SOURCE ALGEBRA EQUIVALENCES FOR BLOCKS OF

FINITE GENERAL LINEAR GROUPS OVER A FIXED FIELD

Radha Kessar

Abstract. T. Jost proved that Donovan’s conjecture holds for the unipotent blocks of the finite general linear groups over a fixed field. In this paper, we show that the Morita equivalences exhibited by Jost are in fact equivalences between the source algebras of the corresponding blocks, and thus that Puig’s conjecture holds for the unipotent blocks of finite general groups over a fixed field.

§1. Introduction

Let q be a prime power, 𝔽_{q} the field of q elements, and r a prime not dividing q. The r-blocks of the finite general linear group Gl(n, q), when r is odd, were described by Fong and Srinivasan in[FS]; In [B1] Broué gave another proof of their result using Brauer pairs which also handled the case r = 2. Roughly speaking, each r-block of Gl(n, q) has a Jordan decomposition into a semi-simple part and a unipotent part, the latter can be described in terms of partitions of ordinary integers; the defect groups of a block are determined by the corresponding weight vector (see Theorem 2.4). Based on Scopes’ proof in [Sc] of Donovan’s conjecture (see [A]) for the blocks of the symmetric groups, in [J], Jost has shown that a certain combinatorial relationship between the unipotent parts of two blocks of two different general linear groups over 𝔽_{q} which have the same weight vector will force them to have equivalent module categories.

In this article, we give an interpretation for Jost’s results similar to the one given by Puig in [P2] for the blocks of the symmetric groups, i.e. we show that the Morita equivalence demonstrated by Jost between two blocks that form a (w_{j} : m) pair (see definition 3.2) comes from an explicit algebra isomorphism (see Theorem3.3), and the nature of this isomorphism is such that it forces the two blocks in question to have isomorphic source algebras. As a consequence we are able to prove that Puig’s conjecture (see Conjecture 1.2) holds for the family of unipotent blocks of finite general linear groups over a fixed finite field.

The paper is divided into four sections. Section 2 contains some known results regarding the blocks of the finite general linear groups. In section 3, we explain the notion of (w_{j} : m) pairs of blocks as given by [J], and then state our main result (Theorem 3.3). The proof of the theorem is given in section 4 and uses the methods introduced by Puig which are also outlined in [K]. The common starting point of Jost’s proof as well as of ours is Lemma 4.1, which is a correspondence, through Harish-Chandra restriction, between the ordinary characters of two blocks that form a (w_{j} : m) pair. The proof of the lemma depends heavily on the relationship between the representations of the finite general linear groups and the representations of the

Typeset by AMS-TEX

corresponding Weyl groups, i.e. the symmetric groups. After this point, our proof becomes quite independent of the specific properties of the finite general linear groups, in particular, unlike in [J] we do not need to use any information about the modular characters or the decomposition matrices of blocks of these groups.

Before beginning section 2, let us recall some general definitions regarding blocks and source algebras of finite groups and give the statement of Puig’s conjecture. All of the definitions and results mentioned below can be found in [T].

Throughout this article, let (O, K, k) be a r-modular system, i.e. O is a complete discrete valuation ring with maximal ideal J(O), k = O/J(O) is its residue field with characteristic r that we assume is algebraically closed and K is its field of quotients with characteristic zero.

If a finite group G acts on an O-algebra A (we assume that A is associative, unitary, finitely generated over A, and either O-free or annihilated by J(O)), we denote by ^{g}a the image of the element a of A under the action of the element g of G; and for a subgroup H of G, A^{H} will denote the space of elements of A that are left fixed under the action of H. We denote by A_{<H}^{H} the smallest ideal of A^{H} containing all elements that can be written as a relative trace for a proper subgroup of H, i.e. containing all elements x such that x = ∑_{x∈Q/H} ^{x}a, for some proper subgroup Q of H, and some element a of A^{Q}. The Brauer homomorphism of A with respect to H, denoted by Br_{H}^{A} is the canonical surjection of A^{H} onto the quotient A^{H}/(A_{<H}^{H} + J(O)A^{H}). A point of H on A is a conjugacy class of primitive idempotents of A^{H}. A local point of H on A is a point of H on A whose elements are not in the kernel of Br_{H}^{A}.

An O-algebra A is an interior G-algebra, where G is a finite group, if there is a homomorphism from G into A^{∗}, the group of units of A. If A is an interior G-algebra, then A is naturally a G-algebra where the action of an element g of G is defined as conjugation by the image of g under the homomorphism defining the interior G-algebra structure of A. If A and B are interior G-algebras and if ϕ : G → A^{∗} and ψ : G → B^{∗} are the homomorphisms that define the interior G-algebra structures of A and B respectively, then a map f : A → B is an interior G-algebra homomorphism if f is a homomorphism of algebras and if

f(ϕ(g)a) = ψ(g)f(a) and f(aϕ(g)) = f(a)ψ(g) ∀g ∈ G, a ∈ A.

A block b of a finite group G is a primitive idempotent of the center of the group algebra OG, the algebra OGb is called the block algebra of b. A defect group R of b is minimal among the subgroups of G such that b is in the image of the relative trace map Tr_{R}^{G}, i.e. R is minimal among subgroups of G such that there is an element a of (OG)^{R} such that b = ∑_{x∈R/G} ^{x}a. Alternatively, the defect groups of b are maximal among subgroups of G having a local point on OGb.

A source algebra of b is an algebra of the form zOGbz, where z is a primitive idempotent of the algebra (OGb)^{R} which cannot be written as a relative trace for any proper subgroup of R, where R is a defect group of b. Alternatively, the primitive idempotent z of (OGb)^{R} can be characterized by the property of belonging to a local point of R on OGb. Since z is fixed by R, the algebra zOGbz of b is naturally an interior R-algebra via the homomorphism which maps an element x of R to the element xz of zOGz.

The source algebra of a block b is Morita equivalent to the block algebra OGb, hence it determines the module category of b over O and over k and K; in addition the interior R-algebra isomorphism class of a source algebra of b determines other local invariants of b, namely its matrix of generalized decomposition numbers, vertices and sources of its indecomposable modules, and its Brauer and Puig categories.

