Descent of splendid Rickard equivalences in ${\rm GL}_n(q)$

TL;DR

This paper proves a refined version of Broué's abelian defect group conjecture for unipotent blocks of GL_n(q) and provides a condition for general blocks, illustrating its limitations with an example.

Contribution

It establishes a refined conjecture for unipotent blocks of GL_n(q) and offers a sufficient condition for other blocks, highlighting its limitations with counterexamples.

Findings

Refined Broué conjecture holds for unipotent blocks of GL_n(q).

A sufficient condition is identified for general blocks to satisfy the conjecture.

Counterexample shows the condition does not always hold.

Abstract

Let $n$ be a positive integer and $q$ a prime power. We prove that a refined version of Brou\'{e}'s abelian defect group conjecture holds for unipotent $\ell$-blocks of ${\rm GL}_n(q)$, where $\ell\nmid q$. We also give a sufficient condition on general $\ell$-blocks of ${\rm GL}_n(q)$ to satisfy the refined abelian defect group conjecture. We explain by an example that this sufficient condition does not hold in general.

Equations

$$V_{n}=\left({\begin{array}[]{*{20}{c}}1&&&*\\ &\ddots&&\vdots\\ &&1&*\\ &&&1\end{array}}\right)_{i\times i},~{}~{}~{}~{}~{}~{}D_{n}=\left({\begin{array}[]{*{20}{c}}1&&&\\ &\ddots&&\\ &&1&\\ &&&*\end{array}}\right)_{i\times i}.$$

$$E_{i,n}^{\prime }:=k^{\prime }{G}_n{e}_{(V_n⋊\cdots ⋊V_{i+1})⋊(D_{i+1}\times \cdots \times D_n)}{\otimes }_{k^{\prime }{G}_i}-:A_i^{\prime }\text{mbox}-\mathrm{mod}\to A_n^{\prime }\text{mbox}-\mathrm{mod}$$

$$F_{i,n}^{\prime }:=e_{(V_n⋊\cdots ⋊V_{i+1})⋊(D_{i+1}\times \cdots \times D_n)}{k}^{\prime }{G}_n{\otimes }_{k^{\prime }{G}_n}-:A_n^{\prime }\text{mbox}-\mathrm{mod}\to A_i^{\prime }\text{mbox}-\mathrm{mod}.$$

$${\hat{X}}_n:=q^{n-1}{e}_{V_n{D}_n}{e}_{{}^t{V}_n}{e}_{V_n{D}_n}.$$

$${\mathcal{E}}^{\prime }:=\underset{n\ge 0}{⨁}{k}^{\prime }{G}_n{e}_{V_n{D}_n}.$$

$${\mathcal{F}}^{\prime }:=\underset{n\ge 0}{⨁}{e}_{V_n{D}_n}{k}^{\prime }{G}_n,$$

$${\hat{X}}_n=q^{n-1}{e}_{V_n{D}_n}{e}_{{}^t{V}_n}{e}_{V_n{D}_n}.$$

$$X_2^{\prime }=C_s^{\prime }{\otimes }_{B_{s-1}^{\prime }}\cdots {\otimes }_{B_1^{\prime }}{C}_1^{\prime }.$$

$${\Theta }_{\lambda }^{\prime }:{\mathrm{Comp}}^b({\mathcal{A}}_{-\lambda }^{\prime })\to {\mathrm{Comp}}^b({\mathcal{A}}_{\lambda }^{\prime }),$$

$$C_j^{\prime }{\otimes }_{B_{j-1}^{\prime }}-:{\mathrm{Comp}}^b(B_{j-1}^{\prime })\to {\mathrm{Comp}}^b(B_j^{\prime }),$$

$$C_j^{\prime }\cong C_j^{\prime }{\otimes }_{B_{j-1}^{\prime }}{B}_{j-1}^{\prime }\cong {\Theta }_{\lambda }^{\prime }(B_{j-1}^{\prime }).$$

$${\Theta }_{\lambda }^{\prime }=\cdots \to ({\Theta }_{\lambda }^{\prime }{)}^i\stackrel{d^{\prime i}}{\to }({\Theta }_{\lambda }^{\prime }{)}^{i+1}\to \cdots ,$$

$$\cdots \to ({\Theta }_{\lambda }^{\prime }{)}^i(B_{j-1}^{\prime })\stackrel{d^{\prime i}(B_{j-1}^{\prime })}{\to }({\Theta }_{\lambda }^{\prime }{)}^{i+1}(B_{j-1}^{\prime })\to \cdots$$

$$E_{i,n}:={\mathbb{F}}_{\ell }{G}_n{e}_{(V_n⋊\cdots ⋊V_{i+1})⋊(D_{i+1}\times \cdots \times D_n)}{\otimes }_{{\mathbb{F}}_{\ell }{G}_i}-:A_i\text{mbox}-\mathrm{mod}\to A_n\text{mbox}-\mathrm{mod}$$

$$F_{i,n}:=e_{(V_n⋊\cdots ⋊V_{i+1})⋊(D_{i+1}\times \cdots \times D_n)}{\mathbb{F}}_{\ell }{G}_n{\otimes }_{{\mathbb{F}}_{\ell }{G}_n}-:A_n\text{mbox}-\mathrm{mod}\to A_i\text{mbox}-\mathrm{mod}.$$

$$\mathcal{E}:=\underset{n\ge 0}{⨁}{\mathbb{F}}_{\ell }{G}_n{e}_{V_n{D}_n}.$$

$$\mathcal{F}:=\underset{n\ge 0}{⨁}{e}_{V_n{D}_n}{\mathbb{F}}_{\ell }{G}_n,$$

$${\Theta }_{\lambda }=\cdots \to ({\Theta }_{\lambda }{)}^i\stackrel{d^i}{\to }({\Theta }_{\lambda }{)}^{i+1}\to \cdots .$$

$$C_j:=\cdots \to ({\Theta }_{\lambda }{)}^i(B_{j-1})\stackrel{d^i(B_{j-1})}{\to }({\Theta }_{\lambda }{)}^{i+1}(B_{j-1})\to \cdots .$$

$$k^{\prime }{G}_1{b}_1{\otimes }_{k^{\prime }}{k}^{\prime }{G}_{2}i\cong k^{\prime }{\otimes }_k(k{G}_1{b}_1{\otimes }_{k}Y)$$

$$Y_1^{\prime }:={\mathrm{Ind}}_{N_{G\times G_1^{\mathrm{op}}}(\Delta P)}^{N_G(P)\times N_{G_1}(P{)}^{\mathrm{op}}}{\mathrm{Br}}_{\Delta P}(X_1^{\prime })$$

$$Y_1:={\mathrm{Ind}}_{N_{G\times G_1^{\mathrm{op}}}(\Delta P)}^{N_G(P)\times N_{G_1}(P{)}^{\mathrm{op}}}{\mathrm{Br}}_{\Delta P}(X_1).$$

$${\mathbf{Y}}_{\mathbf{P}}:=\{g\mathbf{V}\in \mathbf{G}/\mathbf{V}|g^{-1}F(g)\in \mathbf{V}\cdot F(\mathbf{V})\}$$

$$R_{\mathbf{L}\subset \mathbf{P}}^{\mathbf{G}}:G_0(\Lambda {\mathbf{L}}^F)\to G_0(\Lambda {\mathbf{G}}^F)$$

$${\mathrm{Irr}}_{K^{\prime }}({\mathbf{G}}^F)=\coprod _{(s)}{\mathrm{Irr}}_{K^{\prime }}({\mathbf{G}}^F,(s)),$$

$${\mathrm{Irr}}_{K^{\prime }}({\mathbf{L}}^F,(s))\stackrel{\sim }{\to }{\mathrm{Irr}}_{K^{\prime }}({\mathbf{G}}^F,(s)),\psi \mapsto {\epsilon }_{\mathbf{L},\mathbf{G}}{R}_{\mathbf{L}\subset \mathbf{P}}^{\mathbf{G}}(\psi ).$$

$${\mathrm{Irr}}_{K^{\prime }}({\mathbf{L}}^F,e_s^{{\mathbf{L}}^F})\stackrel{\sim }{\to }{\mathrm{Irr}}_{K^{\prime }}({\mathbf{G}}^F,e_s^{{\mathbf{G}}^F}),\psi \mapsto {\epsilon }_{\mathbf{L},\mathbf{G}}{R}_{\mathbf{L}\subset \mathbf{P}}^{\mathbf{G}}(\psi ).$$

$${\mathrm{Irr}}_{K^{\prime }}({\mathbf{L}}^F,e_1^{{\mathbf{L}}^F})\stackrel{\sim }{\to }{\mathrm{Irr}}_{K^{\prime }}({\mathbf{L}}^F,e_s^{{\mathbf{L}}^F}),\psi \to \eta \psi$$

$$M^{\prime }\cong M^{\prime }{\otimes }_{{\mathcal{O}}^{\prime }{\mathbf{L}}^F{e}_1^{{\mathbf{L}}^F}}{\mathcal{O}}^{\prime }{\mathbf{L}}^F{e}_1^{{\mathbf{L}}^F}\cong \Phi ({\mathcal{O}}^{\prime }{\mathbf{L}}^F{e}_1^{{\mathbf{L}}^F})={\mathcal{O}}^{\prime }{\mathbf{L}}^F{e}_1^{{\mathbf{L}}^F}{\otimes }_{{\mathcal{O}}^{\prime }}{S}^{\prime }.$$

$${\mathcal{O}}^{\prime }{\mathbf{L}}^F{\otimes }_{{\mathcal{O}}^{\prime }}{S}^{\prime }={\mathcal{O}}^{\prime }{G}_1{\otimes }_{{\mathcal{O}}^{\prime }}{S}^{\prime }\cong {\mathrm{Ind}}_{\Delta G_1}^{G_1\times G_1^{\mathrm{op}}}(S^{\prime }),$$

Full text

1 Introduction

In [17], Kessar and Linckelmann proposed a refined version of Broué’s abelian defect group conjecture.

Conjecture 1.1** (Kessar and Linckelmann).**

For an arbitrary complete discrete valuation ring $\mathcal{O}$ and a block $b$ of a finite group $G$ over $\mathcal{O}$ with an abelian defect group, there is a splendid Rickard equivalence between $\mathcal{O}Gb$ and its Brauer correspondent.

Note that Broué’s original conjecture is with the assumption that the complete discrete valuation rings have splitting residue fields. For blocks with abelian defect groups, Kessar and Linckelmann ([17, Corollary 1.9]) showed that Conjecture 1.1 implies Navarro’s refinement of the Alperin-McKay conjecture ([21, Conjecture B]). This implication has been generalised by Boltje (see [2, Theorem 1.4]), who proved that Conjecture 1.1 implies Turull’s refinement of the Alperin-McKay conjecture ([29, Conjecture]). Recall that Turull’s refinement of the Alperin-McKay conjecture contains the refined versions of the Alperin-McKay conjecture proposed by Isaacs-Navarro ([16, Conjecture B]) and Navarro ([21, Conjecture B]). In this paper we investigate Conjecture 1.1 for finite general linear groups.

