Mod p Hecke algebras and dual equivariant cohomology I: the case of $GL_2$

TL;DR

This paper explores the relationship between mod p Hecke algebras and equivariant cohomology for the group GL_2 over a p-adic field, revealing new geometric realizations of supersingular modules.

Contribution

It demonstrates that supersingular irreducible modules of the pro-p Iwahori-Hecke algebra for GL_2 can be realized via equivariant cohomology of the flag variety of the mod p Langlands dual group.

Findings

Supersingular modules of dimension 2 are realized through equivariant cohomology.

Provides a geometric interpretation of certain Hecke algebra modules.

Establishes a link between algebraic and geometric structures in the mod p setting.

Abstract

Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $\mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $\mathbb{F}_p$. We show that the supersingular irreducible $\mathcal{H}^{(1)}$-modules of dimension 2 can be realized through the equivariant cohomology of the flag variety of the mod p Langlands dual group of $G$.

Equations

$${\mathcal{H}}^{(1)}={\overline{\mathbb{F}}}_q[I^{(1)}\setminus \mathbf{G}(F)/I^{(1)}].$$

$$\mathbb{Z}[\Lambda ]\stackrel{\simeq }{⟶}{K}^{\hat{\mathbf{G}}}(\hat{\mathcal{B}})$$

$$\mathrm{Sym}(\Lambda )\stackrel{\simeq }{⟶}C{H}^{\hat{\mathbf{G}}}(\hat{\mathcal{B}})$$

$${\mathcal{H}}_R^{(1)}(q):=(R[I^{(1)}\backslash G/I^{(1)}],\star )$$

$$\mathbb{Z}[\frac{1}{q-1},{\mu }_{q-1}].$$

$$\Lambda =\mathrm{Hom}({\mathbb{G}}_m,\mathbf{T})=\mathbb{Z}{\eta }_1\oplus \mathbb{Z}{\eta }_2\simeq \mathbb{Z}\oplus \mathbb{Z}$$

$$s=s_{\alpha }=s_{(1,-1)}:\mathbb{Z}\oplus \mathbb{Z}$$

$$(n_1,n_2)$$

$${\xi }_1:=e^{(1,0)}+e^{(0,1)}\text{and}{\xi }_2:=e^{(1,1)}$$

$$\displaystyle\xi^{+}:\mathbb{Z}[\Lambda^{+}]=\mathbb{Z}\big{[}e^{(1,0)},(e^{(1,1)})^{\pm 1}\big{]}$$

$$e^{(1,0)}$$

$$e^{(1,1)}$$

$$W_{\mathrm{aff}}:=e^{\mathbb{Z}(1,-1)}⋊W_0\subset W:=e^{\Lambda }⋊W_0.$$

$$u:=e^{(1,0)}s=s{e}^{(0,1)}$$

$$\mathbb{T}:=\mathbf{T}({\mathbb{F}}_q)$$

$$R[\mathbb{T}]\subset {\mathcal{H}}_R^{(1)}(q).$$

$$\lambda :\mathbb{T}\to {\mathbb{F}}_q^{\times }$$

$${}^s\lambda (t_1,t_2)=\lambda (t_2,t_1)$$

$${\epsilon }_{\lambda }:=|\mathbb{T}{|}^{-1}\sum _{t\in \mathbb{T}}{\lambda }^{-1}(t)T_t\in R[\mathbb{T}]$$

$${\epsilon }_{\gamma }:=\sum _{\lambda \in \gamma }{\epsilon }_{\lambda }\in R[\mathbb{T}].$$

$${\mathcal{H}}_R^{(1)}(q)=\prod _{\gamma \in {\mathbb{T}}^{\vee }/W_0}{\mathcal{H}}_R^{(1)}(q){\epsilon }_{\gamma }.$$

$$\mathcal{H}(\mathbf{q}):=\underset{(n_1,n_2)\in {\mathbb{Z}}^2}{⨁}\mathbb{Z}[\mathbf{q}]T_{e^{(n_1,n_2)}}\oplus \mathbb{Z}[\mathbf{q}]T_{e^{(n_1,n_2)}s}$$

$$T_w{T}_{w^{\prime }}=T_{w{w}^{\prime }}\text{for w,w′∈W if ℓ(w)+ℓ(w′)=ℓ(ww′)}$$

$$\left\{\begin{array}[]{l}T_{s}^{2}=(\mathbf{q}-1)T_{s}+\mathbf{q}\\ T_{s_{0}}^{2}=(\mathbf{q}-1)T_{s_{0}}+\mathbf{q}.\end{array}\right.$$

$$\mathcal{H}(\mathbf{q})=\mathbb{Z}[\mathbf{q}][S,U^{\pm 1}],S^2=(\mathbf{q}-1)S+\mathbf{q},U^{2}S=S{U}^2$$

$${\mathcal{H}}_0(\mathbf{q})=\mathbb{Z}[\mathbf{q}][S]\subset {\mathcal{H}}_{\mathrm{aff}}(\mathbf{q})=\mathbb{Z}[\mathbf{q}][S_0,S].$$

$$\displaystyle\mathscr{Z}(\mathbf{q}):\mathbb{Z}[\mathbf{q}]\big{[}\Lambda^{+}\big{]}=\mathbb{Z}[\mathbf{q}]\big{[}e^{(1,0)},(e^{(1,1)})^{\pm 1}\big{]}$$

$$e^{(1,0)}$$

$$e^{(1,1)}$$

$$Z(\mathcal{H}(\mathbf{q}))=\mathbb{Z}[\mathbf{q}][US+(1-\mathbf{q})U+SU,U^{\pm 2}]\subset \mathbb{Z}[\mathbf{q}][S,U^{\pm 1}]=\mathcal{H}(\mathbf{q}).$$

$$\mathcal{H}(\mathbf{q}){\otimes }_{\mathbb{Z}[\mathbf{q}]}R\stackrel{\simeq }{⟶}{\mathcal{H}}_R^{(1)}(q){\epsilon }_{\gamma },T_w\mapsto {\epsilon }_{\lambda }{T}_w.$$

$${\mathcal{H}}^{\mathrm{nil}}(\mathbf{q}):=\underset{(n_1,n_2)\in {\mathbb{Z}}^2}{⨁}\mathbb{Z}[\mathbf{q}]T_{e^{(n_1,n_2)}}\oplus \mathbb{Z}[\mathbf{q}]T_{e^{(n_1,n_2)}s}$$

$$T_w{T}_{w^{\prime }}=T_{w{w}^{\prime }}\text{for w,w′∈W if ℓ(w)+ℓ(w′)=ℓ(ww′)}$$

Full text

1. 1 Introduction

2. 2 The pro-$p$-Iwahori-Hecke algebra

3. 2.1 Weyl groups and cocharacters

4. 2.2 Idempotents and component algebras

5. 2.3 The Iwahori-Hecke algebra

6. 2.4 The second Iwahori-Hecke algebra

7. 3 The non-regular case and dual equivariant $K$-theory

8. 3.1 Recollections from algebraic $K^{\widehat{\mathbf{G}}}$-theory

9. 3.2 The morphism from $R(\widehat{\mathbf{G}})[\mathbf{q}]$ to the center of $\mathcal{H}(\mathbf{q})$

10. 3.3 The extended Demazure representation $\mathscr{A}(\mathbf{q})$

11. 3.4 Faithfulness of $\mathscr{A}(\mathbf{q})$

12. 3.5 Supersingular modules

13. 4 The regular case and dual equivariant intersection theory

14. 4.1 Recollections from algebraic $CH^{\widehat{\mathbf{G}}}$-theory

15. 4.2 The morphism from $S(\widehat{\mathbf{G}})[\mathbf{q}]$ to the center of $\mathcal{H}^{\operatorname{nil}}(\mathbf{q})$

16. 4.3 The extended Demazure representation $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}(\mathbf{q})$ at $\mathbf{q}=0$

17. 4.4 Faithfulness of $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$

18. 4.5 The twisted representation $\mathscr{A}_{2,\mathbb{F}_{p}}$

19. 4.6 Faithfulness of $\mathscr{A}_{2,\mathbb{F}_{p}}$

20. 4.7 Supersingular modules

21. 5 Tame Galois representations and supersingular modules

1 Introduction

Let $F$ be a finite extension of $\mathbb{Q}_{p}$ with residue field $\mathbb{F}_{q}$ and let $\mathbf{G}$ be a connected split reductive group over $F$. Let $\mathcal{H}=R[I\setminus\mathbf{G}(F)/I]$ be the Iwahori-Hecke algebra associated to an Iwahori subgroup $I\subset\mathbf{G}(F)$, with coefficients in an algebraically closed field $R$. On the other hand, let $\widehat{\mathbf{G}}$ be the Langlands dual group of $\mathbf{G}$ over $R$, and $\widehat{\mathcal{B}}$ the flag variety of Borel subgroups of $\widehat{\mathbf{G}}$ over $R$.

When $R=\mathbb{C}$, the irreducible $\mathcal{H}$-modules appear as subquotients of the Grothendieck group $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\mathbb{C}}$ of $\widehat{\mathbf{G}}$-equivariant coherent sheaves on $\widehat{\mathcal{B}}$. As such they can be parametrized by the isomorphism classes of irreducible tame $\widehat{\mathbf{G}}(\mathbb{C})$-representations of the absolute Galois group ${\rm Gal}(\overline{F}/F)$ of $F$, thereby realizing the tame local Langlands correspondence (in this setting also called the Deligne-Lusztig conjecture for Hecke modules): Kazhdan-Lusztig [KL87], Ginzburg [CG97]. The idea of studying various cohomological invariants of the flag variety by means of Hecke operators (nowadays called Demazure operators) goes back to earlier work of Demazure [D73, D74]. The approach to the Deligne-Lusztig conjecture is based on the construction of a natural $\mathcal{H}$-action on the whole $K$-group $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\mathbb{C}}$ which identifies the center of $\mathcal{H}$ with the $K$-group of the base point $K^{\widehat{\mathbf{G}}}({\rm pt})_{\mathbb{C}}$. The finite part of $\mathcal{H}$ acts thereby via appropriate $\mathbf{q}$-deformations of Demazure operators.

When $R=\overline{\mathbb{F}}_{q}$ any irreducible $\widehat{\mathbf{G}}(\overline{\mathbb{F}}_{q})$-representation of ${\rm Gal}(\overline{F}/F)$ is tame and the Iwahori-Hecke algebra needs to be replaced by the bigger pro-$p$-Iwahori-Hecke algebra

[TABLE]

Here, $I^{(1)}\subset I$ is the unique pro-$p$ Sylow subgroup of $I$. The algebra $\mathcal{H}^{(1)}$ was introduced by Vignéras and its structure theory developed in a series of papers [V04, V05, V06, V14, V15, V16, V17]. The class of so-called supersingular irreducible $\mathcal{H}^{(1)}$-modules figures prominently among all irreducible $\mathcal{H}^{(1)}$-modules, since it is expected to be related to the arithmetic over the field $F$. For $\mathbf{G}=GL_{n}$, there is a distinguished correspondence between supersingular irreducible $\mathcal{H}^{(1)}$-modules of dimension $n$ and irreducible $GL_{n}(\overline{\mathbb{F}}_{q})$-representations of ${\rm Gal}(\overline{F}/F)$: Breuil [Br03], Vignéras [V04], [V05], Colmez [C10], Grosse-Klönne [GK16], [GK18].

