On the Bonnaf\'e--Dat--Rouquier Morita equivalence

TL;DR

This paper extends Morita equivalence results to new cases involving Deligne-Lusztig varieties, specifically for semisimple elements in type D with non-cyclic centralizer component groups, broadening previous work.

Contribution

It proves a Morita equivalence via cohomology of Deligne-Lusztig varieties in cases not previously covered, notably for type D with non-cyclic centralizers.

Findings

Established Morita equivalence for specific semisimple elements in type D

Extended the applicability of Deligne-Lusztig cohomology in representation theory

Provided new cases where Morita equivalence holds beyond prior known scenarios

Abstract

We prove that the cohomology group of a Deligne-Lusztig variety defines a Morita equivalence in a case which is not covered by the argument by Bonnaf\'e, Dat and Rouquier, specifically we consider the situation for semisimple elements in type $D$ whose centralizer has non-cyclic component group.

Equations

$$({\overline{{\mathbb{F}}_q}}^{\times }{)}^n\to {\mathbf{T}}_0,({\lambda }_1,\dots ,{\lambda }_n)\mapsto \prod _{i=1}^n{\mathbf{h}}_{e_i}({\lambda }_i)$$

$$I_{\mathcal{A}}(\lambda ):=\{j\in \{1,\dots ,n\}\mid {\lambda }_j\in \mathcal{A}\}.$$

$$\{\sigma \in S_{\{\pm 1,\dots ,\pm n\}}\mid \sigma (-i)=-\sigma (i)\text{ for all }i=1,\dots ,n\}$$

$$\epsilon :\overline{W}\to \{\pm 1\},\sigma \mapsto (-1{)}^{|\{i\in \{1,\dots ,n\}\mid \sigma (i)<0\}|}.$$

$${\mathbf{N}}^{\ast }/{\mathbf{L}}^{\ast }\hookrightarrow {\mathrm{C}}_{\mathbf{G}}^{\ast }(s)/{\mathrm{C}}_{{\mathbf{G}}^{\ast }}^{\circ }(s).$$

$${\omega }_s:{\mathrm{C}}_{\mathbf{G}}^{\ast }(s)\to \mathrm{Z}(\mathbf{G}),x\mapsto [y,t]$$

$${\mathrm{C}}_{\mathbf{G}}^{\ast }(s)/{\mathrm{C}}_{{\mathbf{G}}^{\ast }}^{\circ }(s)\hookrightarrow \mathrm{Z}(\mathbf{G}).$$

$$W(s)/W^{\circ }(s)\to {\mathrm{C}}_{{\mathbf{G}}^{\ast }}(s)/{\mathrm{C}}_{{\mathbf{G}}^{\ast }}^{\circ }(s).$$

$$W(s{)}^F/W^{\circ }(s{)}^F\to ({\mathrm{C}}_{\mathbf{G}\ast }(s){)}^{F^{\ast }}/({\mathrm{C}}_{{\mathbf{G}}^{\ast }}^{\circ }(s){)}^{F^{\ast }}$$

$$({\mathrm{C}}_{{\mathbf{G}}^{\ast }}(s)/{\mathrm{C}}_{{\mathbf{G}}^{\ast }}^{\circ }(s){)}^{F^{\ast }}\cong \mathrm{Z}(\mathbf{G}{)}^F\cong C_2\times C_2$$

$$\varnothing =I_{\{\pm 1,\pm {\omega }_4\}}(t)=I_{\{\pm 1,\pm {\omega }_4\}}(t{z}_1)=I_{\{\pm 1,\pm {\omega }_4\}}({}^{w_1}t).$$

$$w:=(x,-x{)}^{|K|}\prod _{k\in K}(k,-k)\in W.$$

$${\mathrm{C}}_{\mathbf{G}}(t)=⟨{\mathbf{T}}_0,{\mathbf{x}}_{\alpha }(r)\mid \alpha \in \Phi \text{ with }\alpha (t)=1,r\in {\overline{{\mathbb{F}}_q}}^{\times }⟩.$$

$$\Phi (t)=\{\pm e_i\pm e_j\mid i,j\in I_{\{\pm 1\}}(t)\}\cup \{\pm e_i\pm e_j\mid i,j\in I_{\{\pm {\omega }_4\}}(t)\}\cup \{e_i-e_j\mid t_i=t_j\}\setminus \{0\}.$$

$$I_{\{\pm 1\}}({}^{w_2}t)=I_{\{\pm 1\}}(t{z}_2)=I_{\{\pm {\omega }_4\}}(t).$$

$$\mathrm{Z}(C_{\mathbf{G}}(t))=\bigcap _{\alpha \in \Phi (t)}\mathrm{Ker}(\alpha ).$$

$$\pi ({\mathrm{Z}}^{\circ }({\mathrm{C}}_{\mathbf{G}}(t)))={\mathrm{Z}}^{\circ }({\mathrm{C}}_{{\mathbf{G}}^{\ast }}^{\circ }(s)).$$

$$\Phi (t)=\{e_i-e_j\mid t_i=t_j\}\setminus \{0\}.$$

$${\mathbf{T}}_1:=⟨{\mathbf{h}}_{\alpha }(r)\mid r\in {\overline{{\mathbb{F}}_q}}^{\times },\alpha \in \{e_1\pm e_n\}⟩$$

$${\mathbf{G}}_2:=⟨{\mathbf{h}}_{\bar{\alpha }}(\bar{r}),{\mathbf{x}}_{\alpha }(r)\mid \bar{\alpha }\in {\overline{\Phi }}^{\prime },\alpha \in \Phi (t),r\in \overline{{\mathbb{F}}_q},\bar{r}\in {\overline{{\mathbb{F}}_q}}^{\times }⟩.$$

$$\mathrm{Res}:\{\sigma \in S_{\{\pm 1,\dots ,\pm n\}}\mid \sigma (1),\sigma (n)\in \{\pm 1,\pm n\}\}\to S_{\{\pm 1,\pm n\}}.$$

$$\mathrm{Res}(v)\in {\mathrm{C}}_{S_{\{\pm 1,\pm n\}}}(⟨(1,-1)(n,-n),(1,-n)(-1,n)⟩).$$

$${\mathbf{n}}_1:={\mathbf{n}}_{e_1}(1){\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}\in {\mathrm{N}}_{\mathbf{G}}({\mathbf{T}}_0),$$

$${\mathbf{n}}_2:={\mathbf{n}}_{e_1-e_n}(1){\mathbf{n}}_2^{\prime }\in {\mathrm{N}}_{\mathbf{G}}({\mathbf{T}}_0)$$

$${\mathbf{n}}_1^{n_2}={\mathbf{n}}_1^{{\mathbf{n}}_{e_1-e_n}(1)}={\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}{\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1{)}^2}={\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}{\mathbf{n}}_{e_1}(1{)}^{{\mathbf{h}}_{e_1-e_n}(-1)}.$$

$${\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}{\mathbf{n}}_{e_1}(1{)}^{{\mathbf{h}}_{e_1-e_n}(-1)}={\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}{\mathbf{n}}_{e_1}(1{)}^{{\mathbf{h}}_{e_1}({\omega }_4)}={\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}{\mathbf{n}}_{e_1}(1)z_1.$$

$$\begin{gathered} {\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}{\mathbf{n}}_{e_1}(1)z_1={\mathbf{n}}_{e_1}(1){\mathbf{n}}_{e_1}(1{)}^{{\mathbf{n}}_{e_1-e_n}(1)}= \\ {n}_1. \end{gathered}$$

$$\mathbf{m}:={\mathbf{m}}_1{\mathbf{m}}_2$$

$$(\mathbf{m}F)({\mathbf{n}}_2){\mathbf{n}}_2^{-1}={}^{\mathbf{m}}{\mathbf{n}}_2{\mathbf{n}}_2^{-1}={}^{{\mathbf{m}}_2}{\mathbf{n}}_2^{\prime }{\mathbf{n}}_2^{\prime -1}.$$

$$(\mathbf{m}F)({\mathbf{n}}_2){\mathbf{n}}_2^{-1}\in {\mathbf{T}}_2=⟨{\mathbf{h}}_{e_i}(r)\mid i\in I,r\in {\overline{{\mathbb{F}}_q}}^{\times }⟩.$$

Full text

Introduction

Let $\mathbf{G}$ be a connected reductive group with Frobenius endomorphism $F:\mathbf{G}\to\mathbf{G}$ defining an $\mathbb{F}_{q}$-structure, where $q$ is a power of a prime $p$. Let $\mathbf{G}^{\ast}$ be a group dual to $\mathbf{G}$ with dual Frobenius $F^{\ast}:\mathbf{G}^{\ast}\to\mathbf{G}^{\ast}$. Let $\ell$ be a prime number different from $p$ and $(\mathcal{O},K,k)$ an $\ell$-modular system as in [2, Section 2.A]. Let $s\in(\mathbf{G}^{\ast})^{F^{\ast}}$ be a semisimple element of $\ell^{\prime}$-order and $\mathbf{L}^{\ast}$ be the minimal $F^{\ast}$-stable Levi subgroup containing $\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ and $\mathbf{L}$ be the Levi subgroup of $\mathbf{G}$ dual to $\mathbf{L}^{\ast}$. In addition, let $e_{s}^{\mathbf{L}^{F}}\in\mathrm{Z}(\mathcal{O}\mathbf{L}^{F})$ be the central idempotent associated to $s\in(\mathbf{L}^{\ast})^{F^{\ast}}$, see [5, Theorem 9.12], and $\mathbf{N}^{F}$ denotes $\mathrm{N}_{\mathbf{G}^{F}}(\mathbf{L}^{F},e_{s}^{\mathbf{L}^{F}})$.