Let us fix some notation. Throughout this paper, $\ell$ is a prime, $n$ is a positive integer, $q$ is a prime power, $k\subseteq k^{\prime}$ are fields of characteristic $\ell$; $\mathcal{O}\subseteq\mathcal{O}^{\prime}$ are complete discrete valuation rings of characteristic [math] with $J(\mathcal{O})\subseteq J(\mathcal{O}^{\prime})$ and with residue fields $k$ and $k^{\prime}$, respectively. Denote by $K^{\prime}$ the quotient field of $\mathcal{O}^{\prime}$. Assume that $K^{\prime}$ contains a primitive $|G|$-th root of unity for every finite group $G$ considered below. So both $K^{\prime}$ and $k^{\prime}$ are splitting fields for all finite groups considered below. Denote by $\mathbb{Q}$ and $\mathbb{F}_{\ell}$ the prime fields of $K^{\prime}$ and $k^{\prime}$, respectively. Denote by $\mathbb{Z}$ the prime ring of $\mathbb{Q}$. Let $\mathbb{Z}_{\ell}$ be the ring of $\ell$-adic integers. By [27, Chapter 2, Theorem 3, 4 and Proposition 1], $\mathbb{Z}_{\ell}$ can be identified with the unique complete discrete valuation ring $R$ contained in $\mathcal{O}$ such that $J(R)=\ell R$ and that the image of $R$ under the canonical surjection $\mathcal{O}\to k$ is $\mathbb{F}_{\ell}$. We always make this identification.

By a block of the a finite group algebra $\Lambda G$, where $\Lambda\in\{\mathcal{O},k\}$, we usually mean a primitive idempotent $b$ of the center of $\Lambda G$, and $\Lambda Gb$ is called a block algebra. Sometimes the term “block” will also be used to mean the correspondent set of irreducible characters. For a subgroup $H$ of $G$, let $(\Lambda Gb)^{H}$ denote the set of $H$-fixed elements of the block algebra $\Lambda Gb$ under the conjugation action. If $H$ is a $p$-subgroup, the Brauer map is the $\Lambda$-algebra homomorphism ${\rm Br}_{H}:(\Lambda Gb)^{H}\to kC_{G}(H)$, $\sum_{g\in G}\alpha_{g}g\mapsto\sum_{g\in C_{G}(H)}\bar{\alpha}_{g}g,$ where $\bar{\alpha}_{g}$ denotes the image of $\alpha_{g}$ in $k$. For a block $b$ of $\Lambda G$, a defect group of $b$ is a maximal $p$-subgroup $P$ of $G$ such that ${\rm Br}_{P}(b)\neq 0$. By Brauer’s first main theorem, there is a unique block $c$ of $\Lambda N_{G}(P)$ with defect group $P$ such that ${\rm Br}_{P}(b)={\rm Br}_{P}(c)$ and the map $b\mapsto c$ is a bijection between the set of blocks of $\Lambda G$ with defect group $P$ and the set of blocks of $\Lambda N_{G}(P)$ with defect group $P$. This bijection is known as the Brauer correspondence.

Let $b$ be a block $\mathcal{O}{\rm GL}_{n}(q)$ with a defect group $P$. Let $b^{\prime}$ be a block of $\mathcal{O}^{\prime}{\rm GL}_{n}(q)$ with $bb^{\prime}=b^{\prime}=b^{\prime}b$, then it is easy to show that $P$ is also a defect group of $b^{\prime}$. Assume that $P$ is abelian. If $\ell\mid q$, then by the last paragraph of Section 4 in [9], $P$ is either the trivial subgroup, or a Sylow $\ell$-subgroup of ${\rm GL}_{n}(q)$. So if $P\neq 1$, then $n=1$ or $n=2$ (otherwise, the Sylow $\ell$-subgroups are not abelian). If $n=1$ and $\ell\mid q$, Conjecture 1.1 is trivially true for the group ${\rm GL}_{n}(q)$. By [15, Theorem 1.1], if $n=2$ and $\ell\mid q$, Conjecture 1.1 is true for ${\rm GL}_{n}(q)$. So for the group ${\rm GL}_{n}(q)$ we only need to consider Conjecture 1.1 in non-defining characteristic. From now on, we assume that $\ell\nmid q$.

For the large enough field $k^{\prime}$, Chuang and Rouquier ([8, Theorem 7.20]) proved that there is a splendid Rickard equivalence between every block algebra of $k^{\prime}{\rm GL}_{n}(q)$ with an abelian defect group and its Brauer correspondent algebra. The main result of this paper is the following.

Theorem 1.2**.**

Let $G$ be the group ${\rm GL}_{n}(q)$, and let $b$ be a unipotent block of $\mathcal{O}^{\prime}G$ with an abelian defect group $P$. Let $c$ be the block of $\mathcal{O}^{\prime}N_{G}(P)$ corresponding to $b$ via the Brauer correspondence. Then $b\in\mathbb{Z}_{\ell}G$, $c\in\mathbb{Z}_{\ell}N_{G}(P)$, and the block algebras $\mathbb{Z}_{\ell}Gb$ and $\mathbb{Z}_{\ell}N_{G}(P)c$ are splendidly Rickard equivalent. More precisely, there is a splendid Rickard complex $X$ of $(\mathbb{Z}_{\ell}Gb,\mathbb{Z}_{\ell}N_{G}(P)c)$-bimodules such that $\mathcal{O}^{\prime}\otimes_{\mathbb{Z}_{\ell}}X$ is isomorphic to Chuang and Rouquier’s complex $X^{\prime}$.

See §2.2 below for the definition of splendid Rickard equivalences. Using a theorem of Boltje ([2, Theorem 1.4]), Theorem 1.2 has the following corollary.

Corollary 1.3**.**

For any prime number $p$, Turull’s refinement of the Alperin-McKay conjecture ([29, Conjecture]) holds for unipotent $\ell$-blocks of ${\rm GL}_{n}(q)$ with an abelian defect group.

By lifting theorem of splendid Rickard equivalences (reviewed in Theorem 2.1 below), to prove Theorem 1.2, we may replace $\mathcal{O}^{\prime}$ by $k^{\prime}$ and replace $\mathbb{Z}_{\ell}$ by $\mathbb{F}_{\ell}$. The main steps of proving Theorem 1.2 are summarized in the following remarks.

Remark 1.4**.**

Let $G$ be ${\rm GL}_{n}(q)$ and $b$ be a unipotent block of $k^{\prime}G$. Let $e$ be the multiplicative order of $q$ in $k^{\times}$. Recall that there is associated a non-negative integer $w$, called the ($e$-)weight of $b$. Let $P$ be a defect group of $b$, and let $c$ be the block of $k^{\prime}N_{G}(P)$ corresponding to $b$ via the Brauer correspondence. The defect group $P$ of $b$ is abelian if and only if $w<\ell$. The proof of [8, Theorem 7.20] consists of three steps:

(i). $k^{\prime}N_{G}(P)c$ is splendid Rickard equivalent to the principal block algebra of $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})$ (by [24, Theorem 10.1] and [19, Theorem 4.3 (b)] ).

(ii). There exists an integer $n^{\prime}\geq 1$ and a unipotent block $b^{\prime}$ of ${\rm GL}_{n^{\prime}}(q)$, with weight $w$, which is splendid Morita equivalent to the principal block algebra of $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})$ (see [20, Theorem 5.0.7] or [28, Theorem 1]).

(iii). The block algebras $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ and $k^{\prime}Gb$ are splendidly Rickard equivalent (see [8, Theorem 7.18]). **

Remark 1.5**.**

Assume that $b$ is a unipotent block of $k^{\prime}G$. By Corollary 2.3 below, we have $b\in\mathbb{F}_{\ell}G$. By Remark 1.4, there is a splendid Rickard complex $X^{\prime}_{1}$ for $k^{\prime}N_{G}(P)b$ and $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$, and a splendid Rickard complex $X^{\prime}_{2}$ for $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ and $k^{\prime}Gb$. We will show (in Section 3 and 4) that there is a complex $X_{1}$ of $(\mathbb{F}_{\ell}N_{G}(P)b,\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime})$-bimodules, and a complex $X_{2}$ of $(\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime},\mathbb{F}_{\ell}Gb)$-bimodules, such that $X^{\prime}_{1}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}X_{1}$ and $X^{\prime}_{2}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}X_{2}$. Then by [17, Proposition 4.5 (a)], $X_{1}$ induces a splendid Rickard equivalence between $\mathbb{F}_{\ell}N_{G}(P)b$ and $\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime}$, and $X_{2}$ induces a splendid Rickard equivalence between $\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ and $\mathbb{F}_{\ell}Gb$. Hence at that time, the proof of Theorem 1.2 is complete.**

In [11] and [12], Broué’s abelian defect group conjecture is proved for unipotent $\ell$-blocks of the groups ${\rm GU}_{n}(q)$, ${\rm Sp}_{2n}(q)$ and ${\rm SO}_{2n+1}(q)$ at linear primes $\ell$. Since the constructions of the derived equivalences for unipotent blocks of these groups have many common properties with the constructions of the equivalences for unipotent blocks of ${\rm GL}_{n}(q)$, the methods in the proof of Theorem 1.2 may be used to prove a similar proposition for these groups instead of ${\rm GL}_{n}(q)$.

One will ask whether the refined abelian defect group conjecture holds for general blocks of $\mathcal{O}{\rm GL}_{n}(q)$, not only unipotent blocks. In Section 5, we investigate this question and give a sufficient condition under which there exists a splendid Rickard equivalence between a block of $\mathcal{O}{\rm GL}_{n}(q)$ and its Brauer correspondent.

2 Preliminaries

2.1 Notation

If $G$ is a finite group, we denote by $G^{\rm op}$ the opposite group, and by $\Delta G$ the subgroup $\{(g,g^{-1})~{}|~{}g\in G\}$ of $G\times G^{\rm op}$. For an additive category $\mathcal{C}$, we denote by ${\rm Comp}^{b}(\mathcal{C})$ the category of bounded complexes of objects of $\mathcal{C}$ and by ${\rm Ho}^{b}(\mathcal{C})$ is homotopy category. For an algebra $A$, we denote by $A^{\rm op}$ the opposite algebra of $A$. Unless specified otherwise, modules in the paper are left modules. We denote by $A$-${\rm mod}$ the category of finite generated $A$-modules, and by $G_{0}(A)$ the Grothendieck group of $A$-${\rm mod}$. Let ${\rm Ho}^{b}(A):={\rm Ho}^{b}(A\mbox{-mod})$.