Our aim is to show that the supersingular irreducible $\mathcal{H}^{(1)}$- modules of dimension $n$ can again be realized as subquotients of some $\widehat{\mathbf{G}}$-equivariant cohomology theory of the flag variety $\widehat{\mathcal{B}}$ over $\overline{\mathbb{F}}_{q}$, although in a way different from the $\mathbb{C}$-coefficient case. Here we discuss the case $n=2$, and we will treat the case of general $n$ in a subsequent article [PS2].

From now on, let $R=\overline{\mathbb{F}}_{q}$ and $\mathbf{G}=GL_{2}$. The algebra $\mathcal{H}^{(1)}$ splits as a direct product of subalgebras $\mathcal{H}^{\gamma}$ indexed by the orbits $\gamma$ of $\mathfrak{S}_{2}$ in the set of characters of $(\mathbb{F}_{q}^{\times})^{2}$, namely the Iwahori components corresponding to trivial orbits, and the regular components. Accordingly, the category of $\mathcal{H}^{(1)}$-modules decomposes as the product of the module categories for the component algebras. In each component sits a unique supersingular module of dimension $2$ with given central character. On the dual side, we have the projective line $\widehat{\mathcal{B}}=\mathbb{P}^{1}_{\overline{\mathbb{F}}_{q}}$ over $\overline{\mathbb{F}}_{q}$ with its natural action by fractional transformations of the algebraic group $\widehat{\mathbf{G}}=GL_{2}(\overline{\mathbb{F}}_{q})$.

For a non-regular orbit $\gamma$, the component algebra $\mathcal{H}^{\gamma}$ is isomorphic to the mod $p$ Iwahori-Hecke algebra $\mathcal{H}=\overline{\mathbb{F}}_{q}[I\setminus\mathbf{G}(F)/I]$ and the quadratic relations in $\mathcal{H}$ are idempotent of type $T^{2}_{s}=-T_{s}$. The $\widehat{\mathbf{G}}$-equivariant $K$-theory $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\overline{\mathbb{F}}_{q}}$ of $\widehat{\mathcal{B}}$ comes with an action of the classical Demazure operator at $\mathbf{q}=0$. Our first result is that this action extends uniquely to an action of the full algebra $\mathcal{H}$ on $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\overline{\mathbb{F}}_{p}}$, which is faithful and which identifies the center $Z(\mathcal{H})$ of $\mathcal{H}$ with the base ring $K^{\widehat{\mathbf{G}}}({\rm pt})_{\overline{\mathbb{F}}_{q}}$. It is constructed from natural presentations of the algebras $\mathcal{H}$ and $Z(\mathcal{H})$ [V04] and through the characteristic homomorphism

[TABLE]

which identfies the equivariant $K$-ring with the group ring of characters $\Lambda$ of a maximal torus in $\widehat{\mathbf{G}}$. In particular, everything is explicit. We finally show that, given a supersingular central character $\theta:Z(\mathcal{H})\rightarrow\overline{\mathbb{F}}_{q}$, the central reduction $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\theta}$ is isomorphic to the unique supersingular $\mathcal{H}$-module of dimension $2$ with central character $\theta$.

For a regular orbit $\gamma$, the component algebra $\mathcal{H}^{\gamma}$ is isomorphic to Vignéras second Iwahori-Hecke algebra $\mathcal{H}_{2}$ [V04]. It can be viewed as a certain twisted version of two copies of the mod $p$ nil Hecke ring $\mathcal{H}^{\rm nil}$ (introduced over the complex numbers by Kostant-Kumar [KK86]). In particular, the quadratic relations are nilpotent of type $T_{s}^{2}=0$. The $\widehat{\mathbf{G}}$-equivariant intersection theory $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\overline{\mathbb{F}}_{q}}$ of $\widehat{\mathcal{B}}$ comes with an action of the classical Demazure operator at $\mathbf{q}=0$. We show that this action extends to a faithful action of $\mathcal{H}^{\rm nil}$ on $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\overline{\mathbb{F}}_{q}}$. To incorporate the twisting, we then pass to the square $\widehat{\mathcal{B}}^{2}$ of $\widehat{\mathcal{B}}$ and extend the action to a faithful action of $\mathcal{H}_{2}$ on $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})_{\overline{\mathbb{F}}_{q}}$. The action identifies a large part $Z^{\circ}(\mathcal{H}_{2})$ of the center $Z(\mathcal{H}_{2})$ with the base ring $CH^{\widehat{\mathbf{G}}}({\rm pt})_{\overline{\mathbb{F}}_{q}}$. As a technical point, one actually has to pass to a certain localization of the Chow groups to realize these actions, but we do not go into this in the introduction. As in the non-regular case, the action is constructed from natural presentations of the algebras $\mathcal{H}_{2}$ and $Z(\mathcal{H}_{2})$ [V04] and through the characteristic homomorphism

[TABLE]

which identifies the equivariant Chow ring with the symmetric algebra on the character group $\Lambda$. So again, everything is explicit. We finally show that, given a supersingular central character $\theta:Z(\mathcal{H}_{2})\rightarrow\overline{\mathbb{F}}_{q}$, the semisimplification of the $Z^{\circ}(\mathcal{H}_{2})$-reduction of (the localization of) $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})_{\overline{\mathbb{F}}_{q}}$ equals a direct sum of four copies of the unique supersingular $\mathcal{H}_{2}$-module of dimension $2$ with central character $\theta$.

In a final section we discuss the aforementioned bijection between supersingular irreducible $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$-modules of dimension $2$ and irreducible smooth $GL_{2}(\overline{\mathbb{F}}_{q})$-representations of ${\rm Gal}(\overline{F}/F)$ in the light of our geometric language.

Notation: In general, the letter $F$ denotes a locally compact complete non-archimedean field with ring of integers $o_{F}$. Let $\mathbb{F}_{q}$ be its residue field, of characteristic $p$ and cardinality $q$. We denote by $\mathbf{G}$ the algebraic group ${\rm GL}_{2}$ over $F$ and by $G:=\mathbf{G}(F)$ its group of $F$-rational points. Let $\mathbf{T}\subset\mathbf{G}$ be the torus of diagonal matrices. Finally, $I\subset G$ denotes the upper triangular standard Iwahori subgroup and $I^{(1)}\subset I$ denotes the unique pro-$p$ Sylow subgroup of $I$. Without further mentioning, all modules will be left modules.

2 The pro-$p$-Iwahori-Hecke algebra

Let $R$ be any commutative ring. The pro-$p$ Iwahori Hecke algebra of $G$ with coefficients in $R$ is defined to be the convolution algebra

[TABLE]

generated by the $I^{(1)}$-double cosets in $G$. In the sequel, we will assume that $R$ is an algebra over the ring

[TABLE]

The first examples we have in mind are $R=\mathbb{F}_{q}$ or its algebraic closure $R=\overline{\mathbb{F}}_{q}$.

2.1 Weyl groups and cocharacters

** 2.1.1****.**

We denote by

[TABLE]

the lattice of cocharacters of $\mathbf{T}$ with standard basis $\eta_{1}(x)=\operatorname{diag}(x,1)$ and $\eta_{2}(x)=\operatorname{diag}(1,x)$. Then $\alpha=(1,-1)\in\Lambda$ is a root and

[TABLE]

is the associated reflection generating the Weyl group $W_{0}=\{1,s\}$. The element $s$ acts on $\Lambda$ and hence also on the group ring $\mathbb{Z}[\Lambda]$. The two invariant elements

[TABLE]

in $\mathbb{Z}[\Lambda]^{s}$ define a ring isomorphism

[TABLE]

where $\Lambda^{+}:=\mathbb{Z}_{\geq 0}(1,0)\oplus\mathbb{Z}(1,1)$ is the monoid of dominant cocharacters.

** 2.1.2****.**

We introduce the affine Weyl group $W_{\operatorname{aff}}$ and the Iwahori-Weyl group $W$ of $G$:

[TABLE]

With

[TABLE]

one has $W=W_{\operatorname{aff}}\rtimes\Omega$ where $\Omega=u^{\mathbb{Z}}\simeq\mathbb{Z}.$ Let $s_{0}=e^{(1,-1)}s=se^{(-1,1)}=usu^{-1}$. Recall that the pair $(W_{\operatorname{aff}},\{s_{0},s\})$ is a Coxeter group and its length function $\ell$ can be inflated to $W$ via $\ell|_{\Omega}=0$.

2.2 Idempotents and component algebras

** 2.2.1****.**

We have the finite diagonal torus

[TABLE]

and its group ring $R[\mathbb{T}]$. As $q-1$ is invertible in $R$, so is $|\mathbb{T}|=(q-1)^{2}$ and hence $R[\mathbb{T}]$ is a semisimple ring. The canonical isomorphism $\mathbb{T}\simeq I/I^{(1)}$ induces an inclusion

[TABLE]

We denote by $\mathbb{T}^{\vee}$ the set of characters

[TABLE]

of $\mathbb{T}$, with its natural $W_{0}$-action given by

[TABLE]

for $(t_{1},t_{2})\in\mathbb{T}$. The number of $W_{0}$-orbits in $\mathbb{T}^{\vee}$ equals $\frac{q^{2}-q}{2}$. Also $W$ acts on $\mathbb{T}^{\vee}$ through the canonical quotient map $W\rightarrow W_{0}$.

** 2.2.2****.**

Definition.* For all $\lambda\in\mathbb{T}^{\vee}$, define*

[TABLE]

and for all $\gamma\in\mathbb{T}^{\vee}/W_{0}$,

[TABLE]

Following the terminology of [V04], we call $|\gamma|=1$ the Iwahori case or non-regular case and $|\gamma|=2$ the regular case.

** 2.2.3****.**

Proposition.* For all $\lambda\in\mathbb{T}^{\vee}$, the element $\varepsilon_{\lambda}$ is an idempotent. For all $\gamma\in\mathbb{T}^{\vee}/W_{0}$, the element $\varepsilon_{\gamma}$ is a central idempotent in $\mathcal{H}^{(1)}_{R}(q)$. The $R$-algebra $\mathcal{H}^{(1)}_{R}(q)$ is the direct product of its sub-$R$-algebras $\mathcal{H}^{(1)}_{R}(q)\varepsilon_{\gamma}$, i.e.*

[TABLE]

Proof.

This follows from [V04, Prop. 3.1] and its proof. ∎

The proposition implies that the category of $\mathcal{H}_{R}^{(1)}(q)$-modules decomposes into a finite product of the module categories for the individual component rings $\mathcal{H}^{(1)}_{R}(q)\varepsilon_{\gamma}$.

2.3 The Iwahori-Hecke algebra

Our reference for the following is [V04, 1.1/2].

** 2.3.1****.**

Definition.* Let $\mathbf{q}$ be an indeterminate. The generic Iwahori-Hecke algebra is the $\mathbb{Z}[\mathbf{q}]$-algebra $\mathcal{H}(\mathbf{q})$ defined by generators*

[TABLE]

and relations:

• •

braid relations

[TABLE]

• •

quadratic relations

[TABLE]

** 2.3.2****.**

Setting $S:=T_{s}$ and $U:=T_{u}$, one can check that

[TABLE]

is a presentation of $\mathcal{H}(\mathbf{q})$. For example, $S_{0}:=T_{s_{0}}=USU^{-1}$. We also have the generic finite and affine Hecke algebras

[TABLE]

The algebra $\mathcal{H}_{0}(\mathbf{q})$ has two characters corresponding to $S\mapsto 0$ and $S\mapsto-1$. Similarly, $\mathcal{H}_{\operatorname{aff}}(\mathbf{q})$ has four characters. The two characters different from the trivial character $S_{0},S\mapsto 0$ and the sign character $S_{0},S\mapsto-1$ are called supersingular.