The following remark concerning the number (up to interior R-algebra isomorphism) of source algebras of a given block b follows trivially from the fact that the set of local points of R on OGb is transitively permuted by the group N_{G}(R).

Remark 1.1. Let b be a block of G and R a defect group of b. Suppose that z and z′ are both elements of (possibly different) local points of R on OGb. Then there is an automorphism σ of R and an algebra isomorphism

ϕ : zOGz → z′OGz′

such that for any r in R, we have ϕ(zr) = z′σ(r).

We now state Puig’s conjecture.

Conjecture 1.2. (Puig)[B2]. Let R be a fixed r-group.There are only finitely many isomorphism classes of interior R-algebras which are source algebras for blocks of finite groups with a defect group isomorphic to R.

§2.The finite general linear groups

Let X be an indeterminate and let I be the set of monic irreducible polynomials in X with coefficients in 𝔽_{q} which are different from X. For i in I denote by d_{i} the degree of i, by q_{i} the integer q^{d_{i}} and by a_{i} the multiplicative order of q_{i} modulo r.

Let V be a finite dimensional vector space over 𝔽_{q} and put G := Gl(V, 𝔽q), the group of invertible 𝔽_{q}-linear maps of V . We assume that all 𝔽_{q}-linear maps of V act from the left. For any element s of G, and any < s >-invariant subspace W of V, we denote by s|_{W} the element of GL(W, 𝔽_{q}) which is defined by the restriction of the action of s to W.

Let s be a semisimple element of G, i.e. the order of s is relatively prime to q. For i in I, let V_{s,i} be the subspace of elements v of V such that i(s)v = 0. Then, V_{i,s} is < s >-invariant and since s is semi-simple we have

V = ⊕_{i∈I}V_{s,i}.  (2.1)

Let s_{i} := s|_{V_{s,i}}. Then clearly, the subring 𝔽_{q}[s_{i}] of End_{𝔽_{q}}(V_{s,i}), generated by s_{i} is the field of q_{i} elements, and considering V_{s,i} as a vector space over 𝔽_{q}[s_{i}], we get that

C_{G}(s) ≃ ∏_{i∈I} Gl(V_{s,i}, 𝔽_{q_{i}}).  (2.2)

Put

m_{s,i} := dim_{𝔽_{q_{i}}} V_{s,i}.  (2.3)

Clearly, the integers m_{s,i} depend only on the G-conjugacy class of s and conversely, the conjugacy class of a semisimple element s is determined by the integers m_{s,i} as i runs over I.

Let P be the set of partitions of positive integers along with the empty partition, denoted ϕ, of zero. For any element λ of P we denote by |λ| the magnitude of λ.

We give below the description of the ordinary irreducible characters of G as given by Green in [G].

Theorem 2.1. The set of ordinary irreducible characters of G are in one to one correspondence with G-conjugacy classes of pairs (u, E) where u is a semi-simple element of G and E : I → P is a map from I to P such that the image E_{i} of i under E is a partition of m_{s,i}. Here, two pairs (s, D) and (s′, E′) are G-conjugate if s and s′ are G-conjugate and E = E′.

We denote by χ_{u,E} the character of G corresponding to the pair (u, E). If u = 1_{G}, the character χ_{u,E} is called a unipotent character of G.

If a is a positive integer, the a-core of a partition λ is the partition obtained by successively removing all hooks of length a from λ until a partition is reached from which no a-hooks can be removed. This is well defined by a theorem of Nakayama (see [N], Paragraph 4). For i ∈ I, denote by P^{i} the subset of P consisting of those partitions (including the empty partition) which are their own a_{i} cores. The r-blocks of G are described in the following theorem.

Theorem 2.2. ( [FS], Theorem [5D] and [B1], (3.4))The r-blocks of G are in one to one correspondence with G-conjugacy classes of pairs (s, B) where s is a semisimple element of G of order prime to r and B : I → P is a map such that denoting by B_{i} the image of i, we have

(i) B_{i} is an element of P^{i} and

(ii)|B_{i}| ≤ m_{s,i}, and a_{i} divides m_{s,i} − |B_{i}|.

Let b_{s,B} be the block of G defined by the pair (s, B) as in the above theorem. We will call s the semisimple part of b_{s,B} and B the unipotent part of b_{s,B}. If B_{i} = ϕ for every i in I, we say that b_{s,B} has trivial unipotent part. Note that if the center of the group G contains elements of order r, or in other words, r divides q − 1, then a_{i} is one for all i in I, and hence all blocks of G have trivial unipotent part.

If s = 1_{G}, then b_{s,B} is called a unipotent block of G. By Theorem 2.5 below, the unipotent blocks of G are precisely those blocks which contain at least one unipotent character.

Let I′ be the subset of those elements of I whose roots (in an algebraic closure of 𝔽_{q}) have order prime to r.

Definition 2.3. ([MO], page 206) Let b := b_{s,B} be a block of G as defined in Theorem 2.2. The weight vector of b is the map

W(b) : I′ → ℤ^{+},

defined by

W(b)(i) = \frac{1}{a_{i}} (m_{s,i} − |B_{i}|).

Note that for s as above, m_{s,i} = 0, unless i is an element of I′.

Now let R be a defect group of the block b_{s,B} chosen such that R commutes with s (see [B1], 3.6 ). Since R acts semi-simply on V, for i in I, we have an R-invariant subspace decomposition

V_{s,i} = V_{s,i}^{R} ⊕ [V_{s,i}, R],  (2.4)

where V_{s,i}^{R} is the R-fixed point space of V_{s,i} and where [V_{s,i}, R] is the unique Rinvariant complement of V_{s,i}^{R} in V_{s,i}.

Theorem 2.4. ( [B1], 3.6) Let b := b_{s,B} and R be chosen as above. Then

dim_{𝔽_{q_{i}}} (V_{s,i}^{R}) = |B_{i}|

and R is a Sylow r-subgroup of

∏_{i∈I} GL([V_{s,i}, R], 𝔽_{q_{i}})

where the above group is regarded as a subgroup of C_{G}(s) via the isomorphism of (2.2).