2.2 Splendid Rickard equivalences

Let $A$ and $B$ be symmetric $\Lambda$-algebras, where $\Lambda\in\{\mathcal{O},k\}$. Let $X$ be a bounded complex of finitely generated $(A,B)$-bimodules which are projective as left $A$-modules and as right $B$-modules, and let $X^{*}:={\rm Hom}_{\Lambda}(X,\Lambda)$ be the dual complex. It is said that $X$ induces a Rickard equivalence and that $X$ is a Rickard complex if there exists a contractible complex of $(A,A)$-bimodules $Y$ and a contractible complex of $(B,B)$-bimodules $Z$ such that $X\otimes_{B}X^{*}=A\oplus Y$ as complexes $(A,A)$-bimodules and $X^{*}\otimes_{A}X=B\oplus Z$ as complexes of $(B,B)$-bimodules. Let $G$ and $H$ be finite groups. Let $b$ (resp. $c$) be a block of $\Lambda G$ (resp. $\Lambda H$). Let $X:=(X_{n})_{n\in\mathbb{Z}}$ be a Rickard complex of $(\Lambda Gb,\Lambda Hc)$-bimodules. If each $X_{n}$ is a direct summand of permutation $\Lambda(G\times H^{\rm op})$-module (i.e., an $\ell$-permutation $\Lambda(G\times H^{\rm op})$-module), then $X$ is said to be splendid; $\Lambda Gb$ and $\Lambda Hc$ are said to be splendidly Rickard equivalent. Chuang and Rouquier remarked in the first paragraph of [8, §7.1.2] that one usually puts some condition on the vertex of $X_{n}$, but this is actually automatic. The following theorem on lifting splendid Rickard equivalences is due to Rickard.

Theorem 2.1** ([23, Theorem 5.2]).**

Let $G$ and $H$ be finite groups. Let $b$ (resp. $c$) be an idempotent in the center of $\mathcal{O}G$ (resp. $\mathcal{O}H$). Denote by $\bar{b}$ (resp. $\bar{c}$) the image of $b$ (resp. $c$) in $kG$ (resp. $kH$). Assume that there is a complex $\bar{X}$ of $(kG\bar{b},kH\bar{c})$-bimodules inducing a splendid Rickard equivalence. Then there is a complex $X$ of $(\mathcal{O}Gb,\mathcal{O}Hc)$-bimodules inducing a splendid Rickard equivalence and satisfying $k\otimes_{\mathcal{O}}X\cong\bar{X}$.

Note that although the statement in [23, Theorem 5.2] is for principal blocks, but the proof carries over nearly verbatim to arbitrary blocks. We also note that the blanket assumption in [12] that the coefficient rings are big enough is not used in the proof of [23, Theorem 5.2].

2.3 Actions of Galois automorphisms on modules

Let $R\subseteq R^{\prime}$ be two commutative domains. Let $A$ be an $R$-algebra and let $A^{\prime}:=R^{\prime}\otimes_{R}A$. Let $\Gamma$ be the group of automorphisms of $R^{\prime}$ which restricts to the identity map on $R$. For an $A^{\prime}$-module $U^{\prime}$ and a $\sigma\in\Gamma$, denote by ${}^{\sigma}U^{\prime}$ the $A^{\prime}$-module which is equal to $U^{\prime}$ as a module over the subalgebra $1\otimes A$ of $A^{\prime}$, such that $x\otimes a$ acts on $U^{\prime}$ as $\sigma^{-1}(x)\otimes a$ for all $a\in A$ and $x\in R^{\prime}$. The $A^{\prime}$-module $U^{\prime}$ is $\Gamma$-stable if ${}^{\sigma}U^{\prime}\cong U^{\prime}$ for all $\sigma\in\Gamma$. $U^{\prime}$ is said to be defined over $R$, if there is an $A$-module $U$ such that $U^{\prime}\cong R^{\prime}\otimes_{R}U$. In this special case, $U^{\prime}$ is $\Gamma$-stable, because for any $\sigma\in\Gamma$, the map sending $x\otimes u$ to $\sigma^{-1}(x)\otimes u$ is an isomorphism $R^{\prime}\otimes_{R}U\cong{}^{\sigma}(R^{\prime}\otimes_{R}U)$, where $u\in U$ and $x\in R^{\prime}$.

2.4 Block idempotents and coefficient rings

Proposition 2.2**.**

Let $G$ be a finite group, $b$ a block of $\mathcal{O}^{\prime}G$, and let $\chi:G\to K^{\prime}$ be the character of a simple $K^{\prime}Gb$-module. Let $\bar{b}$ be the image of $b$ in $k^{\prime}G$. If the values of $\chi$ are contained in $\mathbb{Q}$ (hence in $\mathbb{Z}$), then we have $b\in\mathbb{Z}_{\ell}G$ and $\bar{b}\in\mathbb{F}_{\ell}G$.

Proof. Let $V$ be an $\mathcal{O}^{\prime}G$-module such that the $K^{\prime}G$-module $K^{\prime}\otimes_{\mathcal{O}^{\prime}}V$ affords the character $\chi$ (see e.g. [18, Theorem 4.16.5] for the existence of $V$). Let $\varphi:G\to k^{\prime}$ be the character afforded by the $k^{\prime}G\bar{b}$-module $k^{\prime}\otimes_{\mathcal{O}^{\prime}}V$. The values of $\varphi$ are the images of values of $\chi$ under the canonical surjection $\mathcal{O}\to k$, hence contained in $\mathbb{F}_{\ell}$. For any $\sigma\in{\rm Gal}(k^{\prime}/\mathbb{F}_{\ell})$, $\sigma$ induces a ring automorphism of $k^{\prime}G$ in an obvious way. Hence $\sigma(\bar{b})$ is also a block of $k^{\prime}G$. Since $\varphi$ is invariant under the action of $\sigma$, we see that $\sigma(\bar{b})=\bar{b}$. Because of every finite group has a finite splitting field, we may assume that $k^{\prime}$ is finite. Then we deduce that $\bar{b}\in\mathbb{F}_{\ell}G$ by the Galois theory. By idempotent lifting arguments, we have $b\in\mathbb{Z}_{\ell}G$. $\hfill\square$

Corollary 2.3**.**

Let $b$ be a unipotent block of $\mathcal{O}^{\prime}{\rm GL}_{n}(q)$, and let $\bar{b}$ be the image of $b$ in $k^{\prime}{\rm GL}_{n}(q)$, then $b\in\mathbb{Z}_{\ell}{\rm GL}_{n}(q)$ and $\bar{b}\in\mathbb{F}_{\ell}{\rm GL}_{n}(q)$.

Proof. Let $\chi:{\rm GL}_{n}(q)\to K^{\prime}$ be a unipotent character of ${\rm GL}_{n}(q)$. By [13, Example 1.1], the values of $\chi$ are contained in $\mathbb{Q}$. The statement follows by Proposition 2.2. $\hfill\square$

It is easy to see that Proposition 2.2 also has the following corollary.

Corollary 2.4**.**

Let $G$ be a finite group and let $b$ be the principal block of $\mathcal{O}^{\prime}G$, then $b\in\mathbb{Z}_{\ell}G$ and $\bar{b}\in\mathbb{F}_{\ell}G$.

3 On unipotent blocks of general linear groups with same weights

Keep the notation of Remark 1.4 and 1.5, in this section, we show that the splendid Rickard complex $X^{\prime}_{2}$ is defined over $\mathbb{F}_{\ell}$. The construction of this complex is played back to [8, Theorem 7.18].

We start with the case $\ell|(q-1)$. By [8, Remark 7.19], $k^{\prime}G$ ($=k^{\prime}{\rm GL}_{n}(q)$) has a unique unipotent block $b$, the principal block. The number of isomorphism classes of simple $k^{\prime}Gb$-modules is the number of partitions of $n$. So, if $n\neq n^{\prime}$, $k^{\prime}Gb$ can not Rickard equivalent to a unipotent block of $k^{\prime}{\rm GL}_{n^{\prime}}(q)$. Hence $n=n^{\prime}$. In this case, the complex $X^{\prime}_{2}$ can be taken to be the $(k^{\prime}Gb,k^{\prime}Gb)$-bimodule $k^{\prime}Gb$, and it is obviously defined over $\mathbb{F}_{\ell}$.

It remains to consider the case where $\ell\nmid q(q-1)$. The construction of the complex $X_{2}^{\prime}$ uses the $\mathfrak{sl}_{2}$-categorification. Let $G_{n}:={\rm GL}_{n}(q)$ and let $A^{\prime}_{n}=k^{\prime}G_{n}b_{n}$ be the sum of the unipotent block algebras of $k^{\prime}G_{n}$. Given a finite group $H$ with $\ell\nmid|H|$, put $e_{H}:=\frac{1}{|H|}\sum_{h\in H}h$, an idempotent in $\mathbb{F}_{\ell}H\subseteq k^{\prime}H$. For a matrix $g\in G_{n}$, denote by ${}^{t}g$ the transpose of $g$. Denote by $V_{n}$ the subgroup of upper triangular matrices of $G_{n}$ with diagonal coefficients $1$ whose off-diagonal coefficients vanish outside the $n$-th column. Denote by $D_{n}$ the subgroup of $G_{n}$ of diagonal matrices with diagonal entries $1$ except the $(n,n)$-th one.

[TABLE]

For $i\in\{0,1,\cdots,n-1\}$, we view $G_{i}$ as a subgroup of $G_{n}$ via the first $i$ coordinates. Following Chuang and Rouquier, we put

[TABLE]

and put

[TABLE]

The functors $E^{\prime}_{i,n}$ and $F^{\prime}_{i,n}$ are canonically left and right adjoint. Let $\mathcal{A}^{\prime}:=\bigoplus_{n\geq 0}A^{\prime}_{n}$-mod, $E^{\prime}:=\bigoplus_{n\geq 0}E^{\prime}_{n,n+1}$ and $F^{\prime}:=\bigoplus_{n\geq 0}F^{\prime}_{n,n+1}$. Denote by $X$ the endomorphism of $E^{\prime}$ given on $E^{\prime}_{n-1,n}$ by right multiplication by

[TABLE]

Given $a\in k^{\prime\times}$, let $E^{\prime}_{a}$ be the generalised $a$-eigenspace of $X$ on $E^{\prime}$. We have a decomposition $E^{\prime}=\bigoplus_{a\in k^{\prime\times}}E^{\prime}_{a}$. There is a corresponding decomposition $F^{\prime}=\bigoplus_{a\in k^{\prime\times}}F^{\prime}_{a}$, such that $F^{\prime}_{a}$ is left and right adjoint to $E^{\prime}_{a}$. Note that $E^{\prime}_{a}$ and $F^{\prime}_{a}$ are functors from $\mathcal{A}^{\prime}$ to $\mathcal{A}^{\prime}$, so they induce actions $[E^{\prime}_{a}]$ and $[F^{\prime}_{a}]$ on $\bigoplus_{n\geq 0}G_{0}(A^{\prime}_{n}\mbox{-}{\rm mod})$, respectively. By [8, Lemma 7.16], the action of $[E^{\prime}_{a}]$ and $[F^{\prime}_{a}]$ on $\bigoplus_{n\geq 0}G_{0}(A^{\prime}_{n}\mbox{-}{\rm mod})$ gives a representation of $\mathfrak{sl}_{2}$, and the classes of simple objects are weight vectors. Moreover, these actions give rise to an $\mathfrak{sl}_{2}$-categorification on $\mathcal{A}^{\prime}$ (see [8, §7.3.1]).

