$, with $\alpha_{2}$ and $\alpha_{4}$ fixed.

Taking the quotients of the Dynkin diagrams by the above group actions, we obtain the Dynkin diagrams of Lie algebras of the other type, which is summarized in the following table:

[TABLE]

The above actions lift to automorphisms $\nu$ of $\mathfrak{g}$ defined by $\nu(X_{\alpha})=X_{\nu(\alpha)}$ and $\nu(Y_{\alpha})=Y_{\nu(\alpha)}$. The following theorem is due to Brylinski and Kostant:

Theorem 5.1** ([12]).**

The minimal nilpotent orbit $\mathcal{O}_{min}$ in $D_{n+1}$, $A_{2n-1}$ and $E_{6}$ is the double cover of the minimal special nilpotent orbit $\mathcal{O}_{ms}$ of type $B_{n}$, $C_{2n}$ and $F_{4}$ respectively, while the minimal nilpotent orbit $\mathcal{O}_{min}$ in $D_{4}$ is the 6-fold cover of minimal special nilpotent orbit $\mathcal{O}_{ms}$ in $G_{2}$ with the deck transformation $\mathfrak{S}_{3}$.

The result is summarized by the following table:

[TABLE]

In other words, we have the following isomorphisms:

[TABLE]

In [23], Huang studied the quantization of the minimal special nilpotent orbits in these Lie algebras. For Lie algebras of DE type, the Joseph ideals are maximal and are stable under the actions of $\mathbb{Z}_{2}$ or $\mathfrak{S}_{3}$. For Lie algebras of type A, the Joseph ideals $J^{z}$ are not unique, but we have the following.

Proposition 5.2**.**

Suppose $\mathfrak{g}$ is the Lie algebra $A_{2n-1}$. Let $J^{z}$ be the Joseph ideals parameterized by $z\in\mathbb{C}$. Then there is a unique Joseph ideal which is stable under the action of $\mathbb{Z}_{2}$. More precisely, such an ideal is $J^{-n}$ in Theorem 3.13.

Proof.

For the $A_{2n-1}$ Lie algebra, recall from Theorem 3.13 that its Joseph ideals are generated by three types of elements $\beta(v_{0})$, (3.3.1) and (3.12). It is direct to check that, under the $\mathbb{Z}_{2}$-action, $\beta(v_{0})$ and (3.12) are mapped to elements in $J^{z}$. Applying the nontrivial element of $\mathbb{Z}_{2}$ to (3.3.1), which is now

[TABLE]

we obtain

[TABLE]

Comparing (5.1) with (5.1) we see that (5.1) is an element in $J^{z}$ if and only if $z=-n$. ∎

Convention 5.3** (Compare with Remark 4.25).**

From now on, we take the Joseph ideal for $A_{2n-1}$ to be $J^{-n}$. And on the dual side, the equivariant cohomology of the minimal resolution of the $A_{2n-1}$ singularity is the $\mathbb{C}^{\times}$-equivariant cohomology.

Let us continue our study of the quantization of these nilpotent orbits. The following theorem is due to Huang:

Lemma 5.4** (Huang [23, page 318]).**

The the following algebras

[TABLE]

are the filtered quantizations of $\tilde{\mathcal{O}}_{ms}(B_{n}),\tilde{\mathcal{O}}_{ms}(C_{n}),\tilde{\mathcal{O}}_{ms}(F_{4}),\tilde{\mathcal{O}}_{ms}(G_{2})$ respectively.

Now we study the D-module of graded traces $Q(\mathscr{A})$ for these nilpotent orbits. Let $\Lambda$ be the index of simple roots of an ADE type Lie algebra $\mathfrak{g}$. Denote by $\bar{\mathfrak{g}}$ the Lie algebra corresponding to the Dynkin diagram obtained from the one of $\mathfrak{g}$ by modulo its automorphism described above. Denote the automorphism group by $\Phi$. The simple roots of $\bar{\mathfrak{g}}$ is indexed by $\bar{\Lambda}:=\Lambda/\Phi$. Then there is a projection of roots

[TABLE]

And we have an isomorphism (see [29, Proposition 7.9])

[TABLE]

Now in the $Q(\mathscr{A})$ of the minimal special nilpotent orbit in $\mathfrak{g}$, specifying $q^{\alpha}=q^{[\alpha]}$, we obtain the following.