Let $\mathbf{P}=\mathbf{L}\mathbf{U}$ be a Levi decomposition in $\mathbf{G}$ and denote by $\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}}$ the associated Deligne–Lusztig variety on which $\mathbf{G}^{F}$ acts on the left and $\mathbf{L}^{F}$ acts on the right. Denote $d:=\mathrm{dim}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}})$ and let $H^{d}_{c}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})$ be the $d$th $\ell$-adic cohomology group with compact support of $\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}}$. Suppose that the $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{L}^{F}$-bimodule $H_{c}^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ extends to an $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodule. Then in [2, Theorem 7.7] the authors prove the astonishing result that this extended bimodule induces a Morita equivalence between $\mathcal{O}\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ and $\mathcal{O}\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$. This strengthens an earlier theorem of Bonnafé and Rouquier proving a conjecture of Broué [4].

Note however that [2, Theorem 7.7] was announced without the assumption that this bimodule extends. As the proof of [2, Proposition 7.3] is incomplete, this assumption is necessary at the moment. One case where this assumption is easily seen to be satisfied is when the quotient subgroup $\mathbf{N}^{F}/\mathbf{L}^{F}$ is cyclic, see Lemma 3 below.

Our aim in this note is to remove this technical assumption and therefore extend the results of [2, Theorem 7.7]. From now on we assume that $\mathbf{G}$ is a simple algebraic group. In this case, the quotient group $\mathbf{N}^{F}/\mathbf{L}^{F}$ embeds into $\mathrm{Z}(\mathbf{G})^{F}$. Therefore, a non-cyclic quotient can only appear if $\mathbf{G}$ is simply connected and $\mathbf{G}^{F}$ is of type $D_{n}$ with even $n\geq 4$. Hence we focus on this situation and prove the following statement.

Proposition 1**.**

Let $\mathbf{G}$ be a simple, simply connected algebraic group such that $\mathbf{G}^{F}$ is of type $D_{n}$ with even $n\geq 4$. If $\ell\nmid(q^{2}-1)$ then the $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{L}^{F}$-bimodule $H_{c}^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ extends to an $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodule.

The proof combines group theoretic descriptions of the relevant Levi subgroups and Clifford theoretic arguments tailored to this situation. Unfortunately, the restriction on $\ell$ seems to be necessary with the approach presented, see proof of Proposition 16.

Using an argument from the unpublished note [3] we show that the extended bimodule induces a Morita equivalence and from this we can deduce the validity of [2, Theorem 7.7] in this case.

Theorem 2**.**

Suppose that $\mathbf{G}$ is a simple algebraic group. If $\ell\nmid(q^{2}-1)$ or if $\mathbf{N}^{F}/\mathbf{L}^{F}$ is cyclic then the complex $G\Gamma_{c}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})^{\mathrm{red}}e_{s}^{\mathbf{L}^{F}}$ of $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{L}^{F}$-bimodules extends to a complex $C$ of $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodules. There exists a unique bijection $b\mapsto b^{\prime}$ between blocks of $\mathcal{O}\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$ and blocks of $\mathcal{O}\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ such that $bC\cong Cb^{\prime}$. For each block $b$ the complex $bCb^{\prime}$ induces a splendid Rickard equivalence between $b$ and $b^{\prime}$. Similarily, the bimodule $H^{d}(bCb^{\prime})$ induces a Morita equivalence between $b$ and $b^{\prime}$. The complex $bCb^{\prime}$ induces an isomorphism of the Brauer categories of $k\mathbf{G}^{F}b$ and $k\mathbf{G}^{F}b^{\prime}$. In particular, $b$ and $b^{\prime}$ have the same defect group.

Acknowledgement

Firstly, I would like to thank Cedric Bonnafé and Raphael Rouquier for insightful discussions. I also thank Marc Cabanes and Gunter Malle for helpful comments on a previous version of this paper. Finally, Britta Späth for her continuous support.

This material is partly based upon work supported by the NSF under Grant DMS-1440140 while the author was in residence at the MSRI, Berkeley CA. This research was conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology, which is funded by the DFG.

Notation

We introduce the notation which will be in force until the last section of this paper. Let $\mathbf{G}^{\ast}$ be a simple, adjoint algebraic group of type $D_{n}$ with $n$ even and $F^{\ast}:\mathbf{G}\to\mathbf{G}$ be a Frobenius endomorphism defining an $\mathbb{F}_{q}$-structure on $\mathbf{G}^{\ast}$ such that $\mathbf{G}^{F^{\ast}}$ is of untwisted type $D_{n}$. Fix a semisimple element $s\in(\mathbf{G}^{\ast})^{F^{\ast}}$. Then $\operatorname{C}_{\mathbf{G}^{\ast}}^{\circ}(s)$ is an $F^{\ast}$-stable connected reductive group. Thus, there exists a maximal $F^{\ast}$-stable torus $\mathbf{T}_{0}^{\ast}$ of $\operatorname{C}_{\mathbf{G}^{\ast}}^{\circ}(s)$ contained in an $F^{\ast}$-stable Borel subgroup $\mathbf{B}(s)$ of $\operatorname{C}_{\mathbf{G}^{\ast}}^{\circ}(s)$.

As the dual group $\mathbf{G}$ is of simply connected type, there exists a surjective morphism $\pi:\mathbf{G}\to\mathbf{G}^{\ast}$ with kernel $\operatorname{Z}(\mathbf{G})$. We let $\mathbf{T}_{0}$ be the maximal torus of $\mathbf{G}$ such that $\mathbf{T}_{0}^{\ast}=\pi(\mathbf{T}_{0})$. Let $F:\mathbf{G}\to\mathbf{G}$ be a Frobenius endomorphism stabilizing $\mathbf{T}_{0}$ such that $(\mathbf{G},\mathbf{T}_{0},F)$ is in duality with $(\mathbf{G}^{\ast},\mathbf{T}_{0}^{\ast},F^{\ast})$ via the the map $\pi:\mathbf{T}_{0}\to\mathbf{T}_{0}^{\ast}$.

We denote by $W$ the Weyl group of $\mathbf{G}$ with respect to $\mathbf{T}_{0}$ and by $W^{\ast}$ the Weyl group of $\mathbf{G}^{\ast}$ with respect to $\mathbf{T}_{0}^{\ast}$. The map $\pi$ induces an isomorphism $W\to W^{\ast}$ which allows $W$ to be identified with $W^{\ast}$. Under this identification, the anti-isomorphism ${}^{\ast}:W\to W^{\ast}$, induced by duality, is then given by inversion, i.e. $w^{\ast}=w^{-1}$ for all $w\in W$.

The root system of $\mathbf{G}$ can be described more explicitly as follows. Let $\overline{\Phi}$ be a root system of type $B_{n}$, $n$ even, with base $\{e_{1},e_{i}-e_{i-1}\mid 2\leq i\leq n\}$ where $\{e_{i}\mid 1\leq i\leq n\}$ is the canonical orthonormal basis with respect to the standard scalar product on $\mathbb{R}^{n}$.

Consider the root system $\Phi\subseteq\overline{\Phi}$ consisting of all long roots of $\overline{\Phi}$. Recall that $\Phi$ is a root system of type $D_{n}$. Let $\overline{\mathbf{G}}$ be the associated simple, simply connected algebraic group defined over $\overline{\mathbb{F}_{q}}$. By [10, Section 2.C] there exists an embedding $\mathbf{G}\hookrightarrow\overline{\mathbf{G}}$ such that the image of $\mathbf{T}_{0}$ is a maximal torus of $\overline{\mathbf{G}}$. In particular, we can identify $\Phi$ with the root system of $\mathbf{G}$ with respect to the torus $\mathbf{T}_{0}$ and $\overline{\Phi}$ with the root system of $\overline{\mathbf{G}}$ with respect to $\mathbf{T}_{0}$.

Let $\mathbf{x}_{\bar{\alpha}}(r),\mathbf{n}_{\bar{\alpha}}(r)$ and $\mathbf{h}_{\bar{\alpha}}(r)$ ($r\in\overline{\mathbb{F}_{q}}$ and $\bar{\alpha}\in\overline{\Phi}$) the Chevalley generators associated to the maximal torus $\overline{\mathbf{T}_{0}}$ of $\overline{\mathbf{G}}$ as in [9, Theorem 1.12.1].