The functors $E^{\prime}$ and $F^{\prime}$ are defined by tensoring with bimodules. Since bimodules are more convenient to handle than functors, let us add some notation. Let $\mathscr{A}^{\prime}:=\bigoplus_{n\geq 0}A^{\prime}_{n}$, a $k^{\prime}$-algebra. The functor $E^{\prime}_{n-1,n}$ is defined by the $(k^{\prime}G_{n},k^{\prime}G_{n-1})$-bimodule $k^{\prime}G_{n}e_{V_{n}D_{n}}$, we denote this bimodule by $\mathscr{E}^{\prime}_{n-1,n}$. We can view $\mathscr{E}^{\prime}_{n-1,n}$ as an $(\mathscr{A}^{\prime},\mathscr{A}^{\prime})$-bimodule, so we have an $(\mathscr{A}^{\prime},\mathscr{A}^{\prime})$-bimodule

[TABLE]

Clearly the bimodule $\mathscr{E}^{\prime}$ corresponds to the functor $E^{\prime}$. Similarly, we have an $(\mathscr{A}^{\prime},\mathscr{A}^{\prime})$-bimodule

[TABLE]

which corresponds to the functor $F^{\prime}$. The endomorphism $X$ of $E^{\prime}$ has similar properties with the endomorphism $\mathscr{X}$ of $\mathscr{E}^{\prime}$, given on $\mathscr{E}^{\prime}_{n-1,n}$ by right multiplication by

[TABLE]

Given $a\in k^{\prime\times}$, let $\mathscr{E}^{\prime}_{a}$ be the generalised $a$-eigenspace of $\mathscr{X}$ on $\mathscr{E}^{\prime}$. We have a decomposition $\mathscr{E}^{\prime}=\bigoplus_{a\in k^{\prime\times}}\mathscr{E}^{\prime}_{a}$. Then the $(\mathscr{A}^{\prime},\mathscr{A}^{\prime})$-bimodule $\mathscr{E}^{\prime}_{a}$ corresponds to the functor $E^{\prime}_{a}$. Since the bimodule $\mathscr{F}^{\prime}$ corresponds to the functor $F^{\prime}$, there is a decomposition $\mathscr{F}^{\prime}=\bigoplus_{a\in k^{\prime\times}}\mathscr{F}^{\prime}_{a}$, such that the $(\mathscr{A}^{\prime},\mathscr{A}^{\prime})$-bimodule $\mathscr{F}^{\prime}_{a}$ corresponds to the functor $F^{\prime}_{a}$.

Proposition 3.1**.**

The eigenvalues of $X$ as an endomorphism of $E^{\prime}$ are contained in $\mathbb{F}_{\ell}$. Hence the eigenvalues of $\mathscr{X}$ as an endomorphism of $\mathscr{E}^{\prime}$ are contained in $\mathbb{F}_{\ell}$.

Proof. The set of eigenvalues of $X$ is the union of the sets of eigenvalues of $\hat{X}_{n}$’s. Note that $\hat{X}_{1}$ induces the identical map on $E^{\prime}_{1,2}$, so each eigenvalue of $\hat{X}_{1}$ is $1$. By [14, Lemma 4.7], the eigenvalues of $\hat{X}_{n}$ on $E^{\prime}_{n,n-1}$ are powers of $q$ (considered as elements in $\mathbb{F}_{\ell}$), whence the statement. $\hfill\square$

Since the results on the local block theory of symmetric groups generalise to unipotent blocks of general linear groups [4, §3], we have an analog of Theorem 7.1 in [8], which is omitted in [8].

Theorem 3.2** (an analog of [8, Theorem 7.1]).**

Let $e$ be the multiplicative order of $q$ in $\mathbb{F}_{\ell}$. The functors $[E^{\prime}_{a}]$ and $[F^{\prime}_{a}]$ for $a\in\mathbb{F}_{\ell}$ give rise to an action of the affine Lie algebra $\hat{\mathfrak{sl}}_{e}$ on $\bigoplus_{n\geq 0}G_{0}(A^{\prime}_{n}\mbox{-}{\rm mod})$. The decomposition of $\bigoplus_{n\geq 0}G_{0}(A^{\prime}_{n}\mbox{-}{\rm mod})$ in blocks coincides with its decomposition in weight spaces. Two unipotent blocks of general linear groups have the same weight if and only if they are in the same orbit under the adjoint action of the affine Weyl group.

In Remark 1.4 and 1.5, the unipotent block algebras $k^{\prime}Gb$ and $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ have the same weight. So by Theorem 3.2, there is a sequence of unipotent block algebras $B^{\prime}_{0}=k^{\prime}Gb,B^{\prime}_{1},\cdots,B^{\prime}_{s}=k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ such that $B^{\prime}_{j}$ is the image of $B^{\prime}_{j-1}$ by some simple reflection $\sigma_{a_{j}}$ of the affine Weyl group. By Proposition 3.1, these eigenvalues $a_{1},\cdots,a_{s}$ are contained in $\mathbb{F}_{\ell}$.

By [8, Theorem 6.4], the complex of functors $\Theta^{\prime}$ there associated with $a=a_{j}$ induces a self-equivalence of ${\rm Ho}^{b}(\mathcal{A}^{\prime})$. It restricts to a splendid Rickard equivalence between $B^{\prime}_{j}$ and $B^{\prime}_{j-1}$. The Rickard equivalence between $k^{\prime}Gb$ and $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ is the composition of these equivalences. In other words, if we denote by $C^{\prime}_{j}$ the complex of $(B^{\prime}_{j},B^{\prime}_{j-1})$-bimodules which induces the splendid Rickard equivalence between $B^{\prime}_{j}$ and $B^{\prime}_{j-1}$, then

[TABLE]

By Corollary 2.3, there are block algebras $B_{0}=\mathbb{F}_{\ell}Gb,B_{1},\cdots,B_{s}=\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ such that $B^{\prime}_{j}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}B_{j}$.

Proposition 3.3**.**

There is a complex $X_{2}$ of $(\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime},\mathbb{F}_{\ell}Gb)$-bimodules, such that $X^{\prime}_{2}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}X_{2}$.

To prove Proposition 3.3, it suffices to prove the following statement.

Proposition 3.4**.**

For each $j\in\{1,\cdots,s\}$, there exists a complex $C_{j}$ of $(B_{j},B_{j-1})$-bimodules such that $C^{\prime}_{j}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}C_{j}$.

Proof. Actually, the categories $B^{\prime}_{j-1}$-mod and $B^{\prime}_{j}$-mod are exactly the categories $\mathcal{A}^{\prime}_{-\lambda}$ and $\mathcal{A}^{\prime}_{\lambda}$ in [8, Theorem 6.4] for some $\lambda$. The complex $C^{\prime}_{j}$ of $(B^{\prime}_{j},B^{\prime}_{j-1})$-bimodules is defined by the complex of functors $\Theta^{\prime}_{\lambda}$ described in [8, §6.1]. We can express the complex $C^{\prime}_{j}$ in terms of $\Theta^{\prime}_{\lambda}$. Because of the functor

[TABLE]

coincides with the functor

[TABLE]

we have

[TABLE]

More explicitly, if we write

[TABLE]

then $C^{\prime}_{j}$ is isomorphic to the complex

[TABLE]

(see the first paragraph of [8, §4.1.4]). Note that since the functor $(\Theta^{\prime}_{\lambda})^{i}$ is defined by tensoring with bimodules and since $B^{\prime}_{j-1}$ is a $(B^{\prime}_{j-1},B^{\prime}_{j-1})$-bimodule, $(\Theta^{\prime}_{\lambda})^{i}(B^{\prime}_{j-1})$ is not only a left $B^{\prime}_{j}$-module but also a right $B^{\prime}_{j-1}$-module, hence a $(B^{\prime}_{j},B^{\prime}_{j-1})$-bimodule.

To show that the complex $C^{\prime}_{j}$ is defined over $k$, let us first add some notation. By Corollary 2.3, we have $b_{n}\in\mathbb{F}_{\ell}G_{n}$ (recall that $b_{n}$ denotes the sum of the unipotent block of $k^{\prime}G_{n}$). Put $A_{n}:=\mathbb{F}_{\ell}G_{n}b_{n}$, then we have $A^{\prime}_{n}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}A_{n}$. We put

[TABLE]

and put

[TABLE]

The functors $E_{i,n}$ and $F_{i,n}$ are canonically left and right adjoint.

Let $\mathcal{A}:=\bigoplus_{n\geq 0}A_{n}$-mod, $E:=\bigoplus_{n\geq 0}E_{n,n+1}$ and $F:=\bigoplus_{n\geq 0}F_{n,n+1}$. Let $\mathscr{A}:=\bigoplus_{n\geq 0}A_{n}$, a $k$-algebra. The functor $E_{n-1,n}$ is defined by the $(\mathbb{F}_{\ell}G_{n},\mathbb{F}_{\ell}G_{n-1})$-bimodule $\mathbb{F}_{\ell}G_{n}e_{V_{n}D_{n}}$, we denote this bimodule by $\mathscr{E}_{n,n-1}$. We can view $\mathscr{E}_{n,n-1}$ as an $(\mathscr{A},\mathscr{A})$-bimodule, so we have an $(\mathscr{A},\mathscr{A})$-bimodule

[TABLE]

Clearly the the bimodule $\mathscr{E}$ corresponds to the functor $E$. Similarly, we have an $(\mathscr{A},\mathscr{A})$-bimodule

[TABLE]

which corresponds to the functor $F$. It is obvious that $\mathscr{A}^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{A}$ as $k^{\prime}$-algebras; $\mathscr{E}^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{E}$ and $\mathscr{F}^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{F}$ as $(\mathscr{A}^{\prime},\mathscr{A}^{\prime})$-bimodules.

We identify $\mathscr{A}$ with the subalgebra $1\otimes\mathscr{A}$ of $\mathscr{A}^{\prime}$ and identify $\mathscr{E}$ with the $(\mathscr{A},\mathscr{A})$-submodule $1\otimes\mathscr{E}$ of $\mathscr{E}^{\prime}$. By definition, we see that the endomorphism $\mathscr{X}$ of $\mathscr{E}^{\prime}$ restricts an endomorphism of $\mathscr{E}$. By Proposition 3.1, any eigenvalue $a$ of $\mathscr{X}$ as an endomorphism of the bimodule $\mathscr{E}^{\prime}$ are contained in $\mathbb{F}_{\ell}$. By elementary linear algebra, if we let $\mathscr{E}_{a}$ be the generalised $a$-eigenspace of $\mathscr{X}$ on $\mathscr{E}$, then we have $\mathscr{E}^{\prime}_{a}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{E}_{a}$. Since $E$ and $F$ are left and right adjoint, there is a corresponding decomposition $F=\bigoplus_{a\in\mathbb{F}_{\ell}^{\times}}F_{a}$, such that $F_{a}$ is left and right adjoint to $E_{a}$. Since the bimodule $\mathscr{F}$ corresponds to the functor $F$, there is a decomposition $\mathscr{F}=\bigoplus_{a\in\mathbb{F}_{\ell}^{\times}}\mathscr{F}_{a}$, such that the $(\mathscr{A},\mathscr{A})$-bimodule $\mathscr{F}_{a}$ corresponds to the functor $F_{a}$. Since $\mathscr{E}^{\prime}_{a}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{E}_{a}$, it is easy to see that $\mathscr{F}^{\prime}_{a}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{F}_{a}$ (by the uniqueness of right adjoints, see e.g. [18, Theorem 2.3.7]).