** 2.3.3****.**

The center $Z(\mathcal{H}(\mathbf{q}))$ of the algebra $\mathcal{H}(\mathbf{q})$ admits the explicit description via the algebra isomomorphism

[TABLE]

In particular,

[TABLE]

** 2.3.4****.**

Now let $\gamma\in\mathbb{T}^{\vee}/W_{0}$ such that $|\gamma|=1$, say $\gamma=\{\lambda\}$. The ring homomorphism $\mathbb{Z}[\mathbf{q}]\rightarrow R$, $\mathbf{q}\mapsto q$, induces an isomorphism of $R$-algebras

[TABLE]

2.4 The second Iwahori-Hecke algebra

Our reference for the following is [V04, 2.2], as well as [KK86] for the basic theory of the nil Hecke algebra. We keep the notation introduced above.

** 2.4.1****.**

Definition.* The generic nil Hecke algebra is the $\mathbb{Z}[\mathbf{q}]$-algebra $\mathcal{H}^{\operatorname{nil}}(\mathbf{q})$ defined by generators*

[TABLE]

and relations:

• •

braid relations

[TABLE]

• •

quadratic relations

[TABLE]

** 2.4.2****.**

Setting $S:=T_{s}$ and $U:=T_{u}$, one can check that

[TABLE]

is a presentation of $\mathcal{H}^{\operatorname{nil}}(\mathbf{q})$. Again, $S_{0}:=T_{s_{0}}=USU^{-1}$. The center $Z(\mathcal{H}^{\rm nil}(\mathbf{q}))$ admits the explicit description via the algebra isomorphism

[TABLE]

In particular,

[TABLE]

** 2.4.3****.**

Form the twisted tensor product algebra

[TABLE]

With the formal symbols $\varepsilon_{1}=(1,0)$ and $\varepsilon_{2}=(0,1)$, the ring multiplication is given by

[TABLE]

for all $1\leq i,i^{\prime}\leq 2$. Here, $W$ acts through its quotient $W_{0}$ and $s\in W_{0}$ acts on the set $\{1,2\}$ by interchanging the two elements. The multiplicative unit element in the ring $\mathbb{Z}[\mathbf{q}]\times\mathbb{Z}[\mathbf{q}]$ is $(1,1)=\varepsilon_{1}+\varepsilon_{2}$ and the multiplicative unit element in the ring $\mathcal{H}_{2}(\mathbf{q})$ is $(1,1)\otimes 1$. We identify the rings $\mathbb{Z}[\mathbf{q}]\times\mathbb{Z}[\mathbf{q}]$ and $\mathcal{H}^{\operatorname{nil}}(\mathbf{q})$ with subrings of $\mathcal{H}_{2}(\mathbf{q})$ via the maps $(a,b)\mapsto(a,b)\otimes 1$ and $a\mapsto(1,1)\otimes a$ respectively. In particular, we will write $\varepsilon_{1},\varepsilon_{2},S_{0},S,U\in\mathcal{H}_{2}(\mathbf{q})$ etc.

We also introduce the generic affine Hecke algebra

[TABLE]

It is a subalgebra of $\mathcal{H}_{2}(\mathbf{q})$ and has two supersingular characters $\chi_{1}$ and $\chi_{2}$, namely $\chi_{1}(\varepsilon_{1})=1$ and $\chi_{1}(\varepsilon_{2})=0$ and $\chi_{1}(S_{0})=\chi_{1}(S)=0$. Similarly for $\chi_{2}$.

** 2.4.4****.**

The structure of $\mathcal{H}_{2}(\mathbf{q})$ as an algebra over its center can be made explicit. In fact, there is an algebra isomorphism with an algebra of $2\times 2$-matrices

[TABLE]

which maps the center $Z(\mathcal{H}_{2}(\mathbf{q}))$ to the scalar matrices $\mathcal{Z}(\mathbf{q})$. Under this isomorphism, we have

[TABLE]

[TABLE]

The induced map $Z(\mathcal{H}_{2}(\mathbf{q}))\rightarrow\mathcal{Z}(\mathbf{q})$ satisfies

[TABLE]

In particular, the subring

[TABLE]

lies in fact in the center $Z(\mathcal{H}_{2}(\mathbf{q}))$ of $\mathcal{H}_{2}(\mathbf{q})$.

** 2.4.5****.**

Now let $\gamma\in\mathbb{T}^{\vee}/W_{0}$ such that $|\gamma|=2$, say $\gamma=\{\lambda,^{s}\lambda\}$. The ring homomorphism $\mathbb{Z}[\mathbf{q}]\rightarrow R$, $\mathbf{q}\mapsto q$, induces an isomorphism of $R$-algebras

[TABLE]

** 2.4.6****.**

Remark. We have used the same letters $S_{0},S,U,\zeta_{1},\zeta_{2}$ for the corresponding Hecke operators in the Iwahori Hecke algebra and in the second Iwahori Hecke algebra. This should not lead to confusion, as we will always treat non-regular components and regular components separately in our discussion.

3 The non-regular case and dual equivariant $K$-theory

3.1 Recollections from algebraic $K^{\widehat{\mathbf{G}}}$-theory

For basic notions from equivariant algebraic $K$-theory we refer to [Th87]. A useful introduction may also be found in [CG97, chap. 5].

** 3.1.1****.**

We let

[TABLE]

be the Langlands dual group of $\mathbf{G}$ over the algebraic closure $\overline{\mathbb{F}}_{q}$ of $\mathbb{F}_{q}$. The dual torus

[TABLE]

identifies with the torus of diagonal matrices in $\widehat{\mathbf{G}}.$ A basic object is

[TABLE]

i.e. the Grothendieck ring of the abelian tensor category of all finite dimensional $\widehat{\mathbf{G}}$-representations. It can be viewed as the equivariant $K$-theory $K^{\widehat{\mathbf{G}}}(\operatorname{pt})$ of the base point $\operatorname{pt}=\operatorname{Spec}\;\overline{\mathbb{F}}_{q}$. To compute it, we introduce the representation ring $R(\widehat{\mathbf{T}})$ of $\widehat{\mathbf{T}}$ which identifies canonically, as a ring with $W_{0}$-action, with the group ring of $\Lambda$, i.e.

[TABLE]

The formal character $\chi_{V}\in\mathbb{Z}[\Lambda]^{s}$ of a representation $V$ is an invariant function and is defined by

[TABLE]

for all $\lambda\in\Lambda$ where $V_{\lambda}$ is the $\lambda$-weight space of $V$. The map $V\mapsto\chi_{V}$ induces a ring isomorphism

[TABLE]

The $\mathbb{Z}[\Lambda]^{s}$-module $\mathbb{Z}[\Lambda]$ is free of rank $2$, with basis $\{1,e^{(-1,0)}\}$,

[TABLE]

** 3.1.2****.**

We let

[TABLE]

be the projective line over $\overline{\mathbb{F}}_{q}$ endowed with its left $\widehat{\mathbf{G}}$-action by fractional transformations

[TABLE]

Here, $x$ is a local coordinate on ${\mathbb{P}}^{1}_{\overline{\mathbb{F}}_{q}}$. The stabilizer of the point $x=\infty$ is the Borel subgroup $\widehat{\mathbf{B}}$ of upper triangular matrices and we may thus write $\widehat{\mathcal{B}}=\widehat{\mathbf{G}}/\widehat{\mathbf{B}}$. We denote by

[TABLE]

Given a representation $V$ and an equivariant coherent sheaf $\mathcal{F}$, the diagonal action of $\widehat{\mathbf{G}}$ makes $\mathcal{F}\otimes_{\overline{\mathbb{F}}_{q}}V$ an equivariant coherent sheaf. In this way, $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})$ becomes a module over the ring $R(\widehat{\mathbf{G}})$.

The characteristic homomorphism in algebraic $K^{\widehat{\mathbf{G}}}$-theory is a ring isomorphism

[TABLE]

It maps $e^{\lambda}$ with $\lambda=(\lambda_{1},\lambda_{2})\in\Lambda$ to the class of the $\widehat{\mathbf{G}}$-equivariant line bundle $\mathcal{O}_{{\mathbb{P}}^{1}}(\lambda_{1}-\lambda_{2})\otimes\det^{\lambda_{2}}$ where $\det$ is the determinant character of $\widehat{\mathbf{G}}$. The characteristic homomorphism is compatible with the character morphism $\chi_{\bullet}$ , i.e. $c^{\widehat{\mathbf{G}}}$ is $\mathbb{Z}[\Lambda]^{s}\simeq R(\widehat{\mathbf{G}})$-linear.

** 3.1.3****.**

For the definition of the classical Demazure operators on algebraic $K$-theory we refer to [D73, D74]. The Demazure operators

[TABLE]

are defined by:

[TABLE]

for $a\in R(\widehat{\mathbf{T}})$. They are the projectors on $R(\widehat{\mathbf{T}})^{s}e^{(-1,0)}$ along $R(\widehat{\mathbf{T}})^{s}$, and on $R(\widehat{\mathbf{T}})^{s}$ along $R(\widehat{\mathbf{T}})^{s}e^{(1,0)}$, respectively. In particular $D_{s}^{2}=D_{s}$ and $D^{\prime 2}_{s}=D^{\prime}_{s}$. One sets

[TABLE]

and checks by direct calculation that

[TABLE]

In particular, we obtain a well-defined $\mathbb{Z}[\mathbf{q}]$-algebra homomorphism

[TABLE]

which we call the Demazure representation.

3.2 The morphism from $R(\widehat{\mathbf{G}})[\mathbf{q}]$ to the center of $\mathcal{H}(\mathbf{q})$

In the following we identify the rings

[TABLE]

via the character isomorphism $\chi_{\bullet}$. We have the $\mathbb{Z}[\mathbf{q}]$-algebra isomorphism coming via base change from the isomorphism $\xi^{+}$, cf. 2.1.1:

[TABLE]

On the other hand, the source of $\xi^{+}$ is isomorphic to the center $Z(\mathcal{H}(\mathbf{q}))$ of $\mathcal{H}(\mathbf{q})$ via the isomorphism $\mathscr{Z}(\mathbf{q})$, cf. 2.3.2. The composition

[TABLE]

is then a ring isomorphism.

3.3 The extended Demazure representation $\mathscr{A}(\mathbf{q})$

Recall the Demazure representation $\mathscr{A}_{0}(\mathbf{q})$ of the finite algebra $\mathcal{H}_{0}(\mathbf{q})$ by $R(\widehat{\mathbf{G}})[\mathbf{q}]$-linear operators on the $K$-theory $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})$, cf. 3.1.3. We have the following first main result.

** 3.3.1****.**

Theorem.* There is a unique ring homomorphism*

[TABLE]

which extends the ring homomorphism $\mathscr{A}_{0}(\mathbf{q})$ and coincides on $Z(\mathcal{H}(\mathbf{q}))$ with the isomorphism

[TABLE]

The homomorphism $\mathscr{A}(\mathbf{q})$ is injective.

Proof :

Such an extension exists if and only if there exists

[TABLE]

satisfying

$\mathscr{A}(\mathbf{q})(U)$ is invertible ; 2. 2.

$\mathscr{A}(\mathbf{q})(U)^{2}=\mathscr{A}(\mathbf{q})(U^{2})=\mathscr{A}(\mathbf{q})(\zeta_{2})=\xi_{2}\operatorname{Id}$ ; 3. 3.