Using the embedding of $\mathbf{G}$ into $\overline{\mathbf{G}}$ we obtain a surjective group homomorphism

[TABLE]

with kernel $\{(\lambda_{1},\dots,\lambda_{n})\in\{\pm 1\}^{n}\mid\prod_{i=1}^{n}\lambda_{i}=1\}$. Hence we can write an element $\lambda\in\mathbf{T}_{0}$ (in a non-unique way) as $\lambda=\prod_{i=1}^{n}\mathbf{h}_{e_{i}}(\lambda_{i})$ for suitable $\lambda_{i}\in\overline{\mathbb{F}_{q}}^{\times}$. For a subset $\mathcal{A}\subset\overline{\mathbb{F}_{q}}^{\times}$ with $\mathcal{A}=-\mathcal{A}$ we define

[TABLE]

Note that this does not depend on the choice of the sequence $(\lambda_{1},\dots,\lambda_{n})$ but only on the element $\lambda\in\mathbf{T}_{0}$. Let $\omega_{4}\in\overline{\mathbb{F}_{q}}^{\times}$ be a primitive $4$th root of unity. By [10, Section 2.C] we have $\operatorname{Z}(\mathbf{G})=\langle z_{1},z_{2}\rangle$, where $z_{1}=\mathbf{h}_{e_{1}}(-1)$ and $z_{2}=\prod_{i=1}^{n}\mathbf{h}_{e_{i}}(\omega_{4})$.

We also fix a tuple $(t_{1},\dots,t_{n})\in(\overline{\mathbb{F}_{q}}^{\times})^{n}$ such that $t=\prod_{i=1}^{n}\mathbf{h}_{e_{i}}(t_{i})\in\mathbf{T}_{0}$ satisfies $\pi(t)=s$.

Recall that the Weyl group $\overline{W}=\mathbf{N}_{\overline{\mathbf{G}}}(\mathbf{T}_{0})/\mathbf{T}_{0}$ can be identified with the subgroup

[TABLE]

of $S_{\{\pm 1,\dots,\pm n\}}$. By [8, Proposition 1.4.10] it follows that the natural map $W\hookrightarrow\overline{W}$ identifies the Weyl group $W$ as the kernel of the group homomorphism

[TABLE]

Let $F_{0}:\mathbf{G}\to\mathbf{G}$ be the Frobenius endomorphism defined by $\mathbf{x}_{\alpha}(t)\mapsto\mathbf{x}_{\alpha}(t^{q})$, for $t\in\overline{\mathbb{F}_{q}}$ and $\alpha\in\Phi$. We let $F_{0}^{\ast}:\mathbf{G}^{\ast}\to\mathbf{G}^{\ast}$ be defined as the unique morphism satisfying $\pi\circ F_{0}=F_{0}^{\ast}\circ\pi$. Then the triple $(\mathbf{G}^{\ast},\mathbf{T}_{0}^{\ast},F_{0})$ is in duality with $(\mathbf{G},\mathbf{T}_{0},F_{0})$. There exists an element $v\in W$ with preimage $\mathbf{m}_{v}\in\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})$ of $v$ such that $F=\mathbf{m}_{v}F_{0}$. Since $(\mathbf{G},\mathbf{T}_{0},F)$ is in duality with $(\mathbf{G},\mathbf{T}_{0},F^{\ast})$ there exists some $\mathbf{m}_{v^{\ast}}\mathrm{N}_{\mathbf{G}^{\ast}}(\mathbf{T}_{0}^{\ast})$, a preimage of $v^{\ast}$ in $\mathrm{N}_{\mathbf{G}^{\ast}}(\mathbf{T}_{0}^{\ast})$, such that $F^{\ast}=F_{0}^{\ast}\mathbf{m}_{v^{\ast}}$.

Classifying semisimple conjugacy classes

Let $\mathbf{L}^{\ast}=\operatorname{C}_{\mathbf{G}^{\ast}}(\mathrm{Z}^{\circ}(\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)))$ be the minimal Levi subgroup of $\mathbf{G}^{\ast}$ containing $\mathrm{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ and $\mathbf{N}^{\ast}=\mathrm{C}_{\mathbf{G}^{\ast}}(s)\mathbf{L}^{\ast}$. Let $\mathbf{L}$ be a Levi subgroup of $\mathbf{G}$ containing the maximal torus $\mathbf{T}_{0}$ which is in duality with $\mathbf{L}^{\ast}$. We set $\mathbf{N}$ be the subgroup of $\mathrm{N}_{\mathbf{G}}(\mathbf{L})$ such that $\mathbf{N}/\mathbf{L}\cong\mathbf{N}^{\ast}/\mathbf{L}^{\ast}$ under the canonical isomorphism between $\mathrm{N}_{\mathbf{G}}(\mathbf{L})/\mathbf{L}$ and $\mathrm{N}_{\mathbf{G}^{\ast}}(\mathbf{L}^{\ast})/\mathbf{L}^{\ast}$ induced by duality. We start by recalling the observation already made in the introduction:

Lemma 3**.**

In order to prove Proposition 1 we can assume that $\mathbf{N}^{F}/\mathbf{L}^{F}$ is non-cyclic.

Proof.

If $\mathbf{N}^{F}/\mathbf{L}^{F}$ is cyclic then for instance [11, Lemma 10.2.13] shows that $H_{c}^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ extends to an $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodule. ∎

We will now give a more explicit description of the quotient group $\mathbf{N}^{F}/\mathbf{L}^{F}$. By definition we have an injective morphism

[TABLE]

As in [1, Lemma 2.6] we consider the morphism

[TABLE]

for some $y\in\mathbf{G}$ with $\pi(y)=x$, which by [1, Corollary 2.8], induces an injection

[TABLE]

Thus, we have an embedding $\mathbf{N}/\mathbf{L}\hookrightarrow\mathrm{Z}(\mathbf{G}),$ which induces a map $\mathbf{N}^{F}/\mathbf{L}^{F}\hookrightarrow\mathrm{Z}(\mathbf{G})^{F}$ on fixed points. As $\mathrm{Z}(\mathbf{G})^{F}\cong C^{2}_{\mathrm{gcd}(2,q-1)}$ we can assume by Lemma 3 that $q$ is odd and that $\mathbf{N}^{F}/\mathbf{L}^{F}\cong\mathrm{Z}(\mathbf{G})^{F}$. Let $W(s)$ (resp. $W^{\circ}(s)$) be the Weyl group of $\mathrm{C}_{\mathbf{G}^{\ast}}(s)$ (resp. $\mathrm{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$) with respect to $\mathbf{T}_{0}^{\ast}$. By [7, Remark 2.4] we have a canonical isomorphism

[TABLE]

Recall that $\mathbf{T}_{0}$ is contained in a maximal $F$-stable Borel subgroup $\mathbf{B}(s)$ of $\operatorname{C}_{\mathbf{G}^{\ast}}^{\circ}(s)$. Let $\Phi(s)$ be the root system of $\operatorname{C}_{\mathbf{G}^{\ast}}^{\circ}(s)$ with set of positive roots $\Phi^{+}(s)$ associated to this choice. According to [1, Proposition 1.3] we have $W(s)=A(s)\rtimes W^{\circ}(s)$, where $A(s)=\{w\in W(s)\mid w(\Phi^{+}(s))=\Phi^{+}(s)\}$. Since $A(s)$ is $F$-stable this shows that the map

[TABLE]

is again an isomorphism. As the morphism $\omega_{s}$ induces an isomorphism

[TABLE]

we conclude that there exist $w_{1}^{\ast},w_{2}^{\ast}\in W^{F^{\ast}}$ with ${}^{w_{1}}t=tz_{1}$ and ${}^{w_{2}}t=tz_{2}$. Since $W^{F^{\ast}}=\mathrm{C}_{W^{\ast}}(v)$ we have $w_{1},w_{2}\in\mathrm{C}_{W}(v)$.

Remark 4**.**

The set $I_{\{\pm 1,\pm\omega_{4}\}}(t)$ is non-empty.

Proof.

Suppose that $I_{\{\pm 1,\pm\omega_{4}\}}(t)=\emptyset$. Write ${}^{w_{1}}t=\prod_{i=1}^{n}\mathbf{h}_{e_{i}}(s_{i})$ for suitable $s_{i}\in\overline{\mathbb{F}_{q}}^{\times}$. Then ${}^{w_{1}}tt^{-1}\mathbf{h}_{e_{1}}(-1)=1$ implies that $s_{i}t_{i}^{-1}\in\{\pm 1\}$ for all $i$. Now note that

[TABLE]

Thus, $s_{i},t_{i}\notin\{\pm 1,\pm\omega_{4}\}$ and so $s_{i}=t_{i}$ for all $i$. This leads to the contradiction ${}^{w_{1}}t=t$. ∎

Lemma 5**.**

In order to prove Proposition 1 we may assume that $t$ is of the form $t=\prod_{i=1}^{n}\mathbf{h}_{e_{i}}(t_{i})$ such that $t_{i}=t_{j}$ whenever $t_{j}\in\{\pm t_{i},\pm t_{i}^{-1}\}$.

Proof.

Let $\underline{n}:=\{1,\dots,n\}$. We define the equivalence relation $\sim$ on $\underline{n}$ by saying that $i\sim j$ if $t_{j}\in\{\pm t_{i},\pm t_{i}^{-1}\}$. Let $K$ be a set of representatives for the equivalence classes of $\underline{n}$ under $\sim$.