In [8, §6.1], the construction of $\Theta^{\prime}_{\lambda}$ only uses the functors $E^{\prime}_{a_{j}}$, $F^{\prime}_{a_{j}}$, the co-unit $\varepsilon^{\prime}$ of the adjoint pair $(E^{\prime}_{a_{j}},F^{\prime}_{a_{j}})$ and some elements of the form $c_{i}^{\tau}$ in the affine Hecke algebras, where $i$ is some integer and $\tau\in\{1,{\rm sgn}\}$. (See [8, §3.1.1] for the definition of affine Hecke algebras and see [8, §3.1.4] for the definition of $c_{i}^{\tau}$.) We note that in the definition of affine Hecke algebras, the base field can be any field. We also note that the element $c_{i}^{\tau}$ is an $\mathbb{F}_{\ell}$-linear combination of the generators of an affine Hecke algebra, hence $c_{i}^{\tau}$ can still be defined even if the base field is $\mathbb{F}_{\ell}$.

Next, we explain that the complex of functors $\Theta^{\prime}_{\lambda}:{\rm Comp}(\mathcal{A}^{\prime}_{-\lambda})\to{\rm Comp}(\mathcal{A}^{\prime}_{\lambda})$ in [8, §6.1] can still be defined even if the base field is $\mathbb{F}_{\ell}$ (in our case). Recall that $(\Theta^{\prime}_{\lambda})^{-r}$ is the restriction of $E_{a_{j}}^{\prime({\rm sgn},\lambda+r)}F_{a_{j}}^{\prime(1,r)}$ to $\mathcal{A}^{\prime}_{-\lambda}$ for $r,\lambda+r\geq 0$ and $(\Theta^{\prime}_{\lambda})^{-r}=0$ otherwise. Note that in our case, $\mathcal{A}^{\prime}_{-\lambda}$ is exactly the category $B^{\prime}_{j-1}$-mod and $\mathcal{A^{\prime}}_{\lambda}$ is exactly the category $B^{\prime}_{j}$-mod. Set $\mathcal{A}_{-\lambda}:=B_{j-1}$-mod and $\mathcal{A}_{\lambda}:=B_{j}$-mod. As in [8, §6.1], we denote by $(\Theta_{\lambda})^{-r}$ the restriction of $E_{a_{j}}^{({\rm sgn},\lambda+r)}F_{a_{j}}^{(1,r)}$ to $\mathcal{A}_{-\lambda}$ for $r,\lambda+r\geq 0$ and put $(\Theta_{\lambda})^{-r}=0$ otherwise. Since $E_{a_{j}}^{\prime({\rm sgn},\lambda+r)}F_{a_{j}}^{\prime(1,r)}$ restricts a functor from $\mathcal{A}^{\prime}_{-\lambda}$ to $\mathcal{A}^{\prime}_{\lambda}$, and since we have $\mathscr{E}_{a_{j}}^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{E}_{a_{j}}$ and $\mathscr{F}_{a_{j}}^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{F}_{a_{j}}$, $E_{a_{j}}^{({\rm sgn},\lambda+r)}F_{a_{j}}^{(1,r)}$ must send an object of $\mathcal{A}_{-\lambda}$ to $\mathcal{A}_{\lambda}$. In other words, $E_{a_{j}}^{({\rm sgn},\lambda+r)}F_{a_{j}}^{(1,r)}$ restricts a functor from $\mathcal{A}_{-\lambda}$ to $\mathcal{A}_{\lambda}$. So $(\Theta_{\lambda})^{-r}$ is actually a functor from $\mathcal{A}_{-\lambda}$ to $\mathcal{A}_{\lambda}$.

In the third paragraph of [8, §6.1], Chuang and Rouquier defined a map $d^{\prime-r}:(\Theta^{\prime}_{\lambda})^{-r}\to(\Theta^{\prime}_{\lambda})^{-r+1}$. Using the same way (replacing the co-unit $\varepsilon^{\prime}$ of the adjoint pair $(E^{\prime}_{a_{j}},F^{\prime}_{a_{j}})$ by the co-unit $\varepsilon$ of the adjoint pair $(E_{a_{j}},F_{a_{j}})$), we can define a map $d^{-r}:(\Theta_{\lambda})^{-r}\to(\Theta_{\lambda})^{-r+1}$. So we obtain a complex of functors

[TABLE]

Evaluated $\Theta_{\lambda}$ at the $(B_{j-1},B_{j-1})$-bimodule $B_{j-1}$, we obtain a complex

[TABLE]

Since we have $\mathscr{E}_{a_{j}}^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{E}_{a_{j}}$ and $\mathscr{F}_{a_{j}}^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathscr{F}_{a_{j}}$ and since the co-unit of the adjoint pair $(E^{\prime}_{n,n-1},F^{\prime}_{n,n-1})$ and the co-unit of the adjoint pair $(E_{n,n-1},F_{n,n-1})$ are constructed in the same way (i.e., the construction does not depend on the base field), it easy to see that $C^{\prime}_{j}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}C_{j}$ as complexes of $(B^{\prime}_{j},B^{\prime}_{j-1})$-bimodules. $\hfill\square$

4 Proof of Theorem 1.2

In Section 3, we showed that the splendid Rickard equivalence in Remark 1.4 (iii) descends to $\mathbb{F}_{\ell}$, so we finished half of the task listed in Remark 1.5. To finish the proof of Theorem 1.2, we need to show that the splendid Rickard equivalences in Remark 1.4 (i) and (ii) descend to $\mathbb{F}_{\ell}$.

Keep the notation of Remark 1.4 and 1.5. Recall that $G$ is ${\rm GL}_{n}(q)$, $b$ is a unipotent block of $k^{\prime}G$, $P$ is a defect group of $b$, $c$ is the block of $k^{\prime}N_{G}(P)$ corresponding to $b$ via the Brauer correspondence, $e$ is the multiplicative order of $q$ in $k^{\times}$, and $w$ is the ($e$-)weight of $b$. The block $b$ corresponds uniquely to an $e$-core $\kappa$. The assumption that the defect group $P$ of $b$ is abelian forces $w<\ell$. Let $m$ be the greatest integer such that $\ell^{m}$ divides $q^{e}-1$, then $P\cong\underbrace{C_{\ell^{m}}\times\cdots\times C_{\ell^{m}}}_{w}$ (see [28, §1.5]). (Here the notation $C_{\ell^{m}}$ denotes a cyclic group of order $\ell^{m}$.) We identify $P$ and $\underbrace{C_{\ell^{m}}\times\cdots\times C_{\ell^{m}}}_{w}$ via the isomorphism. Let $t:=n-ew$. Then by [28, §1.4 and §1.5], the block algebra $k^{\prime}N_{G}(P)c$ is isomorphic to $k^{\prime}N_{{\rm GL}_{ew}(q)}(P)c_{ew}\otimes_{k^{\prime}}k^{\prime}{\rm GL}_{t}(q)c_{0}$, where $c_{ew}$ is the principal block of $k^{\prime}N_{{\rm GL}_{ew}(q)}$ and $c_{0}$ is the unipotent block of $k^{\prime}{\rm GL}_{t}(q)$ corresponding to the $e$-core $\kappa$ and having defect group $1$. We identify $k^{\prime}N_{G}(P)c$ and $k^{\prime}N_{{\rm GL}_{ew}(q)}(P)\otimes_{k^{\prime}}k^{\prime}{\rm GL}_{t}(q)c_{0}$ via the isomorphism. Since $c\in\mathbb{F}_{\ell}N_{G}(P)$, $c_{ew}\in\mathbb{F}_{\ell}N_{{\rm GL}_{ew}(q)}(P)$ and $c_{0}\in\mathbb{F}_{\ell}{\rm GL}_{t}(q)$ (see Corollary 2.3 and 2.4), we also have $\mathbb{F}_{\ell}N_{G}(P)c\cong\mathbb{F}_{\ell}N_{{\rm GL}_{ew}(q)}(P)c_{ew}\otimes_{\mathbb{F}_{\ell}}\mathbb{F}_{\ell}{\rm GL}_{t}(q)c_{0}$. We identify this two algebras.

By [7, Lemma 6], there is a splendid Morita equivalence between $k^{\prime}N_{{\rm GL}_{ew}(q)}(P)c_{ew}$ and $k^{\prime}N_{G}(P)c$. To show that this splendid Morita equivalence descends to $\mathbb{F}_{\ell}$, we prove a descent proposition for [7, Lemma 6].

Proposition 4.1**.**

Let $G_{1}$ and $G_{2}$ be finite groups. Let $b_{1}$ and $b_{2}$ be blocks of $k^{\prime}G_{1}$ and $k^{\prime}G_{2}$, and assume that $b_{2}$ has defect group $1$. Assume that $k$ is a subfield of $k^{\prime}$ such that $b_{1}\in kG_{1}$ and $b_{2}\in kG_{2}$, then $kG_{1}b_{1}$ and $kG_{1}b_{1}\otimes_{k}kG_{2}b_{2}$ (a block algebra of $G_{1}\times G_{2}$) are splendidly Morita equivalent.

Proof. Since every finite group has a finite splitting field, we may assume that $k^{\prime}$ is finite. Let $i$ be a primitive idempotent in $k^{\prime}G_{2}b_{2}$. By the proof of [7, Lemma 6], the $(k^{\prime}(G_{1}\times G_{2}),k^{\prime}G_{1})$-bimodule $k^{\prime}G_{1}b_{1}\otimes_{k^{\prime}}k^{\prime}G_{2}i$ induces a splendid Morita equivalence be tween $k^{\prime}G_{1}b_{1}\otimes_{k^{\prime}}k^{\prime}G_{2}b_{2}$ and $k^{\prime}G_{1}b_{1}$. Let $\Gamma:={\rm Gal}(k^{\prime}/k)$. Since $b_{2}$ has defect group $1$, $k^{\prime}G_{2}i$ is the unique projective $k^{\prime}Gb_{2}$-module, up to isomorphism. For any $\sigma\in\Gamma$, ${}^{\sigma}(k^{\prime}G_{2}i)$ is still a projective $k^{\prime}G_{2}b_{2}$-module, so we have ${}^{\sigma}(k^{\prime}G_{2}i)\cong k^{\prime}G_{2}i$. Then by [17, Lemma 6.2 (c)], there is a projective $kG_{2}b_{2}$-module $Y$ such that $k^{\prime}G_{2}i\cong k^{\prime}\otimes_{k}Y$. It follows that

[TABLE]

as $(k^{\prime}(G_{1}\times G_{2})(b_{1}\otimes b_{2}),k^{\prime}G_{1}b_{1})$-bimodule. By [17, Proposition 4.5 (c)], the $(k(G_{1}\times G_{2})(b_{1}\otimes b_{2}),kG_{1}b_{1})$-bimodule $kG_{1}b_{1}\otimes_{k}Y$ induces a Morita equivalence between $kG_{1}b_{1}\otimes_{k}kG_{2}b_{2}$ and $kG_{1}b_{1}$. By [17, Lemma 5.1 and 5.2], this Morita equivalence is splendid. $\hfill\square$