In other words, R is isomorphic to a Sylow r-subgroup of

∏_{i∈I′} GL(W(b)(i), 𝔽_{q_{i}}),

where W(b) is the weight vector of b.

Theorem 2.5. ( [FS], Theorem [7A]) Let R be a defect group of the block b_{s,B} of G chosen such that R commutes with the element s, and let χ_{u,E} be an irreducible character of G. Then χ_{u,E} belongs to the block b_{s,B} if and only if the following two conditions are satisfied.

(i) There is an element y of R such that u is G-conjugate to sy.

(ii) For any i in I, the a_{i}-core of the partition E_{i} is the partition B_{i}.

Again, let R be a defect group of b_{s,B} which commutes with s. We describe next the image of b_{s,B} under the Brauer homomorphism Br_{R}^{OG}.

For any element a of OG, denote by ā its image under the canonical surjection OG → kG. As is explained in Proposition 37.5 of [T], there is a canonical isomorphism, ψ between the image (OG)^{R}/(OG)_{<R}^{R} + J(O)(OG)^{R} of (OG)^{R} under Br_{R}^{OG} with the group algebra kC_{G}(R) of the centralizer of R in G, such that the composite map

ψ ∘ Br_{R}^{OG} : (OG)^{R} → kC_{G}(R)

sends an element ∑_{g∈G} a_{g}g of (OG)^{R} to the element ∑_{g∈C_{G}(R)} ā_{g}g of kC_{G}(R). Henceforth, we shall identify the map Br_{R}^{OG} with the composite map ψ ∘ Br_{R}^{OG}. Then Br_{R}^{OG}(b_{s,B}) is a sum of blocks of C_{G}(R) with defect group Z(R), and

conjugation by elements of N_{G}(R) transitively permutes the set of all blocks of C_{G}(R) that appear as summands of Br_{R}^{OG}(b_{s,B}) (see 40.13 and 40.14 of [T]).

Keeping the notation introduced for (2.1),(2.3) and (2.4), we have that the Rfixed point subspace of V, which is denoted V^{R} is ⊕_{i∈I}V_{s,i}^{R} and the R-invariant complement of V^{R}in V, denoted [V, R] is ⊕_{i∈I}[V_{s,i}, R].

Put

G_{0} := Gl(V^{R}, 𝔽_{q}) and G_{1} := Gl([V, R], 𝔽_{q}).

Then G_{0}, G_{1} and G_{0} × G_{1} can naturally be regarded as subgroups of G and we shall henceforth assume this subgroup structure without comment. We have

C_{G}(R) = G_{0} × C_{G_{1}}(R).  (2.5)

Also, since both V^{R} and [V, R] are N_{G}(R)-invariant subspaces of V,

N_{G}(R) = G_{0} × N_{G_{1}}(R).  (2.6)

Since s is in C_{G}(R), we may write s = s|_{V^{R}}s|_{[V,R]} where s_{V^{R}} is an element of G_{0} and s|_{[V,R]} is an element of G_{1}. Then it is easy to check that the pair (s|_{V^{R}}, B) defines a block (via the correspondence of Theorem 2.2) of defect zero of G_{0}.

Since R acts semisimply on [V, R], there is an indexing set I_{R}, and a decomposition

[V, R] = ⊕_{x∈I_{R}}V_{x}

into R-invariant subspaces of [V, R], such that for any x in I_{R}, V_{x} is a sum of isomorphic irreducible 𝔽_{q}[R] modules and Hom_{𝔽_{q}[R]}(V_{x}, V_{x′}) = 0 for distinct x, x′ in I_{R} ; consequently

C_{G_{1}}(R) = ∏_{x∈I_{R}} C_{Gl(V_{x},𝔽_{q})}(R).  (2.7)

Further, for each x in I_{R}, there is a non-negative power q_{x}, of q such that the center Z𝔽_{q}[R|_{V_{x}}] of the subalgebra of End_{𝔽_{q}}(V_{x}) generated by the restriction of the action of elements of R to V_{x} is the field 𝔽_{q_{x}}, and considering V_{x} as a vector space over 𝔽_{q_{x}} we have

C_{Gl(V_{x},𝔽_{q})}(R) = Gl(V_{x}, 𝔽_{q_{x}}).  (2.8)

For x in I_{R}, let s_{x} denote the restriction of s to V_{x}. By (2.7) and (2.8) a block of C_{G}(R) is defined as product of blocks of Gl(V_{x}, 𝔽_{q_{x}}) as x runs over I_{R}. Let b̃_{s|_{[V,R]},ϕ̃}

be the block of C_{G_{1}}(R), such that for any x in I_{R}, the component of b̃_{s_{|}[V,R],ϕ̃} which comes from Gl(V_{x}, 𝔽_{q_{x}}) has semisimple part s_{x} and trivial unipotent part. Note that since r divides q_{x} − 1, by the remarks following Theorem 2.2 this is the only block of Gl(V_{x}, 𝔽_{q_{x}}) with semisimple part s_{x}.

For a block b_{0} of G_{0}, and a block b_{1} of C_{G_{1}}(R), we denote by b_{1} ⊗ b_{2} the block of C_{G}(R), which corresponds to the idempotent b_{1} ⊗ b_{2} of OG_{0} ⊗_{O} OC_{G_{1}}(R), and we denote by N_{G}(R, b_{1} ⊗ b_{2}) the subgroup of N_{G}(R) which fixes the block b_{1} ⊗ b_{2}.

Proposition 2.6. Keep the notation introduced above and let C_{1} := C_{G_{1}}(s|_{[V,R]}). Then

N_{G}(R, b_{s|_{V ^{R}} ,B} ⊗ b̃_{s|_{[V,R]},ϕ̃}) = G_{0} × C_{G_{1}}(R)N_{C_{1}}(R)  (2.9)

and

Br_{R}^{OG}(b_{s,B}) = b̄_{s|_{V ^{R}} ,B} × ∑_{n} ^{n}¯b̃_{s|_{[V,R]},ϕ̃}  (2.10)

where n runs over a complete system of representatives of left cosets of C_{G_{1}}(R)N_{C_{1}}(R) in N_{G_{1}}(R).