Let $x\in I_{\{\pm 1,\pm\omega_{4}\}}(t)$. Under the identification of the Weyl group we set

[TABLE]

Since $\mathbf{h}_{e_{i}}(-1)=\mathbf{h}_{e_{1}}(-1)=z_{1}$ for all $i\in\{1,\dots,n\}$ we see that either ${}^{w}t$ or ${}^{w}tz_{1}$ is of the desired form. We let $t^{\prime}\in\{{}^{w}t,{}^{w}tz_{1}\}$ be said element. In order to prove Proposition 1 it is therefore harmless to replace $s$ by the $\mathbf{G}^{F}$-conjugate $s^{\prime}:={}^{w}s\in\mathbf{T}_{0}$. Since $\pi(t^{\prime})=s^{\prime}$ this element has a preimage $t^{\prime}\in\mathbf{T}_{0}$ which is of the form as announced in the lemma. ∎

From now on we assume that the element $t$ has the form given in Lemma 5. Recall that

[TABLE]

Let $\alpha=e_{i}\pm e_{j}\in\Phi$ with $\alpha(t)=1$. Then $\alpha(t)=(t_{i}t_{j}^{\pm 1})^{2}=1$ and therefore $t_{i}=\varepsilon t^{\mp 1}_{j}$ for some $\varepsilon\in\{\pm 1\}$. By assumption on $t$, this implies $t_{i}=t_{j}$. In addition, we have $\alpha=e_{i}-e_{j}$ if $t_{i}$ is not a $4$th root of unity. Therefore, the root system $\Phi(t)$ of $\mathrm{C}_{\mathbf{G}}(t)$ is given by

[TABLE]

We write $W(t)$ for the Weyl group of $\mathrm{C}_{\mathbf{G}}(t)$ relative to the torus $\mathbf{T}_{0}$.

Lemma 6**.**

We have $|I_{\{\pm 1\}}(t)|=|I_{\{\pm\omega_{4}\}}(t)|=1$.

Proof.

Recall that $w_{2}\in W$ satisfies ${}^{w_{2}}t=tz_{2}$ with $z_{2}=\prod_{i=1}^{n}\mathbf{h}_{e_{i}}(\omega_{4})$. Therefore, we have

[TABLE]

Thus, $w_{2}$ swaps the sets $I_{\{\pm 1\}}(t)$ and $I_{\{\pm\omega_{4}\}}(t)$. Hence, $|I_{\{\pm 1\}}(t)|=|I_{\{\pm\omega_{4}\}}(t)|$. Note that $I_{\{\pm 1\}}(t)=I_{\{\pm 1\}}({}^{w_{1}}t)$ and $I_{\{\pm\omega_{4}\}}(t)=I_{\{\pm\omega_{4}\}}({}^{w_{1}}t)$.

Suppose that $|I_{\{\pm 1\}}(t)|>1$ and let $a,b\in I_{\{\pm 1\}}(t)$ with $a\neq b$. Fix $c,d\in I_{\{\pm\omega_{4}\}}(t)$ with $c\neq d$ and let $w_{1}^{\prime}:=(a,-a)(d,-d)\in W$. It follows that ${}^{w_{1}^{\prime}}t=tz_{1}$.

Recall that $e_{a}+e_{b},e_{a}-e_{b}\in\Phi(t)$ and

[TABLE]

Thus, for $\lambda=\prod_{i=1}^{n}\mathbf{h}_{e_{i}}(\lambda_{i})\in\mathrm{Z}(\mathrm{C}_{\mathbf{G}}(t))$ we have $(\lambda_{a}\lambda_{b}^{\pm 1})^{2}=1$. This implies that $\lambda_{a}$ and $\lambda_{b}$ are $4$th roots of unity. An analogue argument shows that $\lambda_{c}$ and $\lambda_{d}$ are also $4$th roots of unity. We conclude that ${}^{w_{1}^{\prime}}\lambda=\lambda z_{1}$ or ${}^{w_{1}^{\prime}}\lambda=\lambda$ in this case.

Note that $\pi(\mathrm{C}_{\mathbf{G}}(t))=\mathrm{C}^{\circ}_{\mathbf{G}^{\ast}}(s)$ by [1, (2.2)]. From this we can conclude that

[TABLE]

As ${}^{w_{1}^{\prime}}\pi(\lambda)=\pi(\lambda)$ for all $\lambda\in\mathrm{Z}(\mathrm{C}_{\mathbf{G}}(t))$ we conclude that $w_{1}^{\prime}\in\mathbf{L}^{\ast}=\operatorname{C}_{\mathbf{G}^{\ast}}(\mathrm{Z}^{\circ}(\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)))$. Since ${}^{w_{1}^{-1}w_{1}^{\prime}}t=t$ it follows that $w_{1}^{-1}w_{1}^{\prime}\in W(t)$. From this we deduce that $w_{1}\in\mathbf{L}^{\ast}=\operatorname{C}_{\mathbf{G}^{\ast}}(\mathrm{Z}^{\circ}(\operatorname{C}^{\circ}_{\mathbf{G}^{\ast}}(s)))$. This contradicts the assumption $\mathbf{N}^{\ast}/\mathbf{L}^{\ast}\cong\mathrm{Z}(\mathbf{G})$.

We conclude that $|I_{\{\pm 1\}}(t)|\leq 1$. By Remark 4 we must have $|I_{\{\pm 1\}}(t)|=1$. ∎

By the previous lemma, up to a change of coordinates, we may assume that $I_{\{\pm 1\}}(t)=\{1\}$ and $I_{\{\pm\omega_{4}\}}(t)=\{n\}$.

Computations in the Weyl group

Let us collect the information we have obtained so far. The root system $\Phi(t)$ of $\mathrm{C}_{\mathbf{G}}(t)$ is given by

[TABLE]

Observe that $\mathrm{C}_{\mathbf{G}}(t)$ is an $F$-stable Levi subgroup of $\mathbf{G}$ in duality with $\mathbf{L}^{\ast}$ so that $\mathbf{L}=\mathrm{C}_{\mathbf{G}}(t)$.

Definition 7**.**

Let $I=\{2,\dots,n-1\}$ and define $\overline{\Phi}^{\prime}:=\{\pm e_{i}\pm e_{j}\mid i,j\in I\,\}\setminus\{0\}$. Let

[TABLE]

and

[TABLE]

The roots $\{e_{1}\pm e_{n}\}$ are orthogonal to those in $\bar{\Phi}^{\prime}$ and no non-trivial linear combination of $\{e_{1}\pm e_{n}\}$ and $\Phi^{\prime}$ is a root in $\Phi$. Therefore, we have $\mathbf{T}_{1}\subseteq\mathrm{Z}(\mathbf{L})$. For $\mathbf{T}_{2}:=\mathbf{G}_{2}\cap\mathbf{T}_{0}$ we have $\mathbf{T}_{0}=\mathbf{T}_{1}\mathbf{T}_{2}$. This implies that $\mathbf{L}=\mathbf{T}_{1}\mathbf{G}_{2}$. In addition, we have $\mathbf{T}_{1}\cap\mathbf{T}_{2}=\langle z_{1}\rangle$.

Lemma 8**.**

Consider the restriction map

[TABLE]

We have $\mathrm{Res}(v)\in\langle(1,-1)(n,-n),(1,-n)(-1,n)\rangle$.

Proof.

Firstly, note that ${}^{v}F_{0}(s)=s$ which implies that $I_{\{\pm\omega_{4},\pm 1\}}({}^{v}s)=I_{\{\pm\omega_{4},\pm 1\}}(s)$. Therefore, $\mathrm{Res}(v)$ is well-defined.

Since $w_{2}$ permutes the sets $I_{\{\pm 1\}}(t)=\{1\}$ and $I_{\{\pm\omega\}}(t)=\{n\}$ we have $\mathrm{Res}(w_{2})=(1,-n)(-1,n)$. Let $w_{1}^{\prime}=(1,-1)(n,-n)\in W$. Then we have ${}^{w_{1}^{\prime}}t=tz_{1}$. This implies that $w_{1}^{\prime}w_{1}^{-1}\in W(t)$. Since $W(t)\subseteq\mathrm{Ker}(\mathrm{Res})$ we must have $\mathrm{Res}(w_{1})=(1,-1)(n,-n)$.

As $w_{1},w_{2}\in\mathrm{C}_{W}(v)$ we have $[\mathrm{Res}(w_{i}),\mathrm{Res}(v)]=1$ for $i=1,2$. Thus,

[TABLE]

A short calculation shows that $\langle(1,-1)(n,-n),(1,-n)(-1,n)\rangle$ is self-centralizing in $S_{\{\pm 1,\pm n\}}$. ∎

For the following two lemmas recall that $\mathbf{x}_{\alpha}(t)$, $\mathbf{h}_{\alpha}(t)$ and $\mathbf{n}_{\alpha}(t)$ are not uniquely defined and their relations depend on the choice of certain structure constants. However, the relations simplify in the case where the involved roots are orthogonal, see [12, Remark 2.1.7].