[TABLE]

To find such an operator $\mathscr{A}(\mathbf{q})(U)$, we write

[TABLE]

and use the $R(\widehat{\mathbf{T}})^{s}[\mathbf{q}]$-basis $\{1,e^{(-1,0)}\}$ to identify $\operatorname{End}_{R(\widehat{\mathbf{G}})[\mathbf{q}]}(K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})[\mathbf{q}])$ with the algebra of $2\times 2$-matrices over the ring $R(\widehat{\mathbf{T}})^{s}[\mathbf{q}]$. Then, by definition,

[TABLE]

Hence, if we set

[TABLE]

we get

[TABLE]

and

[TABLE]

[TABLE]

These two conditions together are in turn equivalent to

[TABLE]

Moreover, in this case, the determinant

[TABLE]

is invertible. Specialising to $\mathbf{q}=0$, we find that there is exactly one $R(\widehat{\mathbf{G}})[\mathbf{q}]$-algebra homomorphism

[TABLE]

extending the ring homomorphism $\mathscr{A}_{0}(\mathbf{q})$, corresponding to the matrix

[TABLE]

Note that $a,b,c,d\in R(\widehat{\mathbf{T}})^{s}\subset R(\widehat{\mathbf{T}})^{s}[\mathbf{q}]$. The injectivity of the map $\mathscr{A}(\mathbf{q})$ will be proved in the next subsection. $\square$

3.4 Faithfulness of $\mathscr{A}(\mathbf{q})$

Let us show that the map $\mathscr{A}(\mathbf{q})$ is injective. It follows from 2.3.2 that the ring $\mathcal{H}(\mathbf{q})$ is generated by the elements

[TABLE]

over its center $Z(\mathcal{H}(\mathbf{q}))=\mathbb{Z}[\zeta_{1},\zeta_{2}^{\pm 1}][\mathbf{q}]$. As the latter is mapped isomorphically to the center $R(\widehat{\mathbf{G}})[\mathbf{q}]=\mathbb{Z}[\xi_{1},\xi_{2}^{\pm 1}][\mathbf{q}]$ of the matrix algebra $\operatorname{End}_{R(\widehat{\mathbf{G}})[\mathbf{q}]}(K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})[\mathbf{q}])$ by $\mathscr{A}(\mathbf{q})$, it suffices to check that the images

[TABLE]

of $1,S,U,SU$ by $\mathscr{A}(\mathbf{q})$ are free over $R(\widehat{\mathbf{G}})[\mathbf{q}]$. To ease notation, we will write $\xi$ instead of $\xi_{1}$ in the following calculation. So let $\alpha,\beta,\gamma,\delta\in R(\widehat{\mathbf{T}})^{s}[\mathbf{q}]$ (which is an integral domain) be such that

[TABLE]

This is equivalent to the expression

[TABLE]

being zero, i.e. to the identity

[TABLE]

Then

[TABLE]

As $b\neq 0$, we obtain $\delta=\gamma$ and

[TABLE]

Hence $\alpha=\beta$ and

[TABLE]

The latter system has determinant

[TABLE]

which is nonzero (its specialization at $\mathbf{q}=0$ is equal to $c\neq 0$), whence $\alpha=\gamma=0=\beta=\delta$. This concludes the proof and shows that the map $\mathscr{A}(\mathbf{q})$ is injective. We record the following two corollaries of the proof.

** 3.4.1****.**

Corollary.* The ring $\mathcal{H}(\mathbf{q})$ is a free $Z(\mathcal{H}(\mathbf{q}))$-module on the basis $1,S,U,SU$.*

** 3.4.2****.**

Corollary.* The representation $\mathscr{A}(0)$ is injective.*

3.5 Supersingular modules

In this section we work at $\mathbf{q}=0$ and over the algebraic closure $\overline{\mathbb{F}}_{q}$ of the field $\mathbb{F}_{q}$.

** 3.5.1****.**

Consider the ring homomorphism $\mathbb{Z}[\mathbf{q}]\rightarrow\overline{\mathbb{F}}_{q}$, $\mathbf{q}\mapsto q=0$, and let

[TABLE]

The characters of $\mathcal{H}_{\overline{\mathbb{F}}_{q}}$ are parametrised by the set $\{0,-1\}\times\overline{\mathbb{F}}^{\times}_{q}$ via evaluation on the elements $S$ and $U$. Let $(\tau_{1},\tau_{2})\in\overline{\mathbb{F}}_{q}\times\overline{\mathbb{F}}_{q}^{\times}$. A standard module over $\mathcal{H}_{\overline{\mathbb{F}}_{q}}$ of dimension $2$ is defined to be a module of type

[TABLE]

The center $Z(\mathcal{H}_{\overline{\mathbb{F}}_{q}})=\overline{\mathbb{F}}_{q}[\zeta_{1},\zeta_{2}^{\pm 1}]$ acts on the module $M_{2}(\tau_{1},\tau_{2})$ via the character $\zeta_{1}\mapsto\tau_{1},\zeta_{2}\mapsto\tau_{2}$. The module $M_{2}(\tau_{1},\tau_{2})$ is reducible if and only if $\tau_{1}^{2}=\tau_{2}$. It is called supersingular if $\tau_{1}=0$. A supersingular module is thus irreducible. Any simple finite dimensional $\mathcal{H}_{\overline{\mathbb{F}}_{q}}$-module is either a character or a standard module [V04, 1.4].

** 3.5.2****.**

Now consider the base change of the representation $\mathscr{A}:=\mathscr{A}(0)$ to $\overline{\mathbb{F}}_{q}$

[TABLE]

Recall that the image of $Z(\mathcal{H}_{\overline{\mathbb{F}}_{q}})=\overline{\mathbb{F}}_{q}[\zeta_{1},\zeta_{2}^{\pm 1}]$ is $R(\widehat{\mathbf{G}})_{\overline{\mathbb{F}}_{q}}=\overline{\mathbb{F}}_{q}[\xi_{1},\xi_{2}^{\pm 1}].$

Let us fix a character $\theta:Z(\mathcal{H}_{\overline{\mathbb{F}}_{q}})\rightarrow\overline{\mathbb{F}}_{q}$. Following [V04], we call $\theta$ supersingular if $\theta(\zeta_{1})=0$. Consider the base change of $\mathscr{A}_{\overline{\mathbb{F}}_{q}}$ along $\theta$

[TABLE]

[TABLE]

** 3.5.3****.**

Proposition.* The representation $\mathscr{A}_{\theta}$ is faithful if and only if $\theta(\zeta_{1})^{2}\neq\theta(\zeta_{2})$. In this case, $\mathscr{A}_{\theta}$ is an algebra isomorphism*

[TABLE]

Proof :

The discussion in the preceding section 3.4 shows that $\mathcal{H}_{\theta}$ has $\overline{\mathbb{F}}_{q}$-basis given by $1,S,U,SU$. Moreover, their images

[TABLE]

by $\mathscr{A}_{\theta}$ are linearly independent over $\overline{\mathbb{F}}_{q}$ if and only if the scalar $c=e^{(-1,-1)}\xi_{1}^{2}-1\in R(\widehat{\mathbf{G}})_{\overline{\mathbb{F}}_{q}}$ does not reduce to zero via $\theta$, i.e. if and only if $\zeta_{2}^{-1}\zeta_{1}^{2}-1\notin\ker\theta$. In this case, the map $\mathscr{A}_{\theta}$ is injective and then bijective since $\dim_{\overline{\mathbb{F}}_{q}}K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\theta}=2$. $\square$

** 3.5.4****.**

Corollary.* The $\mathcal{H}_{\overline{\mathbb{F}}_{q}}$-module $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\theta}$ is isomorphic to the standard module $M_{2}(\tau_{1},\tau_{2})$ where $\tau_{1}=\theta(\zeta_{1})$ and $\tau_{2}=\theta(\zeta_{2})$. In particular, if $\theta$ is supersingular, then $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\theta}$ is isomorphic to the unique supersingular $\mathcal{H}_{\overline{\mathbb{F}}_{q}}$-module with central character $\theta$.*

Proof :

In the case $\tau_{1}^{2}\neq\tau_{2}$, the module $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\theta}$ is irreducible by the preceding proposition and hence is standard. In general, it suffices to find $m\in K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\theta}$ with $Sm=-m$ and to verify that $\{m,Um\}$ are linearly independent. For example, $m=e^{\eta_{2}}$ is a possible choice, cf. below. $\square$

A ”standard basis” for the module $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\theta}$ comes from the so-called Pittie-Steinberg basis [St75] of $\overline{\mathbb{F}}_{q}[e^{\pm\eta_{1}},e^{\pm\eta_{2}}]$ over $\overline{\mathbb{F}}_{q}[\xi_{1},\xi_{2}^{\pm 1}]$. It is given by

[TABLE]

It induces a basis of $\overline{\mathbb{F}}_{q}[e^{\pm\eta_{1}},e^{\pm\eta_{2}}]\otimes_{\overline{\mathbb{F}}_{q}[\xi_{1},\xi_{2}^{\pm 1}],\theta}\overline{\mathbb{F}}_{q}$ over $\overline{\mathbb{F}}_{q}$ for any character $\theta$ of $\overline{\mathbb{F}}_{q}[\xi_{1},\xi_{2}^{\pm 1}]$. Let $\tau_{2}=\theta(\xi_{2})$. The matrices of $S$, $U$ and $S_{0}=USU^{-1}$ in the latter basis are

[TABLE]

The two characters of $\mathcal{H}_{0,\overline{\mathbb{F}}_{q}}=\overline{\mathbb{F}}_{q}[S]$ corresponding to $S\mapsto 0$ and $S\mapsto-1$ are realized by $z_{e}$ and $z_{s}$. From the matrix of $S_{0}$, we see in fact that the whole affine algebra $\mathcal{H}_{\operatorname{aff},\overline{\mathbb{F}}_{q}}:=\overline{\mathbb{F}}_{q}[S_{0},S]$ acts on $z_{e}$ and $z_{s}$ via the two supersingular characters of $\mathcal{H}_{\operatorname{aff},\overline{\mathbb{F}}_{q}}$, cf. 2.3.2.

** 3.5.5****.**

We extend this discussion of the component $\gamma=1$ to any other non-regular component as follows. Consider the quotient map

[TABLE]

For any $\gamma\in\mathbb{T}^{\vee}/W_{0}$ define the $\overline{\mathbb{F}}_{q}$-variety

[TABLE]

Suppose $|\gamma|=1$. We have the algebra isomorphism $\mathcal{H}_{\overline{\mathbb{F}}_{q}}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$ from 2.3.4. It identifies the center $Z(\mathcal{H}_{\overline{\mathbb{F}}_{q}})$ with the center of $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$. In this way, we let the component algebra $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$ act on $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\overline{\mathbb{F}}_{q}}$ and we denote this representation by $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{\gamma})_{\overline{\mathbb{F}}_{q}}$. We may then state, in obvious terminology, that any supersingular character $\theta$ of the center of $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$ gives rise to the supersingular irreducible $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$-module $K^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{\gamma})_{\theta}$.

4 The regular case and dual equivariant intersection theory

4.1 Recollections from algebraic $CH^{\widehat{\mathbf{G}}}$-theory

For basic notions from equivariant algebraic intersection theory we refer to [EG96] and [Bri97]. As in the case of equivariant $K$-theory, the characteristic homomorphism will make everything explicit.