Proposition 5.5**.**

Denote by $\tilde{\mathcal{O}}_{ms}(B_{n})$, $\tilde{\mathcal{O}}_{ms}(C_{n})$, $\tilde{\mathcal{O}}_{ms}(F_{4})$, $\tilde{\mathcal{O}}_{ms}(G_{2})$ the normalization of the minimal special nilpotent orbits in Lie algebras of BCFG type respectively. Then we have the following isomorphisms of D-modules of graded traces:

[TABLE]

Proof.

Denote the above isomorphisms to be

[TABLE]

To avoid confusion, we denote by $\mathcal{I}_{[q]}$ the submodule

[TABLE]

(1) Firstly, we prove that $(S\otimes(\mathscr{A}_{ADE})_{0})/\mathcal{I}_{q}\cong(S\otimes(\mathscr{A}_{ADE})_{[0]})/\mathcal{I}_{[q]}$, i.e.,

[TABLE]

In fact, by the weight decomposition, we have $(\mathscr{A}_{ADE})_{[0]}=(\mathscr{A}_{ADE})_{0}\oplus\mathscr{A}^{\prime}$. If we take $a\in S_{reg}\otimes(\mathscr{A}_{ADE})_{[0]}$, we have $a=a_{0}+a^{\prime}$ for $a_{0}\in S_{reg}\otimes(\mathscr{A}_{ADE})_{0}$, $a^{\prime}\in\mathscr{A}^{\prime}$. Now, denote by $(\mathscr{A}^{\prime})^{k}$ the degree $k$ part of $\mathscr{A}^{\prime}$. We claim that $a^{\prime}\in S_{reg}\otimes\mathcal{I}_{[q]}$ and prove this claim by induction on $k$.

1. (1a)

For $k=0$, it is easy to see the claim holds. 2. (1b)

Suppose for $a^{\prime}\in S_{reg}\otimes(\mathcal{A}^{\prime})^{k-1}$, the claim holds. Now consider $a^{\prime}\in S_{reg}\otimes(\mathcal{A}^{\prime})^{k}$. Take $a_{\mu}a_{-\eta}\in S_{reg}\otimes(\mathcal{A}^{\prime})^{k}$, then we have

[TABLE]

Since $[a_{\mu},a_{-\eta}]\in S_{reg}\otimes(\mathcal{A}^{\prime})^{k-1}$ and $a_{\mu}a_{-\eta}-q^{[\mu]}a_{-\eta}a_{\mu}\in I_{[q]}$, the claim holds for $a^{\prime}\in S_{reg}\otimes(\mathcal{A}^{\prime})^{k}$. Then we know that the morphism is surjective. It is easy to see that the morphism is injective.

(2) Now we prove $Q(\mathscr{A}_{BCFG})\cong\left(Q(\mathscr{A}_{ADE})|_{q^{\alpha}=q^{[\alpha]}}\right)^{\nu}$.

Considering the associated graded algebra of the B-algebra of type BCFG, we have 1+rank\ \mathfrak{g}_{BCFG}=dim\ gr(\mathcal{U}_{0})/\big{(}grJ_{0}+gr\mathcal{I}\big{)}\geq dim\ gr(\mathcal{U}_{0})/gr(J_{0}+\mathcal{I})=dim\ grB({\mathscr{A}}_{BCFG})

On the other hand, $dim\ grB({\mathscr{A}}_{BCFG})\geq dim\ grB({\mathscr{A}}_{ADE})^{\nu}=1+rank\ \mathfrak{g}_{BCFG}$. We have $dim\ grB({\mathscr{A}}_{BCFG})=dim\ grB({\mathscr{A}}_{ADE})^{\nu}$, thus $B({\mathscr{A}}_{BCFG})\cong B({\mathscr{A}}_{ADE})^{\nu}$.