Proof. The first assertion is Proposition 4.17 of [B1].

By 3.5 of [B1], the block b̄_{s|_{V ^{R}} ,B} ⊗ b̃_{s|_{[V,R]},ϕ̃} of kC_{G}(R) is a constituent of

Br_{R}^{OG}(b_{s,B}), hence by Proposition 40.13 of [T], a block of kC_{G}(R) is a constituent of Br_{R}^{OG}(b_{s,B}) if and only if it is conjugate to b̄_{s|_{V ^{R}} ,B} ⊗ b̃_{s|_{[V,R]},ϕ̃} by some element of N_{G}(R). (2.10) now follows from (2.6) and (2.9). ■

§3.The main Theorem

Our main result is an algebra isomorphism between two blocks that form a (w_{j} : m) pair for some j in I and some positive integer m. This is a generalization of the (w : m) pairs introduced by Scopes in [S]. In order to define such pairs, we use the treatment of [P2].

The partitions of a non-negative integer n are in one to one correspondence with translation classes of maps ϕ : ℤ → ℤ/2ℤ such that the cardinality of the set

{(t, t′), t, t′ ∈ ℤ; t′ > t; ϕ(t) = 1 + 2ℤ and ϕ(t′) = 2ℤ}

is equal to n. We briefly describe this correspondence:

Suppose ϕ is such a map. Let t_{1}, t_{2}, ⋯, t_{u} be the integers, in increasing order, which appear as the first element of an ordered pair that belongs to the set defined above and for each t_{x}, 1 ≤ x ≤ u, let λ_{x} equal the number of ordered pairs in the set above for which t_{x} is the first element. Then,

λ_{1} ≥ ⋯ ≥ λ_{u}

is the partition defined by ϕ. Conversely, given a partition λ of n with parts λ_{x} as above, the translation class of the map ϕ : ℤ → ℤ/2ℤ such that for any z ∈ ℤ, ϕ(z) = 1 + 2ℤ if and only if either z = x − λ_{x}, for some i, 1 ≤ i ≤ l or z > n, will correspond to λ.

Let a be a positive integer greater than 1 and let us denote by x̃ the residue class of an integer x mod a.

Let σ : ℤ/aℤ → ℤ be a section of the canonical map ℤ → ℤ/aℤ. The map λ_{σ} : ℤ → ℤ/2ℤ defined by setting λ_{σ}(x) = 2ℤ if an only if x ≤ σ(x̃), will give rise to a partition from which no no a-hooks can be removed. In other words if μ is the partition defined by λ_{σ} by the process described above, then μ is its own a-core. Conversely, every partition which is its own a-core is defined by λ_{σ} for some σ. Also, for two sections σ and σ′, λ_{σ} and λ_{σ′} describe the same partition if and only if there is an integer t such that for any x ∈ ℤ, σ′(x̃ + t̃) = σ(x̃) + t.

Definition 3.1. ( [P2], 1.5) Let a be a positive integer and let z̃ be an element of ℤ/aℤ.

(i) The Scopes map at z̃ is the map Scz̃ : ℤ → ℤ such that for any integer x, Scz̃(x) is x − 1 if x is in the class z̃, is x + 1 if x is in the class z̃ − ˜1 and is x otherwise.

(ii) We define also the related map

S̃cz̃ : ℤ/aℤ → ℤ/aℤ

which is the involution that interchanges the classes z̃ and z̃ − ˜1 and fixes all other classes.

Recall the notation introduced for section 2. From now on, until the end of the proof of Theorem 3.3 , let us fix a map

W : I′ → ℤ^{+},

and we denote by w_{i} the integer W(i).

We let b := b_{s,B} be a block of G with weight vector W and fix a j in I such that B_{j} is not the empty partition ( Note that this means in particular that a_{j} is greater than 1.). Further, we fix a section σ : ℤ/a_{j}ℤ → ℤ of the canonical surjection of ℤ onto ℤ/a_{j}ℤ, such that B_{j} is the partition defined by λ_{σ}, and a residue class z̃ in ℤ/a_{j}ℤ.

Let

τ = Scz̃ ∘ σ ∘ S̃cz̃.

Then clearly τ is also a section of the canonical surjection of ℤ onto ℤ/a_{j}ℤ.

Let C_{j} be the element of P^{a_{j}} defined by τ. For i in I, i different from j, we define C_{i} to be the partition B_{i}. Let C : I → P be the map whose image on i is the partition C_{i}.

It is shown in [P2] that

|B_{j}| − |C_{j}| = \frac{1}{a_{j}} (σ(z̃) − (σ(z̃ − ˜1) + 1)

and so, interchanging z̃ and z̃ − ˜1 if necessary, we may assume that

|B_{j}| > |C_{j}|.  (3.1)

We use again the notation introduced for (2.1)-(2.3). It is clear that any direct sum decomposition of V_{s,j} into irreducible 𝔽_{q}[s]-modules will contain m_{s,j} summands and we let W_{j} be a subspace of V_{s,j} which is the sum of any |C_{j}| + a_{j}w_{j} of these. Note that in view of (3.1), this makes sense since by Definition 2.3

m_{s,j} = |B_{j}| + a_{j}w_{j}.

In other words, W_{j} is an < s >-invariant subspace of V_{s,j} and we have

dim_{𝔽_{q_{j}}} (W_{j}) = dim_{𝔽_{q_{j}}} (V_{s,j}) − (|B_{j}| − |C_{j}|)  (3.2)

Let W be the subspace of V defined by

W := W_{j} ⊕ ⊕_{i∈I,i≠j}V_{i}

and put

H := Gl(W, 𝔽_{q}) and t = s|_{W} .