By Proposition 4.1, $\mathbb{F}_{\ell}N_{{\rm GL}_{ew}(q)}(P)c_{ew}$ and $\mathbb{F}_{\ell}N_{G}(P)c$ are splendidly Morita equivalent. By [28, §1.4], we have $N_{{\rm GL}_{ew}(q)}(P)\cong N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})\wr S_{w}$. Let $b_{e}$ be the principal block of $k^{\prime}{\rm GL}_{e}(q)$, then $b_{e}$ is the unique unipotent block of $k^{\prime}{\rm GL}_{e}(q)$ corresponding to the empty $e$-core. So $b_{e}$ has $C_{\ell^{m}}$ as a defect group. The Brauer correspondent of $b_{e}$ in $k^{\prime}N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})$ is the principal block of $k^{\prime}N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})$, we denote it by $c_{e}$. By Rouquier’s result on cyclic blocks ([24, Theorem 10.1]), there is a complex of $C^{\prime}$ of $(k^{\prime}{\rm GL}_{e}(q)b_{e},k^{\prime}N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})c_{e})$-bimodules inducing a splendid Rickard equivalence between $k^{\prime}{\rm GL}_{e}(q)b_{e}$ and $k^{\prime}N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})c_{e}$. Then by a theorem of Marcus ([19, Theorem 4.3]), the complex $C^{\prime}\wr S_{w}$ induces a splendid Rickard equivalence between the principal block algebra of $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})$ and the principal block algebra of $k^{\prime}(N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})\wr S_{w})$. By [17, Theorem 1.10], Rouquier’s complex $C^{\prime}$ descends to $\mathbb{F}_{\ell}$, namely that there is a Rickard complex $C$ of $(\mathbb{F}_{\ell}{\rm GL}_{e}(q)b_{e},\mathbb{F}_{\ell}N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})c_{e})$-bimodules, such that $C^{\prime}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}C$. So we have $C^{\prime}\wr S_{w}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}(C\wr S_{w})$. Still by [19, Theorem 4.3], the complex $C\wr S_{w}$ induces a splendid Rickard equivalence between the principal block algebra of $\mathbb{F}_{\ell}({\rm GL}_{e}(q)\wr S_{w})$ and the principal block algebra of $\mathbb{F}_{\ell}(N_{{\rm GL}_{e}(q)}(C_{\ell^{m}})\wr S_{w})$. Now we have proved the following statement.

Proposition 4.2**.**

$\mathbb{F}_{\ell}N_{G}(P)c$* is splendid Morita equivalent to the principal block algebra of $\mathbb{F}_{\ell}({\rm GL}_{e}(q)\wr S_{w})$.*

Consider an abacus having $w+i(w-1)$ beads on the $i$-th runner, where $i=0,1,\cdots,e-1$, and let $\rho$ be the $e$-core have this abacus representation. Let $r:=|\rho|$, and let $n^{\prime}=:we+r$. By results of Miyachi ([20, Theorem 5.0.7]) and Turner ([28, Theorem 1]), the unipotent block (say $b^{\prime}$) of $k^{\prime}{\rm GL}_{n^{\prime}}(q)$ corresponding to the $e$-core $\rho$ is splendidly Morita equivalent to the principal block algebra of $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})$. The integer $n^{\prime}$ is exactly the integer $n^{\prime}$ mentioned in Remark 1.4, and the block $b^{\prime}$ of $k^{\prime}{\rm GL}_{n^{\prime}}(q)$ is exactly the unipotent block $b^{\prime}$ mentioned in Remark 1.4.

Let us review the construction of the splendid Morita equivalence constructed by Turner. Let $f_{0}$ be the unipotent block of $k^{\prime}{\rm GL}_{r}(q)$ corresponding to the $e$-core $\rho$, so $f_{0}$ has defect group $1$. Denote by $b_{*}$ the principal block of $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})$. By [7, Lemma 6], $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})b_{*}$ is splendidly Morita equivalent to $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})b_{*}\otimes_{k^{\prime}}k^{\prime}{\rm GL}_{r}(q)f_{0}$. Let $L$ be the Levi subgroup $\underbrace{{\rm GL}_{e}(q)\times\cdots\times{\rm GL}_{e}(q)}_{w}\times{\rm GL}_{r}(q)$ of ${\rm GL}_{n^{\prime}}(q)$. Recall that $b_{e}$ denotes the principal block of $k^{\prime}{\rm GL}_{e}(q)$. Let $f:=\underbrace{b_{e}\otimes\cdots\otimes b_{e}}_{w}\otimes f_{0}$, a block idempotent of $k^{\prime}L$. We can view $S_{w}$ as the subgroup of permutation matrices of ${\rm GL}_{n^{\prime}}(q)$ whose conjugation action on $L$ permutes the factors of $\underbrace{{\rm GL}_{e}(q)\times\cdots\times{\rm GL}_{e}(q)}_{w}$. Let $N$ be the semi-direct product of $L$ and $S_{w}$, a subgroup of ${\rm GL}_{n^{\prime}}(q)$ isomorphic to ${\rm GL}_{e}(q)\wr S_{w}\times{\rm GL}_{r}(q)$. Clearly the conjugation action of $N$ stabilise $f$, hence $f$ is a idempotent in the center of $k^{\prime}N$. By [28, Lemma 1 (3)], $f$ is also a block of $k^{\prime}N$. This forces that the idempotent $\underbrace{b_{e}\otimes\cdots\otimes b_{e}}_{w}$ is a block of $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})$, and hence $\underbrace{b_{e}\otimes\cdots\otimes b_{e}}_{w}$ must equal to the principal block $b_{*}$ of $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})$. So the block algebra $k^{\prime}Nf$ is isomorphic to the algebra $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})b_{*}\otimes_{k^{\prime}}k^{\prime}{\rm GL}_{r}(q)f_{0}$. We identify these two algebras via the isomorphism. Let $D:=\underbrace{C_{\ell^{m}}\times\cdots\times C_{\ell^{m}}}_{w}$, a Sylow $\ell$-subgroup of $\underbrace{{\rm GL}_{e}(q)\times\cdots\times{\rm GL}_{e}(q)}_{w}$. By [28, Lemma 1 (4)], $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ and $k^{\prime}Nf$ both have defect group $D$ and are Brauer correspondents.

By Alperin’s description of the Brauer correspondence ([1, Lemma 6.2.7]), the $k^{\prime}({\rm GL}_{n^{\prime}}(q)\times{\rm GL}_{n^{\prime}}(q)^{\rm op})$-module $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ and the $k^{\prime}(N\times N^{\rm op})$-module $k^{\prime}Nf$ both have vertex $\Delta D$ and are Green correspondents. Let $T^{\prime}$ be the Green correspondent of $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ in ${\rm GL}_{n^{\prime}}(q)\times N^{\rm op}$, an indecomposable summand of ${\rm Res}_{{\rm GL}_{n^{\prime}}(q)\times N^{\rm op}}^{{\rm GL}_{n^{\prime}}(q)\times{\rm GL}_{n^{\prime}}(q)^{\rm op}}(k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime})$. Since $k^{\prime}Nf$ is a direct summand of ${\rm Res}_{N\times N^{\rm op}}^{{\rm GL}_{n^{\prime}}(q)\times N^{\rm op}}(T^{\prime})$, we have $T^{\prime}f\neq 0$, thus $T^{\prime}f=T$ and $T^{\prime}$ is a $(k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime},k^{\prime}Nf)$-bimodule. By [28, Proposition 1], the $(k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime},k^{\prime}Nf)$-bimodule $T^{\prime}$ induces a splendid Morita equivalence between $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ and $k^{\prime}Nf$. So, there is a splendid Morita equivalence between $k^{\prime}{\rm GL}_{n^{\prime}}(q)$ and $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})b_{*}$. Next, we prove that this splendid Morita equivalence descends to $\mathbb{F}_{\ell}$.

Proposition 4.3**.**

There is a splendid Morita equivalence between $\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)$ and the principal block algebra of $\mathbb{F}_{\ell}({\rm GL}_{e}(q)\wr S_{w})$.

Proof. By Proposition 4.1, $\mathbb{F}_{\ell}({\rm GL}_{e}(q)\wr S_{w})b_{*}$ is splendidly Morita equivalent to $\mathbb{F}_{\ell}({\rm GL}_{e}(q)\wr S_{w})b_{*}\otimes_{\mathbb{F}_{\ell}}\mathbb{F}_{\ell}{\rm GL}_{r}(q)f_{0}$. Since $b_{e}\in\mathbb{F}_{\ell}{\rm GL}_{e}(q)$ and $f_{0}\in\mathbb{F}_{\ell}{\rm GL}_{r}(q)$ (see Corollary 2.3), we have $f=\underbrace{b_{e}\otimes\cdots\otimes b_{e}}_{w}\otimes f_{0}\in\mathbb{F}_{\ell}N$. Since the block algebra $k^{\prime}Nf$ is isomorphic to $k^{\prime}({\rm GL}_{e}(q)\wr S_{w})b_{*}\otimes_{k^{\prime}}k^{\prime}{\rm GL}_{r}(q)f_{0}$, the block algebra $\mathbb{F}_{\ell}Nf$ is isomorphic to $\mathbb{F}_{\ell}({\rm GL}_{e}(q)\wr S_{w})b_{*}\otimes_{k^{\prime}}\mathbb{F}_{\ell}{\rm GL}_{r}(q)f_{0}$. Hence $\mathbb{F}_{\ell}({\rm GL}_{e}(q)\wr S_{w})b_{*}$ is splendid Morita equivalent to $\mathbb{F}_{\ell}Nf$.

By the definition of Green correspondent, $T^{\prime}$ is the unique (up to isomorphism) direct summand of the $(k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime},k^{\prime}Nf)$-bimodule $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ having $\Delta D$ as a vertex. Noting that $k^{\prime}{\rm GL}_{n^{\prime}}(q)b^{\prime}=k^{\prime}\otimes_{\mathbb{F}_{\ell}}\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime}$, by [17, Lemma 5.1], there is an indecomposable direct summand $T$ of the $(\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime},\mathbb{F}_{\ell}Nf)$-bimodule $\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime}$, such that $T^{\prime}\cong k^{\prime}\otimes_{\mathbb{F}_{\ell}}T$ (use the uniqueness of $T^{\prime}$). By [17, Proposition 4.5], $T$ induces a splendid Morita equivalence between $\mathbb{F}_{\ell}{\rm GL}_{n^{\prime}}(q)b^{\prime}$ and $\mathbb{F}_{\ell}Nf$. This completes the proof. $\hfill\square$

Proof of Theorem 1.2. By Proposition 3.3, 4.2 and 4.3, the task listed in Remark 1.5 is finished, hence the proof of Theorem 1.2 is complete. $\hfill\square$

5 On general blocks of ${\rm GL}_{n}(q)$

By the proof of [17, Theorem 1.12], to answer the question whether Conjecture 1.1 holds for general blocks of ${\rm GL}_{n}(q)$, it suffices to answer the following question.