The calculation of $Q(\mathscr{A}_{BCFG})$ and $\left(Q(\mathscr{A}_{ADE})|_{q^{\alpha}=q^{[\alpha]}}\right)^{\nu}$ is similar to $B({\mathscr{A}}_{BCFG})$ and $B({\mathscr{A}}_{ADE})^{\nu}$.

∎

The quantum action has the form as following:

[TABLE]

in Q\big{(}Rees(\mathcal{U}(\mathfrak{g})/J)\big{)}^{\nu}, where $\gamma^{\vee}=\dfrac{2}{\langle\gamma,\gamma\rangle}\gamma$.

Remark 5.6**.**

In [37], Losev showed that the moduli spaces of these quantizations are isomorphic to $\mathrm{H}^{2}(\widehat{\mathcal{O}}_{ms},\mathbb{C})$, where $\widehat{\mathcal{O}}_{ms}$ denotes the smooth loci of the $\mathbb{Q}$-terminalizations of $\widetilde{\mathcal{O}}_{ms}$. By the works of Fu [17] and Namikawa [41], for Lie algebras of types CDEFG, $\widehat{\mathcal{O}}_{ms}=\mathcal{O}_{ms}$, and therefore the quantizations of these nilpotent orbits are unique since $\mathrm{H}^{2}(\mathcal{O}_{ms},\mathbb{C})\cong\{0\}$ (see [2] and [27]). Thus these quantizations are isomorphic to the ones presented in the paper. For type A Lie algebras, since $\widehat{\mathcal{O}}_{ms}\cong T^{*}\mathbb{P}^{n}$ by [16], the quantizations of their minimal nilpotent orbits are parameterized by $\mathrm{H}^{2}(T^{*}\mathbb{P}^{n},\mathbb{C})\cong\mathbb{C}$, and hence are also isomorphic to the ones given in §3.3.1. For type B Lie algebras, we have only given a quantization of the minimal special nilpotent orbits that comes from the one in Lie algebras of type D. However, in this case, the quantizations are not unique, since the moduli space is $\mathrm{H}^{2}(\widehat{\mathcal{O}}_{ms},\mathbb{C})\cong\mathrm{H}^{2}(T^{*}\mathbb{Q}^{2n-1},\mathbb{C})\cong\mathbb{C}$ by [16] (see also [18, Example 1.1]).

5.2 Equivariant cohomology of the minimal resolutions

Now let us turn to the Slodowy slices in the BCFG type Lie algebras. In [44], Slodowy showed that the intersections of Slodowy slices to the subregular nilpotent orbit with the nilpotent cone are characterized by a pair of subgroups $\Gamma,\Gamma^{\prime}$ in $\mathrm{SL}_{2}(\mathbb{C})$, where $\Gamma$ is a normal subgroup of $\Gamma^{\prime}$. It is given by the following table:

[TABLE]

Slodowy proved the following.

Theorem 5.7** ([44]).**

Suppose $\mathfrak{g}$ is a Lie algebra of BCFG type and $\Gamma,\Gamma^{\prime}\in\mathrm{SL}_{2}(\mathbb{C})$ are given in the above table. Then for any $x$ in the subregular nilpotent orbit, we have the following isomorphism:

[TABLE]

under which the action of $\Gamma^{\prime}/\Gamma$ on $\mathbb{C}^{2}/\Gamma$ corresponds to the action of the stabilizer, denoted by $G_{x,y}$, of $x$ and $y$ (recall from Definition 2.1 that $\{x,h,y\}$ is the $\mathfrak{sl}_{2}(\mathbb{C})$ triple) on $S_{x}\cap\mathcal{N}$.