Then, clearly, we have

W_{t,j} = W_{j} and W_{t,i} = V_{s,i}∀i ∈ I, i ≠ j  (3.3)

and hence

m_{t,j} = m_{s,j} − (|B_{j}| − |C_{j}|) and m_{t,i} = m_{s,i} ∀i ∈ I, i ≠ j.  (3.4)

In particular, a_{i} divides m_{t,i} −|C_{i}| for all i in I, and hence the pair (t, C) satisfies the hypothesis of Theorem 2.2 for the group H.

Let c be the block of H defined by the pair (t, C).

Fix a defect group R of c which commutes with t. Then assuming the natural inclusion of H in G, for any i in I, we have [V_{s,i}, R] = [W_{s,i}, R], and so by Theorem 2.4, R is also a defect group of b. Also, clearly, R commutes with s.

Let W′ be a < s >-invariant complement of W_{j} in V_{s,j} and let

W′ = ⊕_{x=1}^{|B_{j}|−|C_{j}|}Wx^{′}  (3.5)

be a decomposition of W into irreducible 𝔽_{q}[s]-modules. For 1 ≤ x ≤ |B_{j}| − |C_{j}|, let Hx^{′} := Gl(Wx^{′}, 𝔽_{q}) and let cx^{′} be the block of Hx^{′} whose semisimple part is s|_{W′_{x}} and whose unipotent part is the map which sends j to the partition {1} of 1 and which sends all other i in I to the empty partition. Note that since m_{s|_{W′_{x}} ,j} = 1 and m_{s|_{W′_{x}} ,i} = 0 for all other i in I, this makes sense; moreover, since a_{j} is greater

than 1, cx^{′} is a block of defect zero of Hx^{′}.

Put

H′ := ∏_{x=1}^{|B_{j}|−|C_{j}|} H_{x}

and let c′ be the block of H′ which is the product of the cx^{′}’s.

Then L := H × H′ is a Levi subgroup of G. Let d be the block of L which is the product of the block c of H with the block c′ of H′. Note that since c′ is a block of defect zero of H′, d and c have isomorphic source algebras.

Let P be the stabilizer in G of the chain

0 ⊂ W1^{′} ⊂ W1^{′} ⊕ W2^{′} ⊂ ⋯ ⊂ ⊕_{x=1}^{|B_{j}|−|C_{j}|}Wx^{′} ⊂ V  (3.6)

of subspaces of V and let U be the unipotent radical of P; clearly, L is a Levi complement of U in P. Let

e = \frac{1}{|U|} ∑_{u∈U} u  (3.7)

be the central idempotent of KU corresponding to the trivial character of U. Since |U| is a power of q, and q and r are relatively prime, e is defined over O.

Now, clearly the idempotents b, d and e of OG all commute with one another and with the group L, hence bde is an idempotent of OG and moreover, the algebra deOGbde has a natural structure of an interior L -algebra via the map l → lde.

The following definition gives the condition under which the block b of G and the block c of H will be shown to have isomorphic source algebras.

Definition 3.2. ([J], (1.9))Let W, j, b and c be as defined above and assume that (3.1) holds. Then , given a positive integer m, we say that b and c form a (w_{j} : m) pair if

|B_{j}| − |C_{j}| = m  (3.8)

and

m > w_{j}.  (3.9)

We note at this point that in the definition (1.9) of [J], the integer m is also indexed by j, but this is unnecessary and we therefore omit doing so.

We are now ready to state the main theorem:

Theorem 3.3. Suppose that for some m, the blocks b and c form a (w_{j} : m) pair. Then there is an interior L-algebra isomorphism

deOGbde ≃ OLd ⊗_{O} Mat_{m!}(O).  (3.10)

where Mat_{m!}(O) is considered an interior L-algebra via the homomorphism which sends all elements of L to the identity of Mat_{m!}(O).

In particular, there is an interior R-algebra isomorphism between a source algebra of b and a source algebra of d.

Let B_{W} be the set of blocks b in the family of groups {Gl(n, q) : n ≥ 1} with weight vector W.

Corollary 3.4. (i) Let R be a defect group for the elements of B_{W} and let Aut(R) denote the group of automorphisms of R. There are at most

|Aut(R)| ∏_{i∈I′} ∏_{x=1}^{a_{i}}[(x − 1)(w_{i} − 1) + 1]

many isomorphism classes of interior R algebras which are source algebras for elements of B_{W}.

(ii) Puig’s conjecture holds for the unipotent blocks of the family {Gl(n, q) : n ≥ 1} .

§4.Proof of Theorem 3.3

We keep all the notation introduced for Theorem 3.3. In addition, let χ′ denote the unique character of H′ in the block c′ and let l = m!. Let T_{L⊂P}^{G} denote Harish-Chandra restriction of characters from G to L (see [DM], Chapter 4), with respect to the parabolic subgroup P. In other words, for any x in KL, and any character χ in G, T_{L}^{G}(χ)(x) = χ(ex), where e is the idempotent of (3.7).

We start with the following lemma of Jost.

Lemma 4.1. There is a one to one correspondence (denoted by χ → Sc(χ)) between Irr(G, b) and Irr(H, c) such that for any χ in Irr(G, b),

(T_{L}^{G}(χ), Sc(χ) × χ′)_{L} = l  (4.1)

and

(T_{L}^{G}(χ), ψ × χ′)_{L} = 0  (4.2)

for all other ψ in Irr(H, c) . In particular,

rank_{O}deOGbde = l^{2}rank_{O}OLd.  (4.3)

Proof. The first statement is just Lemma 2.4 of [J]. The rank comparison of (4.3) is immediate from this, since for any χ in Irr(G, b), χ(de) measures the O-rank of the component of T_{L}^{G}(χ) which lies in the block d of L.

■

As before, we let G_{0} be the group Gl(V^{R}, 𝔽_{q}) and G_{1} the group Gl([V, R], 𝔽q). Further, put

H_{0} := Gl(W^{R}, 𝔽q).