** 4.1.1****.**

We denote by $\operatorname{Sym}(\Lambda)$ the symmetric algebra of the lattice $\Lambda$ endowed with its natural action of the reflection $s$. The equivariant intersection theory of the base point $\operatorname{pt}=\operatorname{Spec}\;\overline{\mathbb{F}}_{q}$ canonically identifies with the ring of invariants

[TABLE]

cf. [EG96, sec. 3.2]. Recall our basis elements $\eta_{1}:=(1,0)$ and $\eta_{2}:=(0,1)$ of $\Lambda$, so that $\operatorname{Sym}(\Lambda)=\mathbb{Z}[\eta_{1},\eta_{2}]$. We define the invariant elements

[TABLE]

in $\operatorname{Sym}(\Lambda)^{s}$. Then

[TABLE]

and, after inverting the prime $2$, the $\operatorname{Sym}(\Lambda)^{s}$-module $\operatorname{Sym}(\Lambda)$ is free of rank $2$, on the basis $\{1,\frac{\eta_{1}-\eta_{2}}{2}\}$.

** 4.1.2****.**

The equivariant Chern class of line bundles in the algebraic $CH^{\widehat{\mathbf{G}}}$-theory of $\widehat{\mathcal{B}}$ is a map

[TABLE]

which is a group homomorphism. Then, the corresponding characteristic homomorphism is a ring isomorphism

[TABLE]

which maps $\lambda=(\lambda_{1},\lambda_{2})\in\Lambda$ to the equivariant Chern class of the line bundle $\mathcal{O}_{\mathbb{P}^{1}}(\lambda_{1}-\lambda_{2})\otimes\det^{\lambda_{2}}$ on $\widehat{\mathcal{B}}=\mathbb{P}^{1}_{\overline{\mathbb{F}}_{q}}$, i.e.

[TABLE]

Note here that the algebraic group $\widehat{\mathbf{G}}={\rm GL}_{2/\overline{\mathbb{F}}_{q}}$ is special (in the sense of [EG96, 6.3]) and the map $c^{\widehat{\mathbf{G}}}$ is therefore already bijective at the integral level [Bri97, sec. 6.6]. The homomorphism $c^{\widehat{\mathbf{G}}}$ is $\operatorname{Sym}(\Lambda)^{s}\simeq CH^{\widehat{\mathbf{G}}}(\operatorname{pt})$-linear.

To emphasize the duality and the analogy with the case of $K$-theory (and to ease notation), we abbreviate from now on

[TABLE]

** 4.1.3****.**

For the definition of the classical Demazure operators on algebraic intersection theory, we refer to [D73]. The Demazure operators

[TABLE]

are defined by:

[TABLE]

for $a\in S(\widehat{\mathbf{T}})$. Then $D_{s}$ is the projector on $S(\widehat{\mathbf{T}})^{s}\frac{\eta_{1}-\eta_{2}}{2}$ along $S(\widehat{\mathbf{T}})^{s}$, and $(-D_{s})+D_{s}^{\prime}=s$. In particular, $D_{s}^{2}=0$ and $D^{\prime 2}_{s}=\operatorname{id}$. One sets

[TABLE]

and checks by direct calculation that $D_{s}(\mathbf{q})^{2}=\mathbf{q}^{2}$. We obtain thus a well-defined $\mathbb{Z}$-algebra homomorphism

[TABLE]

which we call the Demazure representation.

4.2 The morphism from $S(\widehat{\mathbf{G}})[\mathbf{q}]$ to the center of $\mathcal{H}^{\operatorname{nil}}(\mathbf{q})$

The version of the homomorphism $(\xi^{+})^{-1}$ in the regular case is the $\mathbb{Z}[\mathbf{q}]$-algebra homomorphism

[TABLE]

which becomes an isomorphism after inverting $\xi^{\prime}_{2}$. Its composition with $\mathscr{Z}^{\operatorname{nil}}(\mathbf{q})$, cf. 2.4.2, therefore gives a ring isomorphism

[TABLE]

4.3 The extended Demazure representation $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}(\mathbf{q})$ at $\mathbf{q}=0$

Recall the Demazure representation $\mathscr{A}_{0}^{\operatorname{nil}}(\mathbf{q})$ of the finite algebra $\mathcal{H}_{0}^{\operatorname{nil}}(\mathbf{q})$ by $S(\widehat{\mathbf{G}})[\mathbf{q}]$-linear operators on the intersection theory $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})$, cf. 4.1.3. In this section we work at $\mathbf{q}=0$. We write $\mathscr{A}_{0}^{\operatorname{nil}}$ for the specialization of $\mathscr{A}_{0}^{\operatorname{nil}}(\mathbf{q})$ at $\mathbf{q}=0$. For better readibility we make a slight abuse of notation and denote the elements $\xi^{\prime}_{i}$ by $\xi_{i}$ in this and the following sections. Moreover, $p$ will always be an odd prime.

** 4.3.1****.**

A ring homomorphism

[TABLE]

which extends $\mathscr{A}_{0}^{\operatorname{nil}}$ and which is linear with respect to the above ring homomorphism $S(\widehat{\mathbf{G}})\rightarrow Z(\mathcal{H}^{\operatorname{nil}})$ does not exist, even after inverting $\xi_{2}$. However, there exists a natural good approximation (after inverting the prime $2$). We will explain these points in the following.

** 4.3.2****.**

An extension of $\mathscr{A}_{0}^{\operatorname{nil}}$, linear with respect to $S(\widehat{\mathbf{G}})\rightarrow Z(\mathcal{H}^{\operatorname{nil}})$, exists if and only if there is an operator

[TABLE]

satisfying

$\mathscr{A}^{\rm nil}(U)$ is invertible ; 2. 2.

$\mathscr{A}^{\rm nil}(U)^{2}=\mathscr{A}^{\rm nil}(U^{2})=\xi_{2}\operatorname{Id}$, i.e. $\mathscr{A}^{\operatorname{nil}}(U)^{2}=\xi_{2}\operatorname{Id}$ ; 3. 3.

$\mathscr{A}^{\operatorname{nil}}(U)\mathscr{A}_{0}^{\operatorname{nil}}(S)+\mathscr{A}_{0}^{\operatorname{nil}}(S)\mathscr{A}^{\operatorname{nil}}(U)=\mathscr{A}^{\operatorname{nil}}(US+SU)=\xi_{1}\operatorname{Id}.$

Tensoring by $\mathbb{F}_{p}$, we may write

[TABLE]

and identify $\operatorname{End}_{S(\widehat{\mathbf{G}})_{\mathbb{F}_{p}}}(CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\mathbb{F}_{p}})$ with the algebra of $2\times 2$-matrices over the ring $S(\widehat{\mathbf{G}})_{\mathbb{F}_{p}}$. The analogous statements hold after inverting $\xi_{2}$.

Then, by definition,

[TABLE]

Hence, if we set

[TABLE]

we obtain

[TABLE]

and

[TABLE]

[TABLE]

and then the first system becomes equivalent to the equation

[TABLE]

However, since $\xi_{2}$ has no square root in the ring $\mathbb{F}_{p}[{\xi}_{2}^{\pm 1}]$, this latter equation has no solution (take $\xi_{1}=0$ !). Consequently, there does not exist any matrix $\mathscr{A}_{\mathbb{F}_{p}}^{\rm nil}(U)$ with coefficients in $S(\widehat{\mathbf{G}})_{\mathbb{F}_{p}}[{\xi}_{2}^{-1}]$ satisfying conditions 1, 2, 3, above.

As a best approximation, we keep condition $1$ and also condition $3$ (up to sign), but, because of the square root obstruction above, we modify condition $2$ to $\mathscr{A}_{\mathbb{F}_{p}}^{\rm nil}(\zeta_{2})=\xi_{2}^{2}$. We can then state our second main result.

** 4.3.3****.**

Theorem.* There is a distinguished ring homomorphism*

[TABLE]

which extends the ring homomorphism $\mathscr{A}^{\operatorname{nil}}_{0}$ and coincides on $Z(\mathcal{H}^{\operatorname{nil}})_{\mathbb{F}_{p}}$ with the homomorphism

[TABLE]

The homomorphism $\mathscr{A}^{\operatorname{nil}}_{\mathbb{F}_{p}}$ is injective.

Proof.

The discussion preceding the theorem shows that the matrix

[TABLE]

does satisfy the three conditions

$\mathscr{A}_{\mathbb{F}_{p}}^{\rm nil}(U)$ is invertible ; 2. 2.

$\mathscr{A}_{\mathbb{F}_{p}}^{\rm nil}(U)^{2}=(\xi_{2})^{2}\operatorname{Id}$ ; 3. 3.

$\mathscr{A}_{\mathbb{F}_{p}}^{\rm nil}(US+SU)=-\xi_{1}\operatorname{Id}.$

The injectivity part of the theorem will be shown in the next subsection. ∎

** 4.3.4****.**

Remark. The minus sign before $\xi_{1}$ appearing in the value of $\mathscr{A}_{\mathbb{F}_{p}}^{\rm nil}$ on $\zeta_{1}=US+SU$ could be avoided by setting $\mathscr{A}_{0}^{\operatorname{nil}}(S):=D_{s}$ instead of $-D_{s}$ in the Demazure representation. But we will not do this.

** 4.3.5****.**

Remark. In the Iwahori case, one can check that the action of $U$ coincides with the action of the Weyl element $e^{\eta_{1}}s$. In the regular case, the action of the element $\eta_{1}s$ does not satisfy the conditions $1$-$3$ appearing in the above proof. However, the action of $\eta^{2}_{1}s$ does and, in fact, its matrix is given by matrix $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}(U)$. So the choice of the matrix $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}(U)$ is in close analogy with the Iwahori case. Our chosen extension $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$ of $\mathscr{A}_{0,\mathbb{F}_{p}}^{\operatorname{nil}}$ seems to be distinguished for at least this reason. This observation also shows that the action of $U$ can actually be defined integrally, i.e. before inverting the prime $2$.

4.4 Faithfulness of $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$

Let us show that the map $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$ is injective. It follows from 2.4.2 that the ring $\mathcal{H}_{\mathbb{F}_{p}}^{\operatorname{nil}}$ is generated by the elements

[TABLE]

over its center $Z(\mathcal{H}^{\operatorname{nil}}_{\mathbb{F}})=\mathbb{F}_{p}[\zeta_{1},\zeta_{2}^{\pm 1}]$. The latter is mapped isomorphically to the subring

[TABLE]

of the matrix algebra $\operatorname{End}_{S(\widehat{\mathbf{G}})_{\mathbb{F}_{p}}[\xi_{2}^{-1}]}(CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})_{\mathbb{F}_{p}})$ by $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$. For injectivity, it therefore suffices to show that the images

[TABLE]

of $1,S,U,SU$ under $\mathscr{A}^{\operatorname{nil}}_{\mathbb{F}_{p}}$ are free over $S(\widehat{\mathbf{G}})_{\mathbb{F}_{p}}[\xi_{2}^{-1}]$. To this end, let $\alpha,\beta,\gamma,\delta\in S(\widehat{\mathbf{G}})_{\mathbb{F}_{p}}[\xi_{2}^{-1}]$ (which is an integral domain) be such that

[TABLE]

i.e.

[TABLE]

Then

[TABLE]

with $\alpha,\beta,\gamma,\delta\in S(\widehat{\mathbf{G}})_{\mathbb{F}_{p}}[\xi_{2}^{-1}]$. Now recall our choice

[TABLE]

In particular, $b=\xi_{1}$ implies $\gamma=0$, and then $\alpha=0$, $\delta=0$ and $\beta=0$. This shows that the map $\mathscr{A}^{\operatorname{nil}}$ is injective and concludes the proof. We record the following corollary of the proof.