Lemma 9**.**

Let $A,B\subseteq\overline{\Phi}$ such that $A\perp B$. Let $x=\prod_{\alpha\in A}\mathbf{n}_{\alpha}(r_{\alpha})$ and $y=\prod_{\beta\in B}\mathbf{n}_{\beta}(r_{\beta})$ for $r_{\alpha},r_{\beta}\in\overline{\mathbb{F}_{q}}^{\times}$. If $x,y\in\mathbf{G}$ then $x$ and $y$ commute.

Proof.

Recall that the inclusion map $\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})\hookrightarrow\mathrm{N}_{\overline{\mathbf{G}}}(\mathbf{T}_{0})$ induces the embedding $W\hookrightarrow\overline{W}$ such that $W=\mathrm{Ker}(\varepsilon)$. We note that $\varepsilon(\mathbf{n}_{\alpha}(1)\mathbf{T}_{0})=-1$ for $\alpha\in\bar{\Phi}$ if and only if $\alpha$ is a short root. As $x,y\in\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})$ we deduce that the number of short roots in $A$ resp. $B$ is even.

Let $\alpha\in A$ and $\beta\in B$. By [12, Remark 2.1.7(c)] we have $\mathbf{n}_{\alpha}(r_{\alpha})^{\mathbf{n}_{\beta}(r_{\beta})}=\mathbf{n}_{\alpha}(-r_{\alpha})=\mathbf{h}_{\alpha}(-1)\mathbf{n}_{\alpha}(r_{\alpha})$, if either $\alpha$ or $\beta$ is a long root. On the other hand, we have $\mathbf{n}_{\alpha}(r_{\alpha})^{\mathbf{n}_{\beta}(r_{\beta})}=\mathbf{n}_{\alpha}(r_{\alpha})$ if both $\alpha$ and $\beta$ are short roots. Note that if $\alpha$ is a short root then $\mathbf{h}_{\alpha}(-1)=\mathbf{h}_{e_{1}}(-1)$. The result follows from this. ∎

In the following, we will consider the element

[TABLE]

which is a preimage of $w_{1}^{\prime}=(1,-1)(n,-n)\in W$. By the proof of Lemma 8 it is possible to find $\mathbf{n}_{2}^{\prime}\in\langle n_{e_{i}}(1)\mid i\in I\rangle\cap\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})$ such that the element

[TABLE]

is a preimage of $w_{2}\in W$.

Lemma 10**.**

The elements $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$ commute. In addition, we have $\mathbf{n}_{1}\in\mathrm{C}_{\mathbf{G}}(\mathbf{G}_{2})$.

Proof.

Let us first prove that $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$ commute. By [12, Remark 2.1.7(c)] we have $\mathbf{n}_{e_{j}}(u)^{\mathbf{n}_{e_{i}}(1)}=\mathbf{n}_{e_{j}}(-u)=\mathbf{h}_{e_{j}}(-1)\mathbf{n}_{e_{j}}(u)$ for $u\in\overline{\mathbb{F}_{q}}$, whenever $i\neq j$. By the relation in [12, Theorem 2.1.6(b)] we have $\mathbf{n}_{e_{1}}(1)^{\mathbf{n}_{e_{1}-e_{n}}(1)}\in\{\mathbf{n}_{e_{n}}(1),\mathbf{n}_{e_{n}}(-1)\}$. By Lemma 9,

[TABLE]

According to [12, Remark 2.1.8] we have $\mathbf{h}_{e_{1}-e_{n}}(-1)=\mathbf{h}_{e_{1}}(\omega_{4})\mathbf{h}_{e_{n}}(\omega_{4}^{-1})$ , where $\omega_{4}\in\overline{\mathbb{F}_{q}}^{\times}$ is a fourth root of unity. Using [12, Remark 2.1.7(a)], we obtain

[TABLE]

Since $\mathbf{n}_{e_{1}}(1)^{\mathbf{n}_{e_{1}-e_{n}}(1)}\in\{\mathbf{n}_{e_{n}}(1),\mathbf{n}_{e_{n}}(-1)\}$ we deduce that

[TABLE]

Therefore, $\mathbf{n}_{1}^{n_{2}}=\mathbf{n}_{1}$ and we conclude that $\mathbf{n}_{1}$ and $\mathbf{n}_{2}$ commute.

Finally, note that $\mathbf{n}_{1}\in\mathrm{C}_{\mathbf{G}}(\mathbf{G}_{2})$ by [12, Remark 2.1.7(b)] and [12, Theorem 2.1.6(c)]. ∎

By Lemma 8 there exist $\mathbf{m}_{1}\in\langle\mathbf{n}_{1},\mathbf{n}_{e_{1}-e_{n}}(1)\rangle$ and $\mathbf{m}_{2}\in\langle\mathbf{n}_{e_{i}}(1)\mid i\in I\rangle\cap\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})$ such that

[TABLE]

satisfies $\mathbf{m}\mathbf{T}_{0}=v$ in $W$. Since $w_{2}\in\operatorname{C}_{W}(v)$ we necessarily have $(\mathbf{m}F_{0})(\mathbf{n}_{2})\mathbf{n}_{2}^{-1}\in\mathbf{T}_{0}$. By Lemma 9 it follows that $\mathbf{m}_{2}$ commutes with $\mathbf{n}_{e_{1}-e_{n}}(1)$ and $\mathbf{m}_{1}$ commutes with $\mathbf{n}_{2}^{\prime}$. From this we deduce that

[TABLE]

Since ${}^{\mathbf{m}_{2}}\mathbf{n}_{2}^{\prime}\mathbf{n}_{2}^{\prime-1}$ is purely an expression in the roots $e_{2},\dots,e_{n-1}$ we can deduce that

[TABLE]

By Lang’s theorem there exists $t_{2}\in\mathbf{T}_{2}$ such that $(t_{2}\mathbf{m}F_{0})(\mathbf{n}_{2})=\mathbf{n}_{2}$. Replacing $\mathbf{m}_{2}$ by $t_{2}\mathbf{m}_{2}\in\langle\mathbf{n}_{e_{i}}(r_{i})\mid,r_{i}\in\overline{\mathbb{F}_{q}}^{\times},i\in I\rangle\cap\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})$ we can henceforth assume that $(\mathbf{m}F_{0})(\mathbf{n}_{2})=\mathbf{n}_{2}$.

By Lemma 10 it follows that $\mathbf{m}_{1}\in\langle\mathbf{n}_{1},\mathbf{n}_{e_{1}-e_{n}}(1)\rangle$ commutes with $\mathbf{n}_{1}$. By Lemma 9 we conclude that $\mathbf{m}_{2}$ and $\mathbf{n}_{1}$ commute. As $\mathbf{n}_{1}$ is $F_{0}$-stable it follows that $(\mathbf{m}F_{0})(\mathbf{n}_{1})=\mathbf{n}_{1}$.

If $y\in\mathbf{T}_{0}$ then we have an isomorphism $\mathbf{G}^{F}\to\mathbf{G}^{Fy},\,g\mapsto{}^{y^{-1}}g$ which yields isomorphic fixed-point structures for all relevant subgroups. We may thus fix a nice representative of $v\in W$ in $\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})$ and assume the following:

Assumption 11**.**

From now on, we suppose that $F=\mathbf{m}F_{0}$. In particular, the elements $\mathbf{n}_{1},\mathbf{n}_{2}$ are $F$-stable.

We are now ready to prove the main result of this section.

Proposition 12**.**

We have $\mathbf{L}^{F}\langle\mathbf{n}_{1},n_{2}\rangle=\mathbf{N}^{F}$.

Proof.

The elements $\mathbf{n}_{1},\mathbf{n}_{2}\in\mathrm{N}_{\mathbf{G}}(\mathbf{T}_{0})$ satisfy ${}^{\mathbf{n}_{1}}t=tz_{1}$ and ${}^{\mathbf{n}_{2}}t=tz_{2}$. From this we deduce that $\pi(\mathbf{n}_{1}),\pi(\mathbf{n}_{2})\in\operatorname{C}_{\mathbf{G}^{\ast}}(s)$. By duality we have an isomorphism $\mathbf{N}^{F}/\mathbf{L}^{F}\cong(\mathbf{N}^{\ast})^{F^{\ast}}/(\mathbf{L}^{\ast})^{F^{\ast}}$ from which we can now conclude that $\mathbf{L}^{F}\langle\,\mathbf{n}_{2}\rangle=\mathbf{N}^{F}$. ∎

In the next section we will consider the subgroup $L_{0}$ of $\mathbf{L}^{F}$ defined by $L_{0}=\mathbf{T}_{1}^{F}\mathbf{G}_{2}^{F}$. As $\mathbf{T}_{1}\subseteq\operatorname{C}_{\mathbf{L}}(\mathbf{G}_{2})$ it follows that $L_{0}$ is a central product of $\mathbf{T}_{1}^{F}$ and $\mathbf{G}_{2}^{F}$. The following lemma shows that $\mathbf{L}^{F}/L_{0}\cong C_{2}$.