Since R is chosen to lie in H, [V, R] = [W, R], and hence (2.5) and (2.6) applied to the group H will give

C_{H}(R) = H_{0} × C_{G_{1}}(R)  (4.4)

and

N_{H}(R) = H_{0} × N_{G_{1}}(R).  (4.5)

Also, since t = s|_{W}, and since [V, R] = [W, R], we have t|_{[V,R]} = s|_{[V,R]} and so Proposition 2.6 applied to the block c of H, will give

Br_{R}^{OG}(c) = Br_{R}^{OH}(c) = b̄_{t|_{W^{R}} ,C} ⊗ ∑ ^{n}b̃_{s|_{[V,R]},ϕ̃}  (4.6)

where n runs over a complete system of representatives of left cosets of C_{G_{1}}(R)N_{C_{1}}(R) in N_{G_{1}}(R).

Let b_{0} denote the block b_{s|_{V ^{R}} ,B} of G_{0}, c_{0} the block b_{t|_{W^{R}} ,C} of H_{0} and b_{1} the block b̃_{s|_{[V,R]},ϕ̃} of C_{G_{1}}(R).

Since W′ is a complement of W_{j} in V_{s,j} (see 3.5), and hence a complement of W in V, W′ ⊂ V^{R} and we have by (4.4) and (4.5)

C_{L}(R) = H_{0} × H′ × C_{G_{1}}(R)  (4.7)

and

N_{L}(R) = H_{0} × H′ × N_{G_{1}}(R).  (4.8)

We let

L_{0} := H_{0} × H′

and let d_{0} denote the block c_{0} ⊗ c′ of the group L_{0}.

Then since R commutes with H′,

Br_{R}^{OL}(c′) = c̄′  (4.9)

and so from (4.6) and (4.7)

Br_{R}^{OG}(d) = Br_{R}^{OL}(d) = d̄_{0} × ∑ ^{n}b̄_{1},  (4.10)

where n runs over the same set as in (4.6).

Let P_{0} be the parabolic subgroup of G_{0} which stabilizes the chain of subspaces

0 ⊂ W1^{′} ⊂ W1^{′} ⊕ W2^{′} ⊂ ⋯ ⊂ ⊕_{x=1}^{|B_{j}|−|C_{j}|}Wx^{′} ⊂ V^{R}  (4.11)

where the Wx^{′} are as defined for (3.5).

Let U_{0} be unipotent radical of P_{0}, and put

e_{0} = \frac{1}{|U_{0}|} ∑_{u∈U_{0}} u  (4.12)

the central idempotent of OU_{0} corresponding to the trivial character of U_{0}.

Suppose that u is an element of U ∩ C_{G}(R). Write u as u_{0}u_{1}, where u_{0} ∈ G_{0} and u_{1} ∈ G_{1}, and let x be an element of [V, R]. Then clearly

ux = u_{0}u_{1}x = u_{1}x ∈ [V, R].  (4.13)

On the other hand, since W′ ⊂ V^{R}, and V^{R} ∩ [V, R] = 0, by definition of U (see (3.6))

ux = x + x′  (4.14)

where x′ is some element of W′.

From the above two equations it follows that u_{1} = 1_{G_{1}}, hence U ∩ C_{G}(R) ⊂ G_{0} and then from the definition of U_{0} and by (3.7), we get immediately

U ∩ C_{G}(R) = U_{0} and Br_{R}^{OG}(e) = \frac{1}{|U|} ∑_{u∈U_{0}} ū.  (4.15)

But Br_{R}^{OG}(e) is an idempotent of kC_{G}(R), so

\frac{1}{|U|} ∑_{u∈U_{0}} u = (\frac{1}{|U|} ∑_{u∈U_{0}} u)^{2} = \frac{|U_{0}|}{|U|} ∑_{u∈U_{0}} u.  (4.16)

Therefore, |U_{0}| is congruent to |U| modulo r, and

Br_{R}^{OG}(e) = e_{0}.  (4.17)

Let

A := deOGbde.

¿From (2.10), (4.10), and (4.17), we get that

Br_{R}^{A}(A^{R}) = Br_{R}^{OG}(A^{R}) = d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} ⊗_{k} ∑_{n} kC_{G_{1}}(R) ^{n}b̄_{1},  (4.18)

where n is as in (4.6).

Now, d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} is a simple k-algebra as b_{0} is a block of defect zero of G_{0}. Thus the Jacobson radical J(Br_{R}^{OG}(A^{R})) of Br_{R}^{OG}(A^{R}) is of the form

d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} ⊗_{k} J(∑_{n} kC_{G_{1}}(R) ^{n}b̄_{1})

and

Br_{R}^{A}(A^{R})/J(Br_{R}^{A}(A^{R})) = d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0}⊗_{k}(∑_{n} kC_{G_{1}}(R)^{n}b̄_{1})/J(∑_{n} kC_{G_{1}}(R)^{n}b̄_{1}).

(4.19)

Also, Z(R) is a defect group of the block b_{1} of C_{G_{1}}(R), hence denoting also by b̄_{1} the image of b̄_{1} under the canonical surjection

kC_{G_{1}}(R) → k(C_{G_{1}}(R)/Z(R)),

we have that ^{n}b̄_{1} is a block of defect zero of k(C_{G_{1}}(R)/Z(R)) for any n in N_{G_{1}}(R) and thus that k(C_{G_{1}}(R)/Z(R)) ^{n}b̄_{1} is a simple k-algebra.

On the other hand, the kernel of the surjection above is clearly a nilpotent ideal of kC_{G_{1}}(R). Hence it follows from (4.18) and (4.19), that

Br_{R}^{A}(A^{R})/J(Br_{R}^{A}(A^{R})) = d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} ⊗_{k} ∑_{n} k(C_{G_{1}}(R)/Z(R)) ^{n}b̄_{1} (4.20)

where again n is as in (4.6).

By Lemma 14.5 of [T] the local points of R on A are in one to one correspondence with the points of Br_{R}^{A}(A^{R}) and hence with the points of Br_{R}^{A}(A^{R})/J(Br_{R}^{A}(A^{R})). Hence, (4.19) gives that there are |N_{G_{1}}(R) : C_{G_{1}}(R)N_{C_{1}}(R)| local points of R on A, (this is the same as the number of local points of R on OGb) and we denote by γ the point corresponding to the simple algebra d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} ⊗_{k} b̄_{1}k(C_{G_{1}}(R)/Z(R)) in (4.20) above, i.e. for any element i of γ,

Br_{R}^{A}(i)(d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} ⊗ b̄_{1}) ≠ 0.