** 4.4.1****.**

Corollary.* The ring $\mathcal{H}^{\operatorname{nil}}_{\mathbb{F}_{p}}$ is a free $Z(\mathcal{H}^{\operatorname{nil}}_{\mathbb{F}_{p}})$-module on the basis $1,S,U,SU$.*

4.5 The twisted representation $\mathscr{A}_{2,\mathbb{F}_{p}}$

** 4.5.1****.**

In the algebra

[TABLE]

we have the two subrings $\mathcal{H}^{\operatorname{nil}}$ and $\mathbb{Z}\times\mathbb{Z}$. The aim of this section is to extend the representation $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$ from $\mathcal{H}_{\mathbb{F}_{p}}^{\operatorname{nil}}$ to the whole algebra $\mathcal{H}_{2,\mathbb{F}_{p}}:=\mathcal{H}_{2}\otimes_{\mathbb{Z}}\mathbb{F}_{p}$. To this end, we consider the $\overline{\mathbb{F}}_{q}$-variety

[TABLE]

where $\widehat{\mathcal{B}}_{1}$ and $\widehat{\mathcal{B}}_{2}$ are two copies of $\widehat{\mathcal{B}}$. We have

[TABLE]

After base change to $\mathbb{F}_{p}$, the ring $\mathcal{H}^{\operatorname{nil}}$ acts $S(\widehat{\mathbf{G}})[\xi_{2}^{-1}]$-linearly on $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})[\xi_{2}^{-1}]$ through the map $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$. We extend this action diagonally to $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]$, thus defining a ring homomorphism

[TABLE]

Because of the twisted multiplication in the algebra $\mathcal{H}_{2}$, we need to introduce the permutation action of $W$

[TABLE]

which permutes the two factors of $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})$.

On the other hand, we can consider the projection $p_{i}$ from $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})$ to $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}_{i})$ as an $S(\widehat{\mathbf{G}})$-linear endomorphism of $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})$, for $i=1,2$. The rule $\varepsilon_{i}\mapsto p_{i}$ defines a ring homomorphism

[TABLE]

** 4.5.2****.**

Proposition.* There exists a unique ring homomorphism*

[TABLE]

such that

• •

\quad\mathscr{A}_{2,\mathbb{F}_{p}}|_{\mathcal{H}^{\rm nil}_{\mathbb{F}_{p}}}(T_{w})=\operatorname{diag}(\mathscr{A}^{\rm nil}_{\mathbb{F}_{p}})(T_{w})\circ\operatorname{perm}(w)\quad\textrm{for all w\in W},**

• •

$\quad\mathscr{A}_{2,\mathbb{F}_{p}}|_{\mathbb{F}_{p}\varepsilon_{1}\times\mathbb{F}_{p}\varepsilon_{2}}=\operatorname{proj}.$**

The homomorphism $\mathscr{A}_{2,\mathbb{F}_{p}}$ is injective.

Proof.

Recall that $W_{0}$ acts on the set $\{1,2\}$ by interchanging the two elements and then $W$ acts via its projection to $W_{0}$. As $\{\varepsilon_{i}T_{w},(i,w)\in\{1,2\}\times W\}$ is a $\mathbb{F}_{p}$-basis of $\mathcal{H}_{2,\mathbb{F}_{p}}$, such a ring homomorphism is uniquely determined by the formula

[TABLE]

Conversely, taking this formula as a definition of $\mathscr{A}_{2,\mathbb{F}_{p}}$, we need to check that the resulting $\mathbb{F}_{p}$-linear map is a ring homomorphism, i.e.

[TABLE]

and

[TABLE]

The first condition is clear because $(1,1)=\varepsilon_{1}+\varepsilon_{2}$ and $p_{i}+p_{{}^{s}i}=\operatorname{Id}$. Let us check the second condition. If $i^{\prime}\neq{}^{w^{-1}}i$, i.e. $i\neq{}^{w}i^{\prime}$, then both sides of the claimed equality vanish. Now assume that $i={}^{w}i^{\prime}$. On the left hand side we find

[TABLE]

while on the right hand side, we find

[TABLE]

If $\ell(ww^{\prime})\neq\ell(w)+\ell(w^{\prime})$, then $T_{w}T_{w}^{\prime}=0$ and both sides vanish. Otherwise $T_{w}T_{w}^{\prime}=T_{ww^{\prime}}$, so that the left hand side becomes

[TABLE]

and the right hand side

[TABLE]

These two operators are equal. This proves the existence and the uniqueness of the extension $\mathscr{A}_{2,\mathbb{F}_{p}}$. Its injectivity will be shown in the next subsection. ∎

4.6 Faithfulness of $\mathscr{A}_{2,\mathbb{F}_{p}}$

Let us show that the map $\mathscr{A}_{2,\mathbb{F}_{p}}$ is injective. This is equivalent to show that the family

[TABLE]

is free over $\mathbb{F}_{p}$. So let $\{n_{i,w}\}\in\mathbb{F}_{p}^{(\{1,2\}\times W)}$ such that

[TABLE]

Let us fix $i_{0}\in\{1,2\}$. Composing by $p_{i_{0}}$ on the left, we get

[TABLE]

The left hand side can be rewritten as

[TABLE]

Now let us fix $w_{0}\in W_{0}$. Composing by $p_{{}^{w_{0}^{-1}}i_{0}}$ on the right, we get

[TABLE]

Then, for each $w\in\Lambda w_{0}$, remark that

[TABLE]

in \operatorname{End}\big{(}CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}_{1})[\xi_{2}^{-1}]\times CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}_{2})[\xi_{2}^{-1}]\big{)}, where $\iota_{i_{0},{}^{w_{0}^{-1}}i_{0}}$ is the canonical map

[TABLE]

As the latter is injective, we get

[TABLE]

Finally, as $p_{{}^{w_{0}^{-1}}i_{0}}:CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]\rightarrow CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}_{{}^{w_{0}^{-1}}i_{0}})[\xi_{2}^{-1}]$ is surjective, and as $\mathscr{A}_{\mathbb{F}_{p}}^{\operatorname{nil}}$ is injective, cf. 4.4, we get $n_{i_{0},w}=0$ for all $w\in\Lambda w_{0}$. This concludes the proof that $\mathscr{A}_{2,\mathbb{F}_{p}}$ is injective.

4.7 Supersingular modules

In this section we work over the algebraic closure $\overline{\mathbb{F}}_{q}$ of the field ${\mathbb{F}}_{q}$.

** 4.7.1****.**

Recall from 2.4.4 that

[TABLE]

has the structure of a $2\times 2$-matrix algebra over its center $Z(\mathcal{H}_{2,\overline{\mathbb{F}}_{q}})$. Since $\overline{\mathbb{F}}_{q}$ is algebraically closed, $Z(\mathcal{H}_{2,\overline{\mathbb{F}}_{q}})$ acts on any finite-dimensional irreducible $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module by a character (Schur’s lemma). Let $\theta$ be a character of $Z(\mathcal{H}_{2,\overline{\mathbb{F}}_{q}})$. Then

[TABLE]

is isomorphic to the matrix algebra $M(2,\overline{\mathbb{F}}_{q})$. In particular, it is a semisimple (even simple) ring.

** 4.7.2****.**

The unique irreducible $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module with central character $\theta$ is called the standard module with character $\theta$. Its $\overline{\mathbb{F}}_{q}$-dimension is $2$ and it is isomorphic to the standard representation $\overline{\mathbb{F}}_{q}^{\oplus 2}$ of the matrix algebra $M(2,\overline{\mathbb{F}}_{q})$. The image of the basis $\{(1,0),(0,1)\}$ of $\overline{\mathbb{F}}_{q}^{\oplus 2}$ is called a standard basis. A central character $\theta$ is called supersingular if $\theta(X)=\theta(Y)=0$ (or, equivalently, if $\theta(\zeta_{1})=0$). If $\theta$ is supersingular, then the affine algebra $\mathcal{H}_{2,\operatorname{aff},\overline{\mathbb{F}}_{q}}$ acts on the standard basis of the module via the characters $\chi_{1}$ respectively $\chi_{2}$ and the action of $U$ interchanges the two, cf. 2.4.3 and 2.4.4.

For more details we refer to [V04, 2.3].

** 4.7.3****.**

Now consider the base change of the representation $\mathscr{A}_{2,\mathbb{F}_{p}}$ to $\overline{\mathbb{F}}_{q}$

[TABLE]

Recall that the image under the map $\mathscr{A}_{2,\overline{\mathbb{F}}_{q}}$ of the central subring

[TABLE]

is the subring of scalars

[TABLE]

** 4.7.4****.**

Let us fix a supersingular central character $\theta$ and denote its restriction to $Z^{\circ}:=Z^{\circ}(\mathcal{H}_{2,\overline{\mathbb{F}}_{q}})$ by $\theta$, too. Then consider the $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-action on the base change

[TABLE]

For the base ring, we have

[TABLE]

where $\mathscr{A}_{2,\overline{\mathbb{F}}_{q}}(\zeta_{1})=-\xi_{1}$ and $\mathscr{A}_{2,\overline{\mathbb{F}}_{q}}(\zeta_{2})=\xi_{2}^{2}$. Now put $\theta(\zeta_{2})=:b\in\overline{\mathbb{F}}_{q}^{\times}$. Then

[TABLE]

and so

[TABLE]

Note that the $\overline{\mathbb{F}}_{q}$-algebra $A$ is isomorphic to the direct product $\overline{\mathbb{F}}_{q}\times\overline{\mathbb{F}}_{q}$ (the isomorphism depending on the choice of a square root of $b$ in $\overline{\mathbb{F}}_{q}$). An $A$-basis of $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]_{\theta}$ is given by the four elements $\{1_{i},\frac{\eta_{1}-\eta_{2}}{2}1_{i}\}_{i=1,2}$ where

[TABLE]

is the equivariant Chern class of the structure sheaf on $\widehat{\mathcal{B}}_{i}$, for $i=1,2$. The $\overline{\mathbb{F}}_{q}$-dimension of $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]_{\theta}$ is therefore $8$ and $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$ acts $A$-linearly. The length of the $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]_{\theta}$ is $4$ and the central character of any irreducible subquotient is necessarily equal to $\theta$, since this is true by construction after restriction to $Z^{\circ}$. In the following, we compute explicitly a composition series.

** 4.7.5****.**

Proposition.* The algebra $\mathcal{H}_{2,\operatorname{aff},\overline{\mathbb{F}}_{q}}$ acts on $1_{i}\in CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]_{\theta}$ by the supersingular character $\chi_{i}$, for $i=1,2$.*

Proof :

The action of $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$ on $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})_{\overline{\mathbb{F}}_{q}}[\xi_{2}^{-1}]$ is defined by the map $\mathscr{A}_{2,\overline{\mathbb{F}}_{q}}$. Hence, by definition,

[TABLE]

We calculate

[TABLE]

Moreover,

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

since $\xi_{1}=0$ in $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}_{i})[\xi_{2}^{-1}]_{\theta}$. It follows that $S_{0}\cdot 1_{i}=USU^{-1}\cdot 1_{i}=0$. $\square$

** 4.7.6****.**

Proposition.* A composition series with simple subquotients of the $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module*

[TABLE]

is given by

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Here the direct sums $\oplus$ are taken in the sense of $\overline{\mathbb{F}}_{q}$-vector spaces.

Proof.