More precisely, in local coordinates (recall the table in §2.1),

1. $-$

for the $A_{2n-1}$ singularity $x^{2n}-yz=0$, the $\mathbb{Z}_{2}$-action is given by $x\mapsto-x,\,y\mapsto-y,\,z\mapsto-z.$ 2. $-$

for the $D_{n+1}$ singularity $xy^{2}+x^{n}+z^{2}=0$, the $\mathbb{Z}_{2}$-action is given by $x\mapsto x,\,y\mapsto y,\,z\mapsto-z.$ 3. $-$

for the $E_{6}$ singularity $x^{4}+y^{3}+z^{2}=0$, the $\mathbb{Z}_{2}$-action is given by $x\mapsto-x,\,y\mapsto y,\,z\mapsto-z.$ 4. $-$

for the $D_{4}$ singularity $xy^{2}+x^{3}+z^{2}=0$, the $\mathfrak{S}_{3}$-action is generated by $x\mapsto(-x+\sqrt{-1}y)/2,\,y\mapsto(3\sqrt{-1}x-y)/2,\,z\mapsto z$ and $x\mapsto x,\,y\mapsto-y,\,z\mapsto-z$.

According to Slodowy, these varieties together with the above symmetries are called simple singularities of types $B_{n}$, $C_{n}$, $F_{4}$ and $G_{2}$ respectively. Slodowy also showed in [44, §6.2] that the group actions lift to the minimal resolutions of these singularities, and the lifted actions on the irreducible components on the exceptional fibers are exactly identical to the ones on the associated Dynkin diagrams, described in the previous subsection. The following definition is now reasonable.

Definition 5.8**.**

Let $\mathcal{B}_{n}$, $\mathcal{C}_{n}$, $\mathcal{F}_{4}$ and $\mathcal{G}_{2}$ be the minimal resolutions of $B_{n}$, $C_{n}$, $F_{4}$ and $G_{2}$ respectively. Their equivariant cohomology algebras are defined to be $\mathrm{H}^{\bullet}_{\mathbb{Z}_{2}\times\mathbb{C}^{\times}}(\mathcal{B}_{n})$, $\mathrm{H}^{\bullet}_{\mathbb{Z}_{2}\times\mathbb{C}^{\times}}(\mathcal{C}_{n})$, $\mathrm{H}^{\bullet}_{\mathbb{Z}_{2}\times\mathbb{C}^{\times}}(\mathcal{F}_{4})$ and $\mathrm{H}^{\bullet}_{\mathfrak{S}_{3}\times\mathbb{C}^{\times}}(\mathcal{G}_{2})$ respectively.

Next, we turn to the equivariant quantum cohomology, which we denote by $\mathrm{QH}_{R}^{\bullet}$. Let us first recall that in [11], Bryan and Gholampour construct, for any irreducible and reduced root system, a Frobenius algebra $(\mathrm{QH}^{\bullet}_{R},\star)$ which generalizes Theorem 2.3. Let us go over their construction.

Let $R$ be an irreducible and reduced rank $n$ root system and $\{\alpha_{1},\cdots,\alpha_{n}\}$ be a system of simple roots. Let $\mathrm{H}_{R}=\mathbb{Z}\oplus\mathbb{Z}\alpha_{1}\oplus\cdots\oplus\mathbb{Z}\alpha_{n}$ and let $\mathrm{QH}_{R}^{\bullet}=\mathrm{H}_{R}\otimes\mathbb{Z}[t][\![q_{1},\cdots,q_{n}]\!]$. We associate a group $\Gamma\in\mathrm{SL}_{2}(\mathbb{C})$ listed in the table in §2.1 as follows: the type of $\Gamma$ is the same as the type of a simply-laced Lie algebra which itself is ADE or which gives the Lie algebra of BCFG type via the isomorphism (43).