Question 5.1**.**

Let $G$ be the group ${\rm GL}_{n}(q)$, and let $b$ be a block of $\mathcal{O}^{\prime}G$ with an abelian defect group $P$. Let $c$ be the block of $\mathcal{O}^{\prime}N_{G}(P)$ corresponding to $b$ via the Brauer correspondence. Suppose that $b\in\mathcal{O}G$. Then $c\in\mathcal{O}N_{G}(P)$. Is the block algebras $\mathcal{O}Gb$ and $\mathcal{O}N_{G}(P)c$ are splendidly Rickard equivalent? More precisely, is there a splendid Rickard complex $X$ of $(\mathcal{O}Gb,\mathcal{O}N_{G}(P)c)$-bimodules such that $\mathcal{O}^{\prime}\otimes_{\mathcal{O}}X$ is isomorphic to Chuang and Rouquier’s complex $X^{\prime}$?

By the proof of [8, Theorem 7.20], Chuang and Rouquier first reduced the statement in [8, Theorem 7.20] to unipotent blocks by using results in [3]. More precisely, by results in [3], there is a complex $X^{\prime}_{1}$ inducing a splendid Rickard equivalence between $\mathcal{O}^{\prime}Gb$ and a unipotent block (say $b_{1}$) of $\mathcal{O}^{\prime}G_{1}$, where $G_{1}\cong{\rm GL}_{n_{1}}(q^{d_{1}})\times\cdots\times{\rm GL}_{n_{r}}(q^{d_{r}})$ is a subgroup of $G$. Let $D$ be a defect group of $b_{1}$. By [26, Theorem 1.15], $D$ is also a defect group of $b$. We may assume that $P=D$ (because we can change the choice of $P$). Let $c_{1}$ be the block of $\mathcal{O}^{\prime}N_{G_{1}}(P)$ corresponding to $b_{1}$ via the Brauer correspondence. Then the complex

[TABLE]

induces a splendid Rickard equivalence between $\mathcal{O}^{\prime}N_{G}(P)c$ and $\mathcal{O}^{\prime}N_{G_{1}}(P)c_{1}$ (see [26, Proposition 1.36 and Remark 1.37]). Readers can refer to [18, Definition 5.4.10] for the definition of the Brauer construction ${\rm Br}_{\Delta P}$ and [18, Proposition 5.8] for a property of Brauer construction applied to $\ell$-permutation modules. Hence the statement in [8, Theorem 7.20] convert into proving that $\mathcal{O}^{\prime}G_{1}b_{1}$ and $\mathcal{O}^{\prime}N_{G_{1}}(P)c_{1}$ are splendidly Rickard equivalent. In other words, the statement in [8, Theorem 7.20] is reduced to unipotent blocks.

By Corollary 2.3, we have $b_{1}\in\mathbb{Z}_{\ell}G_{1}\subseteq\mathcal{O}G_{1}$, and hence we have $c_{1}\in\mathbb{Z}_{\ell}N_{G_{1}}(P)\subseteq\mathcal{O}N_{G_{1}}(P)$. Since we have answered Question 5.1 for unipotent blocks of general linear groups (Theorem 1.2), if we can prove that there is a complex $X_{1}$ of $(\mathcal{O}Gb,\mathcal{O}G_{1}b_{1})$-bimodules such that $X^{\prime}_{1}\cong\mathcal{O}^{\prime}\otimes_{\mathcal{O}}X_{1}$, then Question 5.1 has a positive answer. We will give a sufficient condition under which there exists such a complex $X_{1}$ (see Theorem 5.4 below). Then by [17, Proposition 4.5 (a)], $X_{1}$ induces a splendid Rickard equivalence between $\mathcal{O}Gb$ and $\mathcal{O}G_{1}b_{1}$, and $Y_{1}$ induces a splendid Rickard equivalence between $\mathcal{O}N_{G}(P)c$ and $\mathcal{O}N_{G_{1}}(P)c_{1}$, where

[TABLE]

Let us review the construction of the Rickard complex $X^{\prime}_{1}$ constructed by Bonnafé, Dat and Rouquier [3]. So we should first review some material in [3].

5.1 The Deligne-Lusztig induction

Assume that $\Lambda\in\{\mathcal{O},k\}$. Let $\mathbf{G}$ be a connected reductive algebraic group over an algebraic closure of a finite field whose characteristic is not $\ell$, endowed with a Frobenius endomorphism $F$. Let $\mathbf{L}$ be an $F$-stable Levi subgroup of $\mathbf{G}$ contained in a parabolic subgroup $\mathbf{P}$ with unipotent radical $\mathbf{V}$ such that $\mathbf{P}=\mathbf{V}\rtimes\mathbf{L}$. The Deligne-Lusztig variety

[TABLE]

has a left action of $\mathbf{G}^{F}$ and a right action of $\mathbf{L}^{F}$ by multiplication. By works of Rickard ([22]) and Rouquier ([25]), there is an object ${\rm G}\Gamma_{c}(\mathbf{Y}_{\mathbf{P}},\Lambda)$ of ${\rm Ho}^{b}(\Lambda(\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\rm op})\mbox{-perm})$ associated with $\mathbf{Y}_{\mathbf{P}}$, well defined up to isomorphism, where $\Lambda(\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\rm op})\mbox{-perm}$ denotes the category of finite generated $\ell$-permutation $(\Lambda\mathbf{G}^{F},\Lambda\mathbf{L}^{F})$-bimodules.

Proposition 5.2**.**

${\rm G}\Gamma_{c}(\mathbf{Y}_{\mathbf{P}},k)\cong k\otimes_{\mathbb{F}_{\ell}}{\rm G}\Gamma_{c}(\mathbf{Y}_{\mathbf{P}},\mathbb{F}_{\ell})$* as complexes of $(k\mathbf{G}^{F},k\mathbf{L}^{F})$-bimodules.*

Proof. The statement follows from [22, Lemma 2.8]: take the module “$U$” in [22, Lemma 2.8] to be the direct sum of all the permutation $\mathbb{F}_{\ell}(\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\rm op})$-modules of the form $\mathbb{F}_{\ell}(\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\rm op}/H)$, where $H$ is a subgroup of $\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\rm op}$. Then the category of finite generated $\ell$-permutation $\mathbb{F}_{\ell}(\mathbf{G}^{F}\times(\mathbf{L}^{F})^{\rm op})$-modules is exactly the category “${\rm add}$-$U$” in [22, Lemma 2.8]. $\hfill\square$

Let $\Lambda\in\{K^{\prime},k^{\prime}\}$. The corresponding complex of $\ell$-adic cohomology of the complex $R\Gamma_{c}(\mathbf{Y}_{\mathbf{P}},\Lambda)$ induces a morphism

[TABLE]

between Grothendieck groups, which is called the Deligne-Lusztig induction.

5.2 The Jordan decomposition

Let $\mathbf{G}^{*}$ be a group dual to $\mathbf{G}$ with Frobenius endomorphism $F^{*}$. Let ${\rm Irr}_{K^{\prime}}(\mathbf{G}^{F})$ denote the set of characters of irreducible representations of $\mathbf{G}^{F}$ over $K^{\prime}$. Deligne and Lusztig gave a decomposition of ${\rm Irr}_{K^{\prime}}(\mathbf{G}^{F})$ into rational series

[TABLE]

where $(s)$ runs over the set of $\mathbf{G}^{*}{}^{F^{*}}$-conjugacy classes of semi-simple elements of $\mathbf{G}^{*}{}^{F^{*}}$. The unipotent characters of $\mathbf{G}^{F}$ are those in ${\rm Irr}_{K^{\prime}}(\mathbf{G}^{F},1)$.

Let $s$ be a semi-simple element of ${\mathbf{G}^{*}}^{F^{*}}$ of order prime to $\ell$. Broué and Michel ([5]) showed that $\coprod_{(t)}{\rm Irr}_{K^{\prime}}(\mathbf{G}^{F},(t))$, where $(t)$ runs over conjugacy classes of semi-simple elements of $\mathbf{G}^{*}{}^{F^{*}}$ whose $\ell^{\prime}$-part is $(s)$, is a union of blocks of $\mathcal{O}^{\prime}\mathbf{G}^{F}$. The sum of the corresponding block idempotents is an idempotent $e_{s}^{\mathbf{G}^{F}}$ in the center of $\mathcal{O}^{\prime}\mathbf{G}^{F}$, so there is a decomposition $1=\sum_{(s)}e_{s}^{\mathbf{G}^{F}}$, where $(s)$ runs over $\mathbf{G}^{*}{}^{F^{*}}$-conjugacy classes of semi-simple $\ell^{\prime}$-elements of $\mathbf{G}^{*}{}^{F^{*}}$. We denote the set $\coprod_{(t)}{\rm Irr}_{K^{\prime}}(\mathbf{G}^{F},(t))$ above by ${\rm Irr}_{K^{\prime}}(\mathbf{G}^{F},e_{s}^{\mathbf{G}^{F}})$.

Let $\mathbf{L}$ be an $F$-stable Levi subgroup of $\mathbf{G}$ with dual $\mathbf{L}^{*}\subset\mathbf{G}^{*}$ containing $C_{\mathbf{G}^{*}}(s)$. By a result of Lusztig (see [10, Theorem 11.4.3]), there is a sign $\varepsilon_{\mathbf{L},\mathbf{G}}\in\{1,-1\}$, such that $\varepsilon_{\mathbf{L},\mathbf{G}}R_{\mathbf{L}\subset\mathbf{P}}^{\mathbf{G}}$ induces a bijection

[TABLE]

In this situation with $s$ being an $\ell^{\prime}$-element, $\varepsilon_{\mathbf{L},\mathbf{G}}R_{\mathbf{L}\subset\mathbf{P}}^{\mathbf{G}}$ also induces a bijection

[TABLE]

5.3 A sufficient condition

Denote by $\bar{\mathbb{F}}_{q}$ the algebraic closure of $\mathbb{F}_{q}$. Let $\mathbf{G}:={\rm GL}_{n}(\bar{\mathbb{F}}_{q})$, and let $F$ be the map $\mathbf{G}\to\mathbf{G}$ sending every $A=(a_{ij})_{1\leq i,j\leq n}\in{\rm GL}_{n}(\bar{\mathbb{F}}_{q})$ to $(a_{ij}^{q})_{1\leq i,j\leq n}$. Hence $\mathbf{G}^{F}$ is $G:={\rm GL}_{n}(q)$. The pair $(\mathbf{G},F)$ is dual to itself (see e.g. [10, Examples 11.1.13]).

We return to the context of Question 5.1 and keep the notation of the beginning part of this section. We will express the complex $X_{1}^{\prime}$ more explicit. Since $b$ is a block of $k^{\prime}G=k^{\prime}{\rm GL}_{n}(q)=k^{\prime}\mathbf{G}^{F}$, there is a semi-simple $\ell^{\prime}$-element $s$ of $\mathbf{G}^{F}$, such that $be_{s}^{\mathbf{G}^{F}}=b=e_{s}^{\mathbf{G}^{F}}b$. Let $\mathbf{L}:=C_{\mathbf{G}}(s)$, by [10, Theorem 11.7.3], $\mathbf{L}$ is a Levi subgroup of a parabolic subgroup $\mathbf{P}$ of $\mathbf{G}$. Let $\mathbf{V}$ be the unipotent radical of $\mathbf{P}$, then we have $\mathbf{P}=\mathbf{V}\rtimes\mathbf{L}$. Clearly $\mathbf{L}$ is $F$-stable. The group $G_{1}$ is exactly the group $\mathbf{L}^{F}$.