First of all,

[TABLE]

because $0=\xi_{1}=\eta_{1}+\eta_{2}$ and $0=\xi_{1}^{2}=\eta_{1}^{2}+\eta_{2}^{2}+2\xi_{2}$ in $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})[\xi_{2}^{-1}]_{\theta}$. Hence the three first $\oplus$ appearing in the statement of the proposition are indeed direct sums. These three sums are $U$-stable by construction. Moreover, by the preceding proposition, $\mathcal{H}_{2,\operatorname{aff},\overline{\mathbb{F}}_{q}}$ acts by the character $\chi_{i}$ on $1_{i}$, hence by the character $\chi_{{}^{s}i}$ on $U\cdot 1_{i}$. It follows that $\overline{\mathbb{F}}_{q}1_{i}\oplus\overline{\mathbb{F}}_{q}(U\cdot 1_{i})$ realizes the standard $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module with central character $\theta$, and that $A1_{i}\oplus A(U\cdot 1_{i})$ is an $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-submodule of $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]_{\theta}$ of dimension $4$ over $\overline{\mathbb{F}}_{q}$. In fact, if $L\subset A$ is any $\overline{\mathbb{F}}_{q}$-line, the same arguments show that $L1_{i}\oplus L(U\cdot 1_{i})$ realizes the standard $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module with central character $\theta$. In particular, the module $A1_{i}\oplus A(U\cdot 1_{i})$ is semisimple.

Now let us compute the action of $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$ on the element $\frac{\eta_{1}-\eta_{2}}{2}1_{i}$, for $i=1,2$. We have

[TABLE]

Next

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

because $\xi_{1}=0$ and (hence) $\eta_{1}^{2}=-\xi_{2}$ in $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}})[\xi_{2}^{-1}]_{\theta}$, and finally

[TABLE]

which lies in $A^{\times}(\frac{\eta_{1}-\eta_{2}}{2}1_{{}^{s}i})$. Neither of the two elements $\frac{\eta_{1}-\eta_{2}}{2}1_{i}$ and $U\cdot\frac{\eta_{1}-\eta_{2}}{2}1_{i}$ lies in the (semisimple) module $A1_{i}\oplus A(U\cdot 1_{i})$. Hence the three last $\oplus$ appearing in the statement are indeed direct and they form a sub-$\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module of dimension $6$ over $\overline{\mathbb{F}}_{q}$. So the series appearing in the statement is indeed a composition series with irreducible subquotients. ∎

** 4.7.7****.**

Remark. We see from the proof of the preceding proposition that the characters of $\mathcal{H}_{2,\operatorname{aff},\overline{\mathbb{F}}_{q}}$ in the sub-$\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}$-module

[TABLE]

are contained in $A1_{i}\oplus A1_{{}^{s}i}$. Hence this submodule is not semi-simple. A fortiori the whole module $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{2})[\xi_{2}^{-1}]_{\theta}$ is not semisimple and, hence, has no central character.

** 4.7.8****.**

Now we transfer this discussion to any regular component of the algebra $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}$ as follows. Let $\gamma=\{\lambda,{}^{s}\lambda\}\in\mathbb{T}^{\vee}/W_{0}$ be a regular orbit and form the $\overline{\mathbb{F}}_{q}$-variety

[TABLE]

where $\widehat{\mathcal{B}}_{\lambda}$ and $\widehat{\mathcal{B}}_{{}^{s}\lambda}$ are two copies of $\widehat{\mathcal{B}}$. We have the algebra isomorphism $\mathcal{H}_{2,\overline{\mathbb{F}}_{q}}\stackrel{{\scriptstyle\simeq}}{{\rightarrow}}\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$ from 2.4.5. In this way, the representation $\mathscr{A}_{2,\overline{\mathbb{F}}_{q}}$ induces a representation

[TABLE]

We may then state, in obvious terminology, that any supersingular character $\theta$ of the center of $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$ gives rise to the $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$-module $CH^{\widehat{\mathbf{G}}}(\widehat{\mathcal{B}}^{\gamma})[\xi_{2}^{-1}]_{\theta}$ and that the semisimplification of the latter module equals a direct sum of four copies of the unique supersingular $\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{(1)}\varepsilon_{\gamma}$-module with central character $\theta$.

5 Tame Galois representations and supersingular modules

Our reference for basic results on tame Galois representations is [V94].

** 5.1****.**

Let $\varpi\in o_{F}$ be a uniformizer and let $f$ be the degree of the residue field extension $\mathbb{F}_{q}/\mathbb{F}_{p}$, i.e. $q=p^{f}$. Let ${\rm Gal}(\overline{F}/F)$ denote the absolute Galois group of $F$. Let $\mathcal{I}\subset{\rm Gal}(\overline{F}/F)$ be its inertia subgroup. We fix an element $\varphi\in{\rm Gal}(\overline{F}/F)$ lifting the Frobenius $x\mapsto x^{q}$ on ${\rm Gal}(\overline{F}/F)/\mathcal{I}$. The unique pro-$p$-Sylow subgroup of $\mathcal{I}$ is denoted by $\mathcal{P}$ (the wild inertia subgroup) and the quotient $\mathcal{I}/\mathcal{P}$ is pro-cyclic with pro-order prime to $p$. We choose a lift $v\in\mathcal{I}$ of a topological generator for $\mathcal{I}/\mathcal{P}$. Let $\mathcal{W}\subset{\rm Gal}(\overline{F}/F)$ denote the Weil group of $F$. The quotient group $\mathcal{W}/\mathcal{P}$ is topologically generated by (the images of) $\varphi$ and $v$ and the only relation between these two generators is $\varphi v\varphi^{-1}=v^{q}.$ There is a topological isomorphism

[TABLE]

where the projective limit is taken with respect to the norm maps $\mathbb{F}_{p^{nm}}^{\times}\rightarrow\mathbb{F}_{p^{n}}^{\times}$. We denote by $\omega_{n}$ the projection map $\mathcal{W}/\mathcal{P}\rightarrow\mathbb{F}_{p^{n}}^{\times}$ followed by the inclusion $\mathbb{F}_{p^{n}}^{\times}\subseteq\overline{\mathbb{F}}_{q}^{\times}$. We shall only be concerned with the characters $\omega_{f}$ and $\omega_{2f}$. The character $\omega_{f}$ extends from $\mathcal{W}$ to ${\rm Gal}(\overline{F}/F)$ by choosing a root $\sqrt[q-1]{-\varpi}$ and letting ${\rm Gal}(\overline{F}/F)$ act as

[TABLE]

followed by reduction mod $\varpi$. The character

[TABLE]

depends on the choice of $\varpi$ (but not on the choice of $\sqrt[q-1]{-\varpi}$) and equals the reduction mod $\varpi_{F}$ of the Lubin-Tate character $\chi_{L}:{\rm Gal}(\overline{F}/F)\rightarrow o_{F}^{\times}$ associated to the uniformizer $\varpi$. By changing $\varphi$ by an element of $\mathcal{I}$, if necessary, we may assume $\omega_{f}(\varphi)=1$. We normalize local class field theory $\mathcal{W}^{\rm ab}\simeq F^{\times}$ by sending the geometric Frobenius $\varphi^{-1}$ to $\varpi$. We view the restriction of $\omega_{f}$ to $\mathcal{W}$ as a character of $F^{\times}$.

** 5.2****.**

The set of isomorphism classes of irreducible smooth Galois representations

[TABLE]

is in bijection with the set of equivalence classes of pairs $(s,t)\in\widehat{\mathbf{G}}^{2}$ such that

[TABLE]

with $b\in\overline{\mathbb{F}}_{q}^{\times}$ and $y\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}.$ Here, two pairs $(s,t)$ and $(s^{\prime},t^{\prime})$ are equivalent if $s=s^{\prime}$ and $t,t^{\prime}$ are ${\rm Gal}(\mathbb{F}_{q^{2}}/\mathbb{F}_{q})$-conjugate. Note that $\det(s)=b$ and that $sts^{-1}=t^{q}$. The bijection is induced by the map $\rho\mapsto(\rho(\varphi),\rho(v))$. The number of equivalence classes of such pairs $(s,t)$ equals $\frac{q^{2}-q}{2}$ and hence coincides with the number of $W_{0}$-orbits in $\mathbb{T}^{\vee}$.

** 5.3****.**

By the above numerical coincidence (the ”miracle” from [V04]), there exist (many) bijections between the isomorphism classes of irreducible smooth two-dimensional Galois representations and the isomorphism classes of supersingular two-dimensional $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$-modules. In the following we discuss a a certain example of such a bijection in our geometric language.

Let $\rho$ be a two-dimensional irreducible smooth Galois representation with parameters $(s,t)$. Since the element $\omega_{2f}(v)$ generates $\mathbb{F}_{q^{2}}^{\times}$, the element $t$ uniquely determines an exponent $1\leq h\leq q^{2}-1$, such that

[TABLE]

Replacing $\rho$ by an isomorphic representation $\rho^{\prime}$ which replaces $y$ by its Galois conjugate $y^{q}$ replaces $h$ by the rest of the euclidian division of $qh$ by $q^{2}-1$. We call either of the two numbers an exponent of $\rho$.

** 5.4****.**

Lemma.* There is $0\leq i\leq q-2$ such that $\rho\otimes\omega_{f}^{-i}$ has an exponent $\leq q-1$.*

Proof :

This is implicit in the discussion in [V94]. Let $\omega_{2f}(v)^{h}=y$. Then $h\leq q^{2}-2$ since $y\neq 1$. Moreover, $q^{2}-2-(q-2)(q+1)=q$. Since $\omega_{2f}^{q+1}=\omega_{f}$, twisting with $\omega_{f}$ reduces to the case $h\leq q$. Replacing $y$ by its Galois conjugate $y^{q}$, if necessary, reduces then further to $h\leq q-1$. $\square$

By the lemma, we may associate two numbers $1\leq h\leq q-1$ and $0\leq i\leq q-2$ to the representation $\rho$. We form the character

[TABLE]

and restrict to $\mu_{q-1}(F)^{2}.$ This gives rise to an element $\lambda(\rho)$ of $\mathbb{T}^{\vee}$ and we take its $W_{0}$-orbit $\gamma_{\rho}$.

** 5.5****.**

Lemma.* The orbit $\gamma_{\rho}$ depends only on the isomorphism class of $\rho$.*

Proof :

Suppose $\rho^{\prime}\simeq\rho$ with

[TABLE]

By the preceding lemma, there is $0\leq i\leq q-2$ and an exponent $1\leq h\leq q-1$ of $\rho\otimes\omega_{f}^{-i}$. If $1<h$, then by definition $\omega_{2f}^{h}(v)=y\omega_{f}^{-i}(v)$, so that $\omega_{2f}^{qh}(v)=y^{q}\omega_{f}^{-i}(v)$, and hence $\omega_{2f}^{q-(h-1)}(v)=y^{q}\omega_{f}^{-(h-1+i)}(v)$, using $qh=q-(h-1)+(h-1)(q+1)$. Then $1\leq h^{\prime}:=q-(h-1)\leq q-1$ and taking $0\leq i^{\prime}\leq q-2$ congruent to $h-1+i~{}{\rm mod}~{}q-1$, we obtain that $h^{\prime}$ is an exponent for $\rho^{\prime}\otimes\omega_{f}^{-i^{\prime}}$. In particular, $\lambda(\rho^{\prime}):=\omega_{f}^{h^{\prime}-1+i^{\prime}}\otimes\omega_{f}^{i^{\prime}}$, which is $s$-conjugate to $\lambda(\rho)$. If $h=1$, then by definition $\omega_{2f}(v)=y^{q}\omega_{f}^{-i}(v)$, which implies $\lambda(\rho^{\prime})=\lambda(\rho)$ in this case. $\square$

We call $\rho$ (non-)regular if the orbit $\gamma_{\rho}$ is (non-)regular. On the other hand, we view the element $s=\rho(\varphi)$ as a supersingular character $\theta_{\rho}$ of the center $Z(\mathcal{H}_{\overline{\mathbb{F}}_{q}}^{\gamma_{\rho}})$, i.e. $\theta_{\rho}(\zeta_{1})=0$ and $\theta_{\rho}(\zeta_{2})=b$. Finally, we have the $\overline{\mathbb{F}}_{q}$-variety

[TABLE]

coming from the quotient map $\mathbb{T}^{\vee}\rightarrow\mathbb{T}^{\vee}/W_{0}$. These data give rise to the supersingular $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$-module

[TABLE]

Recall that $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$ acts on $\mathcal{M}(\rho)$ via the projection onto $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}\varepsilon_{\gamma_{\rho}}$ followed by the extended Demazure representation $\mathscr{A}^{\gamma_{\rho}}_{\overline{\mathbb{F}}_{q}}$. Recall also that the semisimplification of $\mathcal{M}(\rho)$ is a direct sum of four copies of the supersingular standard module, if $\rho$ is regular. By abuse of notation, we denote a simple subquotient of $\mathcal{M}(\rho)$ again by $\mathcal{M}(\rho)$.