Lemma 13**.**

Let $\mathcal{L}:\mathbf{G}\to\mathbf{G},\,g\mapsto g^{-1}F(g)$, denote the Lang map of $\mathbf{G}$. There exists $x_{1}\in\mathbf{T}_{1}$ and $x_{2}\in\mathbf{T}_{2}$ such that $\mathcal{L}(x_{1})=\mathcal{L}(x_{2})=h_{e_{1}}(-1)$ and $x:=x_{1}x_{2}$ satisfies $\mathbf{L}^{F}=\mathbf{T}_{1}^{F}\mathbf{G}_{2}^{F}\langle x\rangle$.

Proof.

The existence of $x_{1}$ and $x_{2}$ follows by applying Lang’s theorem. Since $\mathbf{T}_{1}\cap\mathbf{G}_{2}=\mathbf{T}_{1}\cap\mathbf{T}_{2}=\langle h_{e_{1}}(-1)\rangle$ the second claim follows. ∎

Representation theory

Let $\hat{s}:\mathcal{O}\mathbf{L}^{F}\to\mathcal{O}^{\times}$ be the character of $\mathbf{L}^{F}$ corresponding to the central element $s\in\mathrm{Z}(\mathbf{L}^{\ast})$, see [5, Equation 8.19].

Lemma 14**.**

The linear character $\hat{s}:\mathbf{L}^{F}\to\mathcal{O}^{\times}$ extends to $\mathbf{N}^{F}$.

Proof.

By [13, Theorem 1.1] the character $\lambda:=\mathrm{Res}^{\mathbf{L}^{F}}_{\mathbf{T}_{0}^{F}}(\hat{s})$ extends to its inertia group in $\mathbf{N}_{\mathbf{G}^{F}}(\mathbf{T}_{0}^{F})$. However, $\mathbf{n}_{1},\mathbf{n}_{2}\in\mathbf{N}_{\mathbf{G}^{F}}(\mathbf{T}_{0}^{F})$ and $\lambda$ is $\mathbf{N}^{F}$-invariant which implies that $\lambda$ extends to a character $\lambda^{\prime}$ of $\mathbf{T}_{0}^{F}\langle\mathbf{n}_{1},\mathbf{n}_{2}\rangle$. We define a character $\hat{s}^{\prime}:\mathbf{N}^{F}\to\mathcal{O}^{\times}$ by $\hat{s}^{\prime}(x):=\hat{s}(l)\lambda^{\prime}(n)$ where $x\in\mathbf{N}^{F}$ with $x=ln$ for $l\in\mathbf{L}^{F}$ and $n\in\mathbf{T}_{0}^{F}\langle\mathbf{n}_{1},\mathbf{n}_{2}\rangle$. Note that this character is well-defined as $\hat{s}$ and $\lambda$ agree on the intersection $\mathbf{L}^{F}\cap\mathbf{T}_{0}^{F}\langle\mathbf{n}_{1},\mathbf{n}_{2}\rangle=\mathbf{T}_{0}^{F}$. ∎

The following lemma is a module theoretic generalization of [13, Lemma 4.1].

Lemma 15**.**

Let $\widetilde{Y}$ be a finite group with normal subgroup $\widetilde{X}$ and subgroup $Y$ such that $\widetilde{Y}=Y\widetilde{X}$. Denote $X:=Y\cap\widetilde{X}$ and suppose that $\ell\nmid[Y:X]$. Suppose that $M$ is an indecomposable $\mathcal{O}X$-module which extends to an $\mathcal{O}Y$-module and suppose that $\widetilde{M}$ is an $\mathcal{O}\widetilde{X}$-module such that $M=\mathrm{Res}^{\widetilde{X}}_{X}(\widetilde{M})$. If $\widetilde{M}$ is $\widetilde{Y}$-invariant then $\widetilde{M}$ extends to $\widetilde{Y}$.

Proof.

Let us recall some basic facts about Clifford theory, see [2, Section 7.B] (over $k$) and [6] (over $\mathcal{O}$). We follow the notation in [2, Section 7.B].

Firstly, for $y\in Y$, define

[TABLE]

and let $N:=\cup_{y\in Y}N_{y}$. Note that $N$ is a group with normal subgroup $N_{1}$. Since $M$ is $Y$-invariant we have a surjective morphism $Y\to N/N_{1}$ given by $y\mapsto yN_{1}$. We form the group

[TABLE]

We let $A:=\operatorname{End}_{\mathcal{O}X}(M)$. Consider the following exact sequence:

[TABLE]

The $\mathcal{O}X$-module $M$ extends to an $\mathcal{O}Y$-module if and only if this sequence splits, see [6, 1.7]

The action of $X$ on $M$ defines $\phi_{x}\in N_{x}$ for every $x\in X$. We identify $X$ with its image under the diagonal embedding $X\hookrightarrow\widehat{Y},x\mapsto(x,\phi_{x})$. As $\ell\nmid[Y:X]$ it follows (see [6, Theorem 4.5]) that $M$ extends to a $\mathcal{O}Y$-module if and only if the following exact sequence splits:

[TABLE]

Similarly, we can look at $\widetilde{M}$ instead of $M$. We denote the corresponding objects with a tilde. Analogously, the module $\widetilde{M}$ extends to an $\mathcal{O}\widetilde{Y}$ module if and only if the following exact sequence splits:

[TABLE]

Let $\pi:\widetilde{Y}/\widetilde{X}\to Y/X$ be the inverse map of the natural isomorphism $Y/X\to\widetilde{Y}/\widetilde{X}$. Restriction defines a homomorphism $\widetilde{A}^{\times}\to A^{\times}$.

Now we define a map $\widehat{\widetilde{Y}}\to\widehat{Y}$ as follows. For $(\widetilde{y},\phi)\in\widehat{\widetilde{Y}}$ we let $\tilde{x}\in\widetilde{X}$ such that $y:=\tilde{y}\tilde{x}\in Y$. Let $\phi_{\tilde{x}}$ be the natural action of $\widetilde{x}$ on $\widetilde{M}$. Then it follows that $\phi\phi_{\tilde{x}}\in\widetilde{N}_{y}\subseteq N_{y}$. We define

[TABLE]

Note that if $\tilde{x}^{\prime}\in\widetilde{X}$ with $y^{\prime}:=\tilde{y}\tilde{x}^{\prime}\in Y$ then $x:=\tilde{x}^{-1}\tilde{x}^{\prime}\in X$ and we have $y^{\prime}=yx$. From this we deduce that $(y^{\prime},\phi\phi_{\tilde{x}^{\prime}})=(y,\phi\phi_{x})$ in $\tilde{Y}/X$ which shows that the map $\widehat{\pi}$ is well-defined. We can therefore consider the following commutative diagram:

{1}$${\widetilde{A}^{\times}/(1+J(\widetilde{A}))}$${\widehat{\widetilde{Y}}/\widetilde{X}(1+J(\widetilde{A}))}$${\widetilde{Y}/\widetilde{X}}$${1}$${1}$${A^{\times}/(1+J(A))}$${\widehat{Y}/X(1+J(A))}$${Y/X}$${1}$$\widehat{\pi}$$\pi

Now note that $\pi$ is an isomorphism. Also as $M$ and $\widetilde{M}$ are indecomposable we have that $A^{\times}/(1+J(A))\cong k^{\times}$ resp. $\widetilde{A}^{\times}/(1+J(\widetilde{A}))\cong k^{\times}$. Thus, the first and the third vertical map are isomorphisms. By the five lemma, it follows that $\widehat{\pi}:\widehat{\widetilde{Y}}/\widetilde{X}(1+J(\widetilde{A}))\to\widehat{Y}/X(1+J(A))$ is also an isomorphism. Thus, the two extensions are isomorphic. However, by assumption we already know that $M$ extends to an $\mathcal{O}Y$-module which implies that the sequence in the second row splits. Thus, also the sequence of the first row splits and $\widetilde{M}$ extends to an $\mathcal{O}\widetilde{Y}$-module. ∎

We are now ready to prove the main statement of this section.

Proposition 16**.**

Let $M$ be a $\mathbf{N}^{F}$-invariant indecomposable $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}}$-bimodule. If $\ell$ does not divide $q^{2}-1$ then $M$ extends to an $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodule.

Proof.

By Lemma 14, it follows that $M$ extends to $\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}$ if and only if $M\otimes_{\mathcal{O}}\hat{s}^{-1}$ extends to $\mathcal{O}\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}$. We may therefore assume from now on that $M$ is an indecomposable $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}}$-bimodule.

Since $\ell\nmid[\mathbf{L}^{F}:L_{0}]$ there exists an indecomposable $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}L_{0}$-bimodule $M_{0}$ such that $M$ is a direct summand of $\operatorname{Ind}^{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times L_{0}^{\mathrm{opp}}}(M_{0})$. As $1\times(\mathbf{T}^{F}_{1})^{\mathrm{opp}}$ is central in $\mathbf{G}^{F}\times{L_{0}}^{\mathrm{opp}}$ we deduce that

[TABLE]

for some simple $\mathcal{O}(\mathbf{T}_{1})_{\ell^{\prime}}^{F}$-module $S$. Let $\lambda:(\mathbf{T}_{1})_{\ell^{\prime}}^{F}\to\mathcal{O}^{\times}$ be the character corresponding to $S$. Since $\mathrm{Res}^{\mathbf{G}^{F}\times{\mathbf{L}^{F}}^{\mathrm{opp}}}_{1\times{\mathbf{L}^{F}}}$ is a $\mathcal{O}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}}$-module it follows that $\lambda$ is a character in a unipotent block, which implies that $\lambda$ is the trivial character.