Recall that for a subgroup Y of L, and a point α of Y on A, the multiplicity of α, denoted m_{α}, is the number of elements of α that appear in a decomposition of the identity 1_{A}, of A into primitive idempotents of A^{Y}. A pointed group, Y_{α} on A is a pair (Y, α), where Y is a subgroup of L and α is a point of Y on A. For two pointed groups Y_{α} and Yα′^{′} on A, we say that Y_{α} is contained in Yα′^{′} if Y is a subgroup of Y′ and if the decomposition of any element of α′ into orthogonal primitive idempotents in A^{Y} contains elements of α.

Lemma 4.2. There is exactly one point, say α of L on A for which the corresponding pointed group L_{α} contains R_{γ}, and this point is of multiplicity l.

Proof. Denote by A(R_{γ}) the simple algebra d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} ⊗_{k} k(C_{G_{1}}(R)/Z(R))b̄_{1} and by S_{γ} the composition of Br_{R}^{A} with the canonical surjection of Br_{R}^{A}(A^{R}) onto A(R_{γ}) (see 4.20). Clearly, the restriction of the natural action of N_{L}(R)/R on Br_{R}^{A}(A^{R})/J(Br_{R}^{A}(A^{R})) (see [T], section 11)to the subgroup N_{L}(R_{γ})/R will induce an action of N_{L}(R_{γ})/R on A(R_{γ}). Let A_{R}^{L} denote the set of all elements a of A that are in the image, Tr_{R}^{L}(A^{R}), of the relative trace map

Tr_{R}^{L} : A^{R} → A^{L}.

By Proposition 1.3 of [P1], applied to the interior L-algebra A and to the local pointed group R_{γ} on A, we know that the restriction of S_{γ} to the ideal A_{R}^{L} of A^{L} induces a surjection of A_{R}^{L} onto A(R_{γ})_{1}^{N_{L}(R_{γ})/R}. But R being a defect group of c, and hence of d we have that d ∈ Tr_{R}^{L}(OLd) and hence that d ∈ Tr_{R}^{L}(OGd). Since the idempotents b and e both commute with L, this gives immediately that bde ∈ Tr_{R}^{L}(deOGbde). In other words, A_{R}^{L} is A^{L}, and so we get that the restriction of S_{γ} to A^{L} induces a surjection of A^{L} onto A(R_{γ})^{N_{L}(R_{γ})/R} = A(R_{γ})^{N_{L}(R_{γ})}. Since elements of a point α of L on A will be in the kernel of S_{γ} if and only if L_{α} does not contain R_{γ}, this shows that there is a multiplicity preserving bijection between the points of L on A for which the corresponding pointed groups contain R_{γ} and the points of N_{L}(R_{γ}) on A(R_{γ}).

So, in order to prove the lemma, it suffices to determine the points of N_{L}(R_{γ}) on A(R_{γ}).

By (2.10) and Proposition 40.13 of [T], we have that

N_{H}(R_{γ}) = C_{H}(R)N_{C_{1}}(R) = H_{0} × C_{G_{1}}(R)N_{C_{1}}(R).  (4.21)

Since R commutes with H′, (4.21) will give

N_{L}(R_{γ}) = L_{0} × C_{G_{1}}(R)N_{C_{1}}(R).  (4.22)

Then clearly,

A(R_{γ})^{N_{L}(R_{γ})} = (d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0})^{L_{0}} ⊗_{k} (k(C_{G_{1}}(R)/Z(R))b̄_{1})^{C_{G_{1}} (R)N_{C_{1}} (R)}. (4.23)

But since b̄_{1} is a block of defect zero of C_{G_{1}}(R)/Z(R), we have

(k(C_{G_{1}}(R)/Z(R))b̄_{1})^{C_{G_{1}} (R)} ≃ k  (4.24)

and therefore in particular that

(k(C_{G_{1}}(R)/Z(R))b̄_{1})^{C_{G_{1}} (R)N_{C_{1}} (R)} ≃ k.  (4.25)

Thus

A(R_{γ})^{N_{L}(R_{γ})} ≃ (d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0})^{L_{0}}.  (4.26)

Now, the natural inclusion of L_{0} into G_{0} induces an interior L_{0}-algebra homomorphism

kL_{0}d̄_{0} → d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0}

which sends an element x of kL_{0}d̄_{0} to the element xē_{0}b̄_{0}. Since kL_{0}d̄_{0} and d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} are both matrix algebras over k, this map is injective (it is nonzero by (4.28) below), and identifying kL_{0}d̄_{0} with its image in d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} via the above map, we get by Proposition 7.3 of [T] that

d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} ≃ kL_{0}d̄_{0} ⊗_{k} (d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0})^{L_{0}}  (4.27)

and since d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0} is a simple k-algebra,we also get that (d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0})^{L_{0}} is a matrix algebra over k.

Now, by definition b_{0} = b_{s|V^{R},B} and c_{0} = b_{t|W^{R},C} are related to one another through a Scopes map just as the blocks b and c are. Further, since b_{0} and c_{0} are of defect zero, their weight vector is the map that takes the value 0 for every i in I, and so trivially b_{0} and c_{0} form a (0 : l) pair. In particular, Lemma 4.1 applies and we get

dim_{k}(d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0}) = l^{2}dim_{k}(kL_{0}d̄_{0}).  (4.28)

Hence, we get that (d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0})^{L_{0}} is a matrix algebra of dimension l^{2}. In particular, (d̄_{0}ē_{0}kG_{0}b̄_{0}d̄_{0}ē_{0})^{L_{0}} has only one point and it is of multiplicity l. Thus, by (4.26), we get that there is only one point of N_{L}(R_{γ}) on A(R_{γ}), and this is of multiplicity l. This completes the proof of the lemma.