** 5.6****.**

Proposition.* The map $\rho\mapsto\mathcal{M}(\rho)$ gives a bijection between the isomorphism classes of two-dimensional irreducible smooth $\overline{\mathbb{F}}_{q}$-representations of ${\rm Gal}(\overline{F}/F)$ and the isomorphism classes of two-dimensional supersingular $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$-modules.*

Proof :

By construction, the restriction of $\omega_{f}^{h-1}$ to $\mu_{q-1}(F)\simeq\mathbb{F}_{q}^{\times}$ is given by the exponentiation $x\mapsto x^{h-1}$. Given $0\leq i\leq q-2$ and $1\leq h\leq q-1$, and $b\in\overline{\mathbb{F}}^{\times}_{q}$, the parameter $y:=\omega_{2f}(v)^{h}$ lies in $\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ and the pair $(s,t)$ determines a Galois representation $\rho$ having $h$ comme exponent. Hence, $\rho\otimes\omega_{f}^{i}$ gives rise to the character $\omega_{f}^{h-1+i}\otimes\omega_{f}^{i}$. The elements of type $\gamma_{\rho}$ exhaust therefore all orbits in $\mathbb{T}^{\vee}/W_{0}$. Since a two-dimensional supersingular $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$-module is determined by its $\gamma$-component and its central character, the map $\rho\mapsto\mathcal{M}(\rho)$ is seen to be surjective. It is then bijective, since source and target have the same cardinality. $\square$

** 5.7****.**

Let $F$ be a finite extension of $\mathbb{Q}_{p}$. A distinguished natural bijection between irreducible two-dimensional ${\rm Gal}(\overline{F}/F)$-representations and supersingular two-dimensional $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$-modules is established by Breuil [Br03] for $F=\mathbb{Q}_{p}$ (see [Be11] for its relation to the $p$-adic local Langlands correspondence for ${\rm GL}_{2}(\mathbb{Q}_{p})$) and by Grosse-Klönne [GK18] for general $F/\mathbb{Q}_{p}$. In this final paragraph we will show that the bijection $\rho\mapsto\mathcal{M}(\rho)$ from 5.6 coincides in this case with the bijections [Br03] and [GK18].

The case $F=\mathbb{Q}_{p}$ follows directly from the explicit formulae given in [Be11, 1.3]. For the general case, we briefly recall the main construction from [GK18] in the case of standard supersingular modules of dimension $2$. Let $F_{\phi}$ be the special Lubin-Tate group with Frobenius power series $\phi(t)=\varpi t+t^{q}$. Let $F_{\infty}/F$ be the extension generated by all torsion points of $F_{\phi}$ and let $\Gamma={\rm Gal}(F_{\infty}/F)$. We identify in the following $\Gamma\simeq o_{F}^{\times}$ via the character $\chi_{L}$.

Let $k/\mathbb{F}_{q}$ be a finite extension and let $\mathcal{H}^{(1)}_{k}:=\mathcal{H}^{(1)}(\mathbf{q})\otimes_{\mathbb{Z}[\mathbf{q}]}k$ via $\mathbf{q}\mapsto q=0$. Let $M$ be a two-dimensional standard supersingular $\mathcal{H}^{(1)}_{k}$-module, arising from a supersingular character $\chi:\mathcal{H}^{(1)}_{{\rm aff},k}\rightarrow k$. Let $e_{0}\in M$ such that $\mathcal{H}^{(1)}_{{\rm aff},k}$ acts on $e_{0}$ via $\chi$ and put $e_{1}=T^{-1}_{\omega}e_{0}$ (where $\omega=u^{-1}$ in our notation).111For example, if $M$ is an $\mathcal{H}_{\theta}$-module on which $U^{2}=\zeta_{2}$ acts via the scalar $\theta(\zeta_{2})=\tau_{2}$, then $U=U^{-1}\cdot\tau_{2}$ on $M$ and $m:=\tau_{2}^{-1}e_{1}$ satisfies $Um=T_{\omega}e_{1}=e_{0}$, i.e. $\{m,Um\}$ is a standard basis for $M$ in the sense of 3.5.1. The character $\chi$ determines two numbers $0\leq k_{0},k_{1}\leq q-1$ with $(k_{0},k_{1})\neq(0,0),(q-1,q-1)$. One considers $M$ a $k[[t]]$-module with $t=0$ on $M$. Let $\Gamma=o_{F}^{\times}$ act on $M$ via

[TABLE]

for $\gamma\in o_{F}^{\times}$ with reduction $\overline{\gamma}\in\mathbb{F}_{q}^{\times}$ and $\eta_{1}(\overline{\gamma})^{-1}=\operatorname{diag}(\overline{\gamma}^{-1},1)\in\mathbb{T}$. The $k[[t]][\varphi]$-submodule $\nabla(M)$ of

[TABLE]

is then generated by the two elements $h(e_{j})=t^{k_{j}}\varphi\otimes T_{\omega}^{-1}(e_{j})+1\otimes e_{j}$ thereby defining the relation between the Frobenius $\varphi$ and the Hecke action of $T_{\omega}$. Note that in the case of ${\rm GL}_{2}$, the cocharacter $e^{*}$ of [GK18, 2.1] is equal to $\eta_{1}.$

The module $\nabla(M)$ is stable under the $\Gamma$-action and thus the quotient

[TABLE]

defines a $k[[t]][\varphi,\Gamma]$-module. It is torsion standard cyclic with weights $(k_{0},k_{1})$ in the sense of [GK18, 1.3], according to [GK18, Lemma 5.1]. Let $\Delta(M)^{*}=\operatorname{Hom}_{k}(\Delta(M),k)$. By a general construction (which goes back to Colmez and Emerton in the case $F=\mathbb{Q}_{p}$ and $\phi(t)=(1+t)^{p}-1$, as recalled in [Br15, 2.6]) the $k((t))$-vector space

[TABLE]

is in a natural way an étale Lubin-Tate $(\varphi,\Gamma)$-module of dimension $2$. The correspondence $M\mapsto\Delta(M)^{*}\otimes_{k[[t]]}k((t))$ extends in fact to a fully faithful functor from a suitable category of supersingular $\mathcal{H}^{(1)}_{k}$-modules to the category of étale $(\varphi,\Gamma)$-modules over $k((t))$. The composite functor to the category of continuous ${\rm Gal}(\overline{F}/F)$-representations over $k$ is denoted by $M\mapsto V(M)$. It induces the aforementioned bijection between irreducible two-dimensional ${\rm Gal}(\overline{F}/F)$-representations and supersingular two-dimensional $\mathcal{H}^{(1)}_{\overline{\mathbb{F}}_{q}}$-modules.

** 5.8****.**

Proposition.* The inverse map to the bijection $M\mapsto V(M)$ is given by the map $\rho\mapsto\mathcal{M}(\rho)$.*

Proof :

The correspondence $M\mapsto V(M)$ is compatible with the twist by a character of $F^{\times}$ and local class field theory, such that the determinant corresponds to the central character restricted to $F^{\times}$. By its very construction, the same is true for the correspondence $\rho\mapsto\mathcal{M}(\rho)$. It therefore suffices to compare them on irreducible Galois representations having parameters $b=1$ and $i=0$. Let $k=\mathbb{F}_{q^{2}}$ in the following. Let $\operatorname{ind}(\omega_{2f}^{h})$ be the Galois representation with exponent $1\leq h\leq q-1$ and $b=1$ and $i=0$. Let $D$ be the $(\varphi,\Gamma)$-module associated to $\rho:=\operatorname{ind}(\omega_{2f}^{h})$ and let $M$ be a supersingular $\mathcal{H}^{(1)}_{k}$-module such that $\Delta(M)^{*}\otimes_{k[[t]]}k((t))\simeq D$. According to the main result of [PS3] for $n=2$, the module $D$ has a basis $\{g_{0},g_{1}\}$ such that

[TABLE]

for all $\gamma\in\Gamma$ and $\varphi(g_{0})=g_{1}$ and $\varphi(g_{1})=-t^{-h(q-1)}g_{0}$. Here, $\overline{f}_{\gamma}(t)=\omega_{f}(t)t/\gamma(t)\in k[[t]]^{\times}$. Define the triple $(k_{0},k_{1},k_{2})=(h-1,q-h,h-1)$ and let $i_{j}:=q-1-k_{2-j}$, so that $i_{0}=i_{2}=q-h$ and $i_{1}=2q-h-1$. Define the triple $(h_{0},h_{1},h_{2})=(0,i_{1},i_{0}+i_{1}q)$. Note that $h_{2}=h(q-1).$ Put $f_{j}=t^{h_{j}}g_{j}$ for $j=0,1$ and let $D^{\sharp}\subset D$ be the $k[[t]]$-submodule generated by $\{f_{0},f_{1}\}$. Let $(D^{\sharp})^{\ast}$ be the $k$-linear dual. Define $e_{i}^{\prime}\in(D^{\sharp})^{\ast}$ via $e^{\prime}_{i}(f_{j})=\delta_{ij}$ and $e^{\prime}_{i}=0$ on $tD^{\sharp}$. Using the explicit formulae for the $\psi$-operator on $k((t))$ as described in [GK18, Lemma 1.1] one may follow the argument of [GK16, Lemma 6.4] and show that $D^{\sharp}$ is a $\psi$-stable lattice in $D$ and that $\{e^{\prime}_{0},e^{\prime}_{1}\}$ is a $k$-basis of the $t$-torsion part of $(D^{\sharp})^{\ast}$ satisfying

[TABLE]

But according to [GK18, 1.15] there is only one $\psi$-stable lattice in $\Delta(M)^{*}\otimes_{k[[t]]}k((t))$, namely $\Delta(M)^{*}$. It follows that $\Delta(M)\simeq(D^{\sharp})^{\ast}$ and so the weights of the torsion standard cyclic $k[[t]][\varphi,\Gamma]$-module $\Delta(M)$ are $(k_{0},k_{1})$. Since $k_{0}=h-1$, one deduces from [GK18, Lemma 4.1/5.1] that $\epsilon_{1}\equiv h-1\mod(q-1).$ This means $\lambda\circ\alpha^{\vee}(x)^{-1}=x^{h-1}$ for the character $\lambda\in\mathbb{T}^{\vee}$ of $M$. Since $i=0$ and hence $a=0$ (in the notation of [GK16, 2.2]), we arrive therefore at

[TABLE]

Hence the image of $\lambda$ in $\mathbb{T}^{\vee}/W_{0}$ coincides with $\gamma_{\rho}$. This implies $M\simeq\mathcal{M}(\rho)$, as claimed. $\square$