Note that $|\mathbf{T}_{1}^{F}|\in\{(q-1)^{2},(q+1)^{2}\}$ and therefore $\ell\nmid|\mathbf{T}_{1}^{F}|$ by assumption. We conclude that

[TABLE]

where $\mathcal{O}$ is the trivial $\mathcal{O}{(\mathbf{T}_{1}^{F})}^{\mathrm{opp}}$-module. Since $L_{0}/\mathbf{T}_{1}^{F}\cong\mathbf{G}^{F}_{2}/\langle z_{1}\rangle$ we may consider $M_{0}$ as an indecomposable $\mathcal{O}[\mathbf{G}^{F}\times(\mathbf{G}^{F}_{2}/\langle z_{1}\rangle)^{\mathrm{opp}}]$-module.

The element $\mathbf{n}_{1}$ centralizes $\mathbf{G}^{F}_{2}$ and hence we can extend $M_{0}$ to an $\mathcal{O}\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1}\rangle)^{\mathrm{opp}}$-module by letting $\mathbf{n}_{1}$ act trivially on $M_{0}$. We denote this extension by $M_{1}$.

Since $M$ is a direct summand of $\operatorname{Ind}^{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times L_{0}^{\mathrm{opp}}}(M_{0})$ it follows that $\operatorname{Res}^{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times(L_{0})^{\mathrm{opp}}}M$ is a direct summand of

[TABLE]

where $x=x_{1}x_{2}\in\mathbf{L}^{F}$ as in Lemma 13. As the quotient group $\mathbf{L}^{F}/L_{0}$ is cyclic of $\ell^{\prime}$ order it follows by [11, Lemma 10.2.13] that either $\operatorname{Res}^{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times(L_{0})^{\mathrm{opp}}}(M)\cong M_{0}$ or $\operatorname{Res}^{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times(L_{0})^{\mathrm{opp}}}(M)\cong M_{0}\oplus M_{0}^{x}$. We treat these two cases separately.

Case 1: Assume that $\operatorname{Res}^{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times(L_{0})^{\mathrm{opp}}}(M)\cong M_{0}$.

Since $M$ is $\mathbf{N}^{F}$-invariant it follows that $M_{0}$ is $\mathbf{N}^{F}$-invariant.

We have $[\mathbf{n}_{1},\mathbf{n}_{2}]=1$. Thus, the action of $\mathbf{n}_{1}$ on $M_{1}^{\mathbf{n}_{2}}$ is equal to the action of $\mathbf{n}_{2}\mathbf{n}_{1}\mathbf{n}_{2}^{-1}=\mathbf{n}_{1}$ on $M_{1}$. However, $\mathbf{n}_{1}$ acts trivially on $M_{0}$. Since $M_{0}$ is $\mathbf{n}_{2}$-invariant there exists an isomorphism $\phi:M_{0}\to M_{0}^{\mathbf{n}_{2}}$ of $\mathbf{G}^{F}\times(L_{0})^{\mathrm{opp}}$-modules. Since $\mathbf{n}_{1}$ acts trivially on $M_{1}$ it follows that $\phi:M_{1}\to M_{1}^{\mathbf{n}_{2}}$ is an isomorphism of $\mathcal{O}\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1}\rangle)^{\mathrm{opp}}$-modules or in other words $M_{1}$ is $\mathbf{n}_{2}$-invariant. From this we conclude that $M_{1}$ extends to $\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1},\mathbf{n}_{2}\rangle)^{\mathrm{opp}}$.

Applying Lemma 15 to $\tilde{X}=\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}$ and $Y=\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1},\mathbf{n}_{2}\rangle)^{\mathrm{opp}}$) implies that $M$ extends to a $\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}$-module.

Case 2: Assume that $\operatorname{Res}^{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times(L_{0})^{\mathrm{opp}}}(M)\cong M_{0}\oplus{}^{x}M_{0}$.

We note that $M_{0}^{\mathbf{n}_{1}}\cong M_{0}$. On the other hand, we either have $M_{0}^{\mathbf{n}_{2}}\cong M_{0}$ or $M_{0}^{\mathbf{n}_{2}x}\cong M_{0}$.

Suppose that $M_{0}^{\mathbf{n}_{2}}\cong M_{0}$. Then $M_{0}$ is $\mathbf{N}^{F}$-invariant. Using the same proof as in case 1 we deduce that $M_{0}$ extends to a $\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1},\mathbf{n}_{2}\rangle)^{\mathrm{opp}}$-module.

Suppose that $M_{0}^{\mathbf{n}_{2}x}\cong M_{0}$. We have $h_{e_{1}}(-1)^{x}=h_{e_{1}}(-1)$ as $\mathcal{L}(x_{1})=\mathcal{L}(x_{2})=h_{e_{1}}(-1)$. Since $x_{2}\in\mathbf{G}_{2}$ we conclude that $\mathbf{n}_{1}^{x_{2}}=\mathbf{n}_{1}$. Therefore, $\mathbf{n}_{1}^{\mathbf{n}_{2}x}=\mathbf{n}_{1}^{x}=\mathbf{n}_{1}^{x_{1}}.$

Clearly, $x_{1}x_{1}^{\mathbf{n}_{1}^{-1}}\in\mathbf{T}_{1}^{F}$ which implies that $\mathbf{n}_{1}^{x_{1}}\mathbf{n}_{1}^{-1}\in\mathbf{T}_{1}^{F}$. From this we deduce that $\mathbf{n}_{1}^{\mathbf{n}_{2}x}\mathbf{n}_{1}^{-1}\in\mathbf{T}_{1}^{F}$. Now $\mathbf{n}_{1}$ acts on $M_{1}^{\mathbf{n}_{2}x}$ as $\mathbf{n}_{1}^{\mathbf{n}_{2}x}$ acts on $M_{1}$. Since $\mathbf{T}^{F}_{1}$ and $\mathbf{n}_{1}$ act trivially on $M_{1}$ it follows that $\mathbf{n}_{1}$ acts trivially on $M_{1}^{\mathbf{n}_{2}x}$. Since $M_{0}$ is $\mathbf{n}_{2}x$-invariant it follows that $M_{1}$ is $\mathbf{n}_{2}x$-invariant. Thus, $M_{0}$ extends to a $\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1},\mathbf{n}_{2}x\rangle)^{\mathrm{opp}}$-module.

It follows that $M_{0}$ extends to a $\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1},\mathbf{n}_{2}^{\prime}\rangle)^{\mathrm{opp}}$-module $M^{\prime}$, where $\mathbf{n}_{2}^{\prime}\in\{\mathbf{n}_{2},\mathbf{n}_{2}x\}$. By Mackey’s formula we deduce that

[TABLE]

Thus, $\mathrm{Ind}_{\mathbf{G}^{F}\times(L_{0}\langle\mathbf{n}_{1},\mathbf{n}_{2}^{\prime}\rangle)^{\mathrm{opp}}}^{\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}}M^{\prime}$ is an extension of $M$ to $\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}$. This finishes the proof. ∎

Using a standard argument in Clifford theory we can now deduce Proposition 1 from the previous proposition.

Proof of Proposition 1.

According to [2, Theorem 7.2] the bimodule $H^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ is $\mathcal{O}\mathbf{N}^{F}$-invariant. Let $H^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}=\oplus_{i=1}^{k}M_{i}$ be a decomposition into $\mathbf{N}^{F}$-orbits of indecomposable direct summands of $H^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$. Let $N_{i}$ be an indecomposable direct summand of $M_{i}$ and $T_{i}$ be its inertia group in $\mathbf{N}^{F}$. If $T_{i}$ is a proper subgroup of $\mathbf{N}^{F}$ then $T_{i}/\mathbf{L}^{F}$ is cyclic of $\ell^{\prime}$-order so that $N_{i}$ extends to $\mathbf{G}^{F}\times(T_{i})^{\mathrm{opp}}$. If $T_{i}=\mathbf{N}^{F}$ then $N_{i}$ extends to $\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}$ by Proposition 16. Let $N_{i}^{\prime}$ be an extension of $N_{i}$ to $\mathbf{G}^{F}\times(T_{i})^{\mathrm{opp}}$. By Clifford theory, it follows that $\mathrm{Ind}^{\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}}_{\mathbf{G}^{F}\times(T_{i})^{\mathrm{opp}}}N_{i}^{\prime}$ is an extension of $M_{i}$. This shows that $H^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ extends to $\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}$. ∎

Proof of Morita equivalence

In this final section we prove that the extended bimodule induces a Morita equivalence. The following section borrows arguments from [3].

From now on let $\mathbf{G}$ be a connected reductive group. We keep the notation as in [2, Section 7.C]. In particular we fix a regular embedding $\mathbf{G}\hookrightarrow\mathbf{\tilde{G}}$. We let $\mathbf{\tilde{L}}=\mathbf{L}\mathrm{Z}(\mathbf{\tilde{G}})$ and $\tilde{\mathbf{N}}=\mathbf{N}\mathbf{\tilde{L}}$.