Denote by $C^{\prime}$ the complex ${\rm G}\Gamma_{c}(\mathbf{Y}_{\mathbf{P}},\mathcal{O}^{\prime})$ of $(\mathcal{O}^{\prime}G,\mathcal{O}^{\prime}G_{1})$-bimodules. By [3, Theorem 7.7], there is a block $b^{\prime}$ of $\mathcal{O}^{\prime}G_{1}$, such that the complex $bC^{\prime}b^{\prime}$ induces a splendid Rickard equivalence between $\mathcal{O}^{\prime}Gb$ and $\mathcal{O}^{\prime}G_{1}b^{\prime}$. (Note that in our case, $\mathbf{L}^{F}$ and $\mathbf{N}^{F}$ in [3, Theorem 7.7] are equal.)

Since $s\in Z(\mathbf{L})$, there is a bijection

[TABLE]

where $\eta$ is the one-dimensional character of $\mathbf{L}^{F}$ corresponding to $s$ (see e.g. [6, Proposition 8.26]). Let $S^{\prime}$ an $\mathcal{O}^{\prime}\mathbf{L}^{F}$-module, which affords the character $\eta$. Then it is easy to see that the functor (say $\Phi$) sending an $\mathcal{O}^{\prime}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}}$-module $V$ to the $\mathcal{O}^{\prime}\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}}$-module $V\otimes_{\mathcal{O}^{\prime}}S^{\prime}$ induces a Morita equivalence between $\mathcal{O}^{\prime}\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}}$ and $\mathcal{O}^{\prime}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}}$. Assume that this Morita equivalence is induced by an $(\mathcal{O}^{\prime}\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}},\mathcal{O}^{\prime}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}})$-bimodule $M^{\prime}$. Then as $(\mathcal{O}^{\prime}\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}},\mathcal{O}^{\prime}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}})$-bimodules,

[TABLE]

Since $\mathcal{O}^{\prime}\mathbf{L}^{F}e_{s}^{\mathbf{L}^{F}}$ and $\mathcal{O}^{\prime}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}}$ are direct sums of block algebras, there is a unique block $b_{1}$ of $\mathcal{O}^{\prime}\mathbf{L}^{F}=\mathcal{O}^{\prime}G_{1}$, such that $b^{\prime}M^{\prime}\cong M^{\prime}b_{1}$ and $b^{\prime}M^{\prime}b_{1}$ induces a Morita equivalence between $\mathcal{O}^{\prime}G_{1}b^{\prime}$ and $\mathcal{O}^{\prime}G_{1}b_{1}$. (The $b_{1}$ in the beginning part of this section is the $b_{1}$ here.) So the complex $bC^{\prime}b^{\prime}\otimes_{\mathcal{O}^{\prime}G_{1}b^{\prime}}b^{\prime}M^{\prime}b_{1}=bC^{\prime}\otimes_{\mathcal{O}^{\prime}G_{1}}M^{\prime}b_{1}$ induces a Rickard equivalence between $\mathcal{O}^{\prime}Gb$ and $\mathcal{O}^{\prime}G_{1}b_{1}$. Set $\bar{C}^{\prime}:={\rm G}\Gamma_{c}(\mathbf{Y}_{\mathbf{P}},k^{\prime})\cong k^{\prime}\otimes_{\mathcal{O}^{\prime}}C^{\prime}$, $\bar{S}^{\prime}:=k^{\prime}\otimes_{\mathcal{O}^{\prime}}S$ and $\bar{M}^{\prime}:=k^{\prime}\otimes_{\mathcal{O}^{\prime}}M^{\prime}$. Denote by $\bar{b}$ (resp. $\bar{b_{1}}$) the image of $b$ (resp. $b_{1}$) in $k^{\prime}G$ (resp. $k^{\prime}G_{1}$).

Proposition 5.3**.**

The complex $\bar{b}\bar{C}^{\prime}\otimes_{k^{\prime}G_{1}}\bar{M}^{\prime}\bar{b}_{1}$ is a splendid Rickard complex for $k^{\prime}G\bar{b}$ and $k^{\prime}G_{1}\bar{b}_{1}$, and it lifts uniquely (up to isomorphism) to a splendid Rickard complex $X^{\prime}_{1}$ for $\mathcal{O}^{\prime}Gb$ and $\mathcal{O}^{\prime}G_{1}b_{1}$.

Proof. Since $M^{\prime}\cong\mathcal{O}^{\prime}\mathbf{L}^{F}e_{1}^{\mathbf{L}^{F}}\otimes_{\mathcal{O}^{\prime}}S^{\prime}$, $M^{\prime}$ is isomorphic to a direct summand of

[TABLE]

where the isomorphism is by [18, Proposition 2.8.19]. It follows that $\bar{M}^{\prime}$ is isomorphic to a direct summand of ${\rm Ind}_{\Delta G_{1}}^{G_{1}\times G_{1}^{\rm op}}(\bar{S}^{\prime})$. Note that $\bar{S}^{\prime}$ is a 1-dimension $k^{\prime}G_{1}$-module, hence the restriction of $\bar{S}^{\prime}$ to any $p$-subgroup of $\Delta G_{1}$ is a trivial module. Using the Mackey formula, we easily see that $\bar{M}^{\prime}$ is an $\ell$-permutation $k^{\prime}(G_{1}\times G_{1}^{\rm op})$-module. So the complex $\bar{b}\bar{C}^{\prime}\otimes_{k^{\prime}G_{1}}\bar{M}^{\prime}\bar{b}_{1}$ is a complex of $\ell$-permutation $k^{\prime}(G\times G_{1}^{\rm op})$-modules, hence it induces a splendid Rickard equivalence between $k^{\prime}G\bar{b}$ and $k^{\prime}G_{1}\bar{b_{1}}$. By Theorem 2.1, there is a unique (up to isomorphism) complex $X^{\prime}_{1}$ of $(\mathcal{O}^{\prime}Gb,\mathcal{O}^{\prime}G_{1}b_{1})$-bimodules inducing a splendid Rickard equivalence between $\mathcal{O}^{\prime}Gb$ and $\mathcal{O}^{\prime}G_{1}b_{1}$ such that $k^{\prime}\otimes_{\mathcal{O}^{\prime}}X^{\prime}_{1}\cong\bar{b}\bar{C}^{\prime}\otimes_{k^{\prime}G_{1}}\bar{M}^{\prime}\bar{b}_{1}$. $\hfill\square$

In Question 5.1, we assumed that $b\in\mathcal{O}G$. Under this assumption, we don’t know whether the values of the character $\eta$ is contained in $\mathcal{O}$. Let $\bar{\eta}$ be the $k^{\prime}$-character afforded by the $kG_{1}$-module $\bar{S}$. If we assume further that the values of $\eta$ are contained in $\mathcal{O}$, alternatively, if we assume further that the values of $\bar{\eta}$ are contained in $k$, then we can give a positive answer to Question 5.1 for the block $b$.

Theorem 5.4**.**

Keep the notation above. Assume further that the values of $\eta$ (resp. $\bar{\eta}$) are contained in $\mathcal{O}$ (resp. $k$), then there is a complex $X^{\prime}_{1}$ of $(\mathcal{O}Gb,\mathcal{O}G_{1}b_{1})$-bimodules inducing a Rickard equivalence between $\mathcal{O}Gb$ and $\mathcal{O}G_{1}b_{1}$, and satisfying $X^{\prime}_{1}\cong\mathcal{O}^{\prime}\otimes_{\mathcal{O}}X_{1}$.

Proof. Let $\bar{C}:={\rm G}\Gamma_{c}(\mathbf{Y}_{\mathbf{P}},k)$, a complex of $(kG,kG_{1})$-bimodule. By Proposition 5.2, we have $\bar{C}^{\prime}\cong k^{\prime}\otimes_{k}\bar{C}$. Hence $\bar{b}\bar{C}^{\prime}\cong k^{\prime}\otimes_{k}\bar{b}\bar{C}$ as complexes of $(k^{\prime}Gb,k^{\prime}G_{1})$-bimodules. Since the values of $\bar{\eta}$ are contained in $k$ (by the assumption), there exists a $kG_{1}$-module $\bar{S}$ such that $\bar{S}^{\prime}\cong k^{\prime}\otimes_{k}\bar{S}$. Since $e_{1}^{\mathbf{L}^{F}}$ is a sum of unipotent blocks of $\mathcal{O}^{\prime}\mathbf{L}^{F}=\mathcal{O}^{\prime}G_{1}$, by Corollary 2.3, we have $e_{1}^{\mathbf{L}^{F}}\in\mathbb{F}_{\ell}G_{1}\subseteq\mathcal{O}G_{1}$. Let $\bar{e}_{1}^{\mathbf{L}^{F}}$ be the image of $e_{1}^{\mathbf{L}^{F}}$ in $k^{\prime}G_{1}$, then we have $\bar{e}_{1}^{\mathbf{L}^{F}}\in kG_{1}$. Taking $\bar{M}:=k\mathbf{L}^{F}\bar{e}_{1}^{\mathbf{L}^{F}}\otimes_{k}\bar{S}$, we see that $\bar{M}^{\prime}\cong k^{\prime}\otimes_{k}\bar{M}$. So we have

[TABLE]

as complexes of $(k^{\prime}G,k^{\prime}G_{1})$-bimodules. The same argument in the proof of Proposition 5.3 shows that $\bar{b}\bar{C}\otimes_{kG_{1}}\bar{M}\bar{b}_{1}$ is a complex of $\ell$-permutation $k(G\times G_{1}^{\rm op})$-modules. By [17, Proposition 4.5 (a)], $\bar{b}\bar{C}\otimes_{kG_{1}}\bar{M}\bar{b}_{1}$ induces a splendid Rickard equivalence between $kGb$ and $kG_{1}b_{1}$. By Theorem 2.1, there is a unique (up to isomorphism) complex $X_{1}$ of $(\mathcal{O}G_{1}b,\mathcal{O}G_{1}b_{1})$-bimodules inducing a splendid Rickard equivalence between $\mathcal{O}Gb$ and $\mathcal{O}G_{1}b_{1}$ such that $k\otimes_{\mathcal{O}}X_{1}\cong\bar{b}\bar{C}\otimes_{kG_{1}}\bar{M}\bar{b}_{1}$. Since

[TABLE]

by 2.1, we have $\mathcal{O}^{\prime}\otimes_{\mathcal{O}}X_{1}\cong X^{\prime}_{1}$. $\hfill\square$

By Theorem 5.4, if the values of $\eta$ (resp. $\bar{\eta}$) are contained in $\mathcal{O}$ (resp. $k$), then we can give an affirmative answer to Question 5.1 of the block $b$ of $\mathcal{O}^{\prime}G$. For unipotent blocks, this is the case.

Acknowledgement. We wish to thank Professor Joseph Chuang for suggesting that unipotent blocks of finite general linear groups could be considered for the refined abelian defect group conjecture.