■

Let α be as in Lemma 4.2 and suppose that f is an idempotent of A^{L} such that any primitive idempotent decomposition of f in A^{L} only contains elements of α and no element of α appears in a primitive idempotent decomposition of 1_{A} − f in A^{L}.

Consider the interior L algebra fAf. By Theorem 7.3 of [T], we may find a subalgebra T of fA^{L}f, such that T is a product of matrix algebras over O and T is maximal such. As α is the only point of L on fAf and α has multiplicity l, T has actually one simple factor, i.e. T is isomorphic to the l × l matrix algebra over O. Let C := C_{fAf}(T). T is a unitary O-simple subalgebra of fAf. Hence, by Proposition 7.5 of [T],

C ⊗_{O} T ≃ fAf, via x ⊗ y → xy  (4.29)

and

C ≃ zfAfz = zAz, via x → zxz  (4.30)

where z is any primitive idempotent of T.

Note that T is contained in (fAf)^{L}. So, C contains the image of L in fAf and we give C ⊗_{O} T an interior L algebra structure via h → hf ⊗ f = h1_{C} ⊗ 1_{T} . Also, for a primitive idempotent z of T, zAz has a natural interior L algebra structure as z commutes with L, and it is clear that as such, the isomorphisms of (4.29) and (4.30) are interior L algebra isomorphisms. In particular, passing to the subalgebras fixed by L in the isomorphism of (4.29), we get

fA^{L}f = (fAf)^{L} ≃ (C ⊗_{O} T)^{L} = C^{L} ⊗_{O} T ≃ C^{L} ⊗_{O} Mat_{l}(O).  (4.31)

Consequently, the number of idempotents appearing in a primitive idempotent decomposition of the identity in fA^{L}f is equal to the number of idempotents appearing in a primitive idempotent decomposition of the identity in C^{L} ⊗_{O} Mat_{l}(O).

This shows that 1_{C} ⊗_{O} z is primitive even in (C ⊗_{O} T)^{L}. Thus z · 1_{C} = z is in α. Now the interior L-algebra structure of zAz induces a homomorphism

OLd → zAz,

where clearly

d → bez = z,

making the homomorphism unitary. This, along with (4.29) induces a sequence of interior L-algebra homomorphisms:

OLd ⊗_{O} Mat_{l}(O) → zAz ⊗_{O} Mat_{l}(O) ≃ fAf.  (4.32)

The group R stabilizes an O-basis of OG (via the diagonal action of R) and zAz is a R module summand of OG. Thus by Corollary 27.2 of [T], R stabilizes an O-basis of zAz. We have Br_{R}(z) ≠ 0 as z is in α and L_{α} ⊃ R_{γ}, and thus zAz(P) = Br_{R}(z)kC_{G}(P)Br_{R}(z) is a non-zero projective kZ(R) module via the left multiplication action of Z(R). The groups R, L and the algebra zAz satisfy therefore the hypothesis of Proposition 3.8 of [P2], and hence the map OLd → zAz as defined above is a direct inclusion of OL bimodules and in particular, of O modules. But then the composed map between the extreme terms of (4.32) is clearly also a direct inclusion of O-modules.

On the other hand, by Lemma 4.1 and Lemma 4.2,

rank_{O}(OLd ⊗_{O} Mat_{l}(O)) = rank_{O}(OLd)l^{2} = rank_{O}(A) ≥ rank_{O}(fAf). (4.33)

This rank comparison yields that the map in (4.31) is actually an isomorphism (since it is a direct inclusion) and also that the idempotent f must be the identity of A. In other words, fAf is A and so the isomorphism of (4.32) gives

OLd ⊗_{O} Mat_{l}(O) ≃ A,  (4.34)

as interior L-algebras. This is the isomorphism of (3.10).

The isomorphism between a source algebra of b and a source algebra of d as interior R algebras is immediate as we now indicate:

Let z belong to a local point of R on OLd and let z′ be any primitive idempotent of Mat_{l}(O) . Further, let y be the image of z ⊗ z′ in A under the isomorphism (3.10). Then clearly,

zOLz ≃ (z ⊗ z′)(OLd ⊗_{O} Mat_{l}(O))(z ⊗ z′) ≃ yAy = yOGby  (4.35)

as interior R algebras. Also, y must belong to a local point of R on fAf and hence to a local point of R on OGb. As R is maximal among groups having a local point on OGb, the two extreme terms of (4.35) are source algebras for d and b respectively. This completes the proof of Theorem 3.3.

Proof of Corollary 3.4. Since c′ is a block of defect zero of H′, the blocks d and c have isomorphic source algebras, and hence by Theorem 3.3, there is an interior R-algebra isomorphism between a source algebra of b and a source algebra of c. We divide B_{W} into sub families as follows: Two blocks b and b′ of B_{W} are in the same family if there is a finite sequence

b = b^{0}, b^{1} ⋯, b^{y} = b′

of elements of B_{W}, such that for any successive elements b^{x} and b^{x+1} of the sequence, with b^{x} a block of Gl_{n_{x}}(q) and b^{x+1} a block of Gl_{n_{x+1}}(q), there is an i in I and a positive integer m such that either b^{x} and b^{x+1} form a (w_{i} : m) pair or b^{x+1} and b^{x} form a (w_{i} : m) pair.

By iterating Theorem 3.3, we get that for any two blocks b and b′ of B_{W} lying in the same family, there is an interior R-algebra isomorphism between a source algebra of b and a source algebra of b′.

By Theorem 6.3 of [J], the number of such families is at most

∏_{i∈I} ∏_{x=1}^{a_{i}}[(x − 1)(w_{i} − 1) + 1].

Part (i) of the corollary now follows from Remark 1.1.

The family B_{W} will contain unipotent blocks if and only if W_{i} = 0 for all i in I except for the polynomial X − 1, and in this case R is a Sylow r-subgroup of Gl(|W_{X−1}|, q). Clearly, for a fixed R there are only finitely many choices for |W_{X−1}|, and hence there are only finitely maps W : I′ → ℤ′ for which B_{W} contains unipotent blocks. Part (ii) is now immediate from part (i).

■

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E-mail address: kessar@maths.ox.ac.uk