Proposition 17**.**

Suppose that the $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{L}^{F}$-bimodule $H_{c}^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ extends to an $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodule $M^{\prime}$. Then the bimodule $M^{\prime}$ induces a Morita equivalence between $\mathcal{O}\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ and $\mathcal{O}\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$.

Proof.

Let $M^{\prime}$ be an $\mathcal{O}\mathbf{G}^{F}\times(\mathbf{N}^{F})^{\mathrm{opp}}$-bimodule extending $M:=H^{d}_{c}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$. Recall that $M$ is projective as $\mathcal{O}\mathbf{G}^{F}$-module and projective as $\mathcal{O}\mathbf{L}^{F}$-module. As $\ell\nmid[\mathbf{N}^{F}:\mathbf{L}^{F}]$ it follows that $M^{\prime}$ is projective as $\mathcal{O}\mathbf{N}^{F}$-module. Note that $\operatorname{Ind}_{\mathbf{G}^{F}}^{\mathbf{\tilde{G}}^{F}}M$ is a faithful $\mathcal{O}\mathbf{\tilde{G}}^{F}e_{s}^{\mathbf{G}^{F}}$-module, see proof of [2, Theorem 7.5]. Thus, $M$ is a faithful $\mathcal{O}\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$-module.

By [4, Theorem 0.2] it suffices to prove that $M^{\prime}\otimes_{\mathcal{O}}K$ induces a bijection between irreducible characters of $K\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ and $K\mathbf{G}^{F}e_{s}^{\mathbf{L}^{F}}$. As $M$ is a faithful $\mathcal{O}\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$-module it suffices to prove that $\operatorname{End}_{K\mathbf{G}^{F}}(M^{\prime}\otimes_{\mathcal{O}}K)\cong K\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$. As in the proof of [2, Theorem 7.5] let $\widetilde{M}=\operatorname{Ind}_{\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\mathrm{opp}}\Delta\mathbf{\tilde{L}}^{F}}^{\mathbf{\tilde{G}}^{F}\times(\mathbf{\tilde{L}}^{F})^{\mathrm{opp}}}(M)$ be the $\mathcal{O}\mathbf{\tilde{G}}^{F}\times(\widetilde{\mathbf{L}}^{F})^{\mathrm{opp}}$-module. We have $\operatorname{Ind}_{\mathbf{G}^{F}}^{\mathbf{\tilde{G}}^{F}}M^{\prime}\cong\widetilde{M}$ as $\mathbf{\tilde{G}}^{F}$-modules . Since $M$ is $\mathbf{\tilde{G}}^{F}$-invariant this implies

[TABLE]

In addition, the bimodule $\widetilde{M}$ extends to an $\mathcal{O}\mathbf{\tilde{G}}^{F}\times(\widetilde{\mathbf{N}}^{F})^{\mathrm{opp}}$-bimodule $\widetilde{M}^{\prime}$, see proof of [2, Theorem 7.5], which induces a Morita equivalence between $\mathcal{O}\widetilde{\mathbf{G}}^{F}e_{s}^{\mathbf{G}^{F}}$ and $\mathcal{O}\widetilde{\mathbf{N}}^{F}e_{s}^{\mathbf{L}^{F}}$. This shows that $\mathrm{dim}(\operatorname{End}_{K\mathbf{\tilde{G}}^{F}}(\widetilde{M}))=\mathrm{dim}(K\widetilde{\mathbf{N}}^{F}e_{s}^{\mathbf{L}^{F}})$. Moreover, we have

[TABLE]

From these calculations using $\mathbf{\tilde{G}}^{F}/\mathbf{G}^{F}\cong\widetilde{\mathbf{N}}^{F}/\mathbf{N}^{F}$ we deduce that

[TABLE]

Lemma 18**.**

The natural map $K\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}\to\operatorname{End}_{K\mathbf{G}^{F}}(M^{\prime})$ is injective.

Proof.

Let $\dot{n}$ be a representative of $n\in\mathbf{N}^{F}/\mathbf{L}^{F}$ in $\mathbf{N}^{F}$. Let $\alpha_{n}\in\mathcal{O}\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}}$, $n\in\mathbf{N}^{F}/\mathbf{L}^{F}$, such that $\sum_{n\in\mathbf{N}^{F}/\mathbf{L}^{F}}\alpha_{n}\dot{n}=0$ on $M^{\prime}$. Let $\theta_{\dot{n}}$ be the automorphism on $\widetilde{M}$ induced by the action of $\dot{n}$. More concretely, we have

[TABLE]

for $(g,l)\in\mathbf{\tilde{G}}^{F}\times(\mathbf{\tilde{L}}^{F})^{\mathrm{opp}}$ and $m\in M$.

For $(g,l)\in\mathbf{\tilde{G}}^{F}\times(\mathbf{\tilde{L}}^{F})^{\mathrm{opp}}$ we have $\theta_{\dot{n}}\circ(g,l)\circ\theta_{\dot{n}}^{-1}=(g,{}^{n}l)$ on $\widetilde{M}$. Let $e\in\mathrm{Z}(\mathcal{O}\mathbf{\tilde{L}}^{F})$ be the central idempotent as in [2, Theorem 7.5] such that $e_{s}^{\mathbf{L}^{F}}=\sum_{n\in\mathbf{N}^{F}/\mathbf{L}^{F}}{}^{n}e$. We have

[TABLE]

For $\widetilde{m}\in\widetilde{M}e$ we therefore have

[TABLE]

As $\theta_{\dot{n}}(\widetilde{m})\in\widetilde{M}{}^{n}e$ we have $\alpha_{n}\theta_{\dot{n}}(\widetilde{m})=0$ for all $n\in\mathbf{N}^{F}/\mathbf{L}^{F}$ and $\widetilde{m}\in\widetilde{M}e$. This means that $\alpha_{n}\theta_{\dot{n}}$ vanishes on $\widetilde{M}e$. Composing with $\theta_{\dot{y}}^{-1}$ for $y\in\mathbf{N}^{F}/\mathbf{L}^{F}$ shows that $\alpha_{n}\theta_{\dot{n}}$ vanishes on $\widetilde{M}{}^{y}e$ as well. We conclude that $\alpha_{n}\theta_{n}=0$ on $\widetilde{M}$. As $\theta_{\dot{n}}$ is an isomorphism we must have $\alpha_{n}=0$ on $\widetilde{M}$ and therefore, $\alpha_{n}=0$ on $M$. As $M$ is faithful as $K\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}}$-module it follows that $K\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}\to\operatorname{End}_{K\mathbf{G}^{F}}(M^{\prime})$ is injective. ∎

Now let us finish the proof of Proposition 17. Since $\mathrm{dim}(\operatorname{End}_{K\mathbf{G}^{F}}(M))=\mathrm{dim}(K\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}})$ it follows that the natural map

[TABLE]

is an isomorphism. Thus, the bimodule $M^{\prime}\otimes_{\mathcal{O}}K$ induces a Morita equivalence between $K\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ and $K\mathbf{G}^{F}e_{s}^{\mathbf{L}^{F}}$. As we have argued above this implies that $M^{\prime}$ induces a Morita equivalence between $\mathcal{O}\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ and $\mathcal{O}\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$. ∎

The previous statement is now sufficient to prove Theorem 2:

Proof of Theorem 2.

The quotient $\mathbf{N}^{F}/\mathbf{L}^{F}$ is of $\ell^{\prime}$-order and embeds into $\mathrm{Z}(\mathbf{G})^{F}$, see for instance [5, Lemma 13.16(i)]. Thus, the quotient $\mathbf{N}^{F}/\mathbf{L}^{F}$ is cyclic of $\ell^{\prime}$-order unless possibly if $\mathbf{G}$ is simply connected and $\mathbf{G}^{F}$ is of untwisted type $D_{n}$, $n$ even. If $\mathbf{N}^{F}/\mathbf{L}^{F}$ is cyclic (of $\ell^{\prime}$-order) then it follows by [11, Lemma 10.2.13] that the $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{L}^{F}$-bimodule $H_{c}^{d}(\mathbf{Y}_{\mathbf{U}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ extends to an $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodule. In the remaining cases Proposition 1 asserts that the bimodule $H_{c}^{d}(\mathbf{Y}_{\mathbf{U}}^{\mathbf{G}},\mathcal{O})e_{s}^{\mathbf{L}^{F}}$ extends to an $\mathcal{O}\mathbf{G}^{F}$-$\mathcal{O}\mathbf{N}^{F}$-bimodule.

By Proposition 17 the extended bimodule induces a Morita equivalence between $\mathcal{O}\mathbf{N}^{F}e_{s}^{\mathbf{L}^{F}}$ and $\mathcal{O}\mathbf{G}^{F}e_{s}^{\mathbf{G}^{F}}$. In particular, one observes that the conclusion of [2, Theorem 7.6] and therefore of [2, Theorem 7.7] holds true in this case. This proves Theorem 2. ∎

Index