Definition 5.9** (Bryan and Gholampour [11]).**

Define a product $\star$ on $\mathrm{QH}_{R}^{\bullet}$ as follows:

[TABLE]

where $\gamma^{\vee}=\dfrac{2}{\langle\gamma,\gamma\rangle}\gamma$, $e_{\gamma}=c_{1}e_{1}+\cdots+c_{n}e_{n}$ if the root $\gamma=c_{1}\alpha_{1}+\cdots+c_{n}\alpha_{n}$ with $\alpha_{1},\cdots,\alpha_{n}$ being the simple roots, $\alpha,\alpha^{\prime}$ being the positive roots corresponding to $e_{\alpha}$ and $e_{\alpha^{\prime}}$, and $\langle-,-\rangle$ is the inner product in the root system.

Bryan and Gholampour showed that the product thus defined is associative and $(\mathrm{QH}_{R}^{\bullet},\star)$ forms a Frobenius algebra (see [11, Theorem 6]). Observe that if $R$ is of ADE type, $\mathrm{QH}_{R}^{\bullet}$ is exactly the $\mathbb{C}^{\times}$-equivariant quantum cohomology of the minimal resolution of the ADE singularity (see Theorem 2.3).

Now if $R$ is of BCFG type, to specify the particular root system, let us denote $\mathrm{QH}_{R}^{\bullet}$ by $\mathrm{QH}_{R}^{\bullet}(\mathcal{B}_{n})$, $\mathrm{QH}_{R}^{\bullet}(\mathcal{C}_{n})$, $\mathrm{QH}_{R}^{\bullet}(\mathcal{F}_{4})$ and $\mathrm{QH}_{R}^{\bullet}(\mathcal{G}_{2})$. We first show that they are the “equivariant quantum” cohomology of $\mathcal{B}_{n}$, $\mathcal{C}_{n}$, $\mathcal{F}_{4}$ and $\mathcal{G}_{2}$ respectively; that is, they are the deformations of the equivariant cohomology:

Proposition 5.10**.**

There are the following isomorphisms

[TABLE]

of algebras over $\mathbb{C}[\hbar]$.

Proof.

We show the first isomorphism. In fact,

[TABLE]

The rest isomorphisms are proved similarly, and we leave them to the interested reader. ∎

Next we relate the these equivariant quantum cohomology with those of ADE resolutions. Take $\mathcal{B}_{n}$ and $\mathcal{A}_{2n-1}$, for example. Identify the generators of their equivariant quantum cohomology with the simple roots in the Dynkin diagram. Recall that $\Lambda=\{1,2,\cdots,2n-1\}$ be the index set for the simple roots of $A_{2n-1}$, and let $\bar{\Lambda}=\Lambda/\langle\nu\rangle$, where $\nu$ is the nontrivial automorphism of the Dynkin diagram. Specialize the quantum variables $\{q_{i}\}_{i\in\Lambda}$ to variables $\{q_{[i]}\}_{[i]\in\bar{\Lambda}}$ by setting $q_{i}=q_{[i]}$. Then we obtain a map

[TABLE]

For the other singularities, proceed analogously and we obtain the following.

Proposition 5.11** ([11, §4.3]).**

There are the following isomorphisms

[TABLE]

which are compatible with the quantum product $\star$.

Proof.

See [11, Theorem 6]. ∎

5.3 Proof of Theorems 1.4 and 1.5

We are now ready to prove Theorem 1.4.

Proof of Theorem 1.4.

Combining Theorem 1.2, Propositions 5.5 and 5.11, we get the desired isomorphisms. ∎

Now we consider the extremal situation where $q=0$. First, we have the following.

Proposition 5.12**.**

There are isomorphisms of $B$-algebras:

[TABLE]

Proof.

Analogous to the proof of Proposition 5.5. ∎

Proof of Theorem 1.5.

In Theorem 1.4, let $q=0$, then by Propositions 5.10 and 5.12, the specialized quantum D-module reduces to the equivariant cohomology, and the D-module of graded traces $Q({\mathscr{A}}(-))$ reduces to the ring $B({\mathscr{A}}(-))$. The theorem follows. ∎