A PROGENERATOR FOR FINITE REDUCTIVE GROUPS IN TRANSVERSE CHARACTERISTIC

IN TRANSVERSE CHARACTERISTIC

OLIVIER DUDAS AND GUNTER MALLE

Abstract. We exhibit a progenerator for the category of representations of a finite reductive group coming from generalised Gelfand–Graev representations. We show that cuspidal representations appear in the head of generalised Gelfand–Graev representations attached to cuspidal unipotent classes, as defined and studied in [20]. As a consequence, we prove that cuspidal unipotent characters remain irreducible afterℓ-reduction, which solves a long-standing conjecture of Geck. We use our progenerator to give a neces- sary condition for a simple module to be cuspidal, or more generally to get information on which Harish-Chandra series it can lie in. As a further application, we determine a unitriangular approximation to part of the unipotent decomposition matrix of even dimensional orthogonal groups.

1. Introduction

Let G be a connected reductive group defined over a finite field F

, with corresponding Frobenius endomorphism F . In this paper we construct a progenerator of the category of representations of the finite reductive group G

over a field of positive characteristic ℓ with ℓ ∤ q (non-defining characteristic). We produce such a progenerator using generalised Gelfand–Graev representations associated to suitably chosen unipotent classes.

Generalized Gelfand–Graev representations (GGGRs) are a family of projective repre- sentations { Γ

} labelled by unipotent classes of G = G

. They are well-defined whenever the characteristic of F

is good for G. The sum of all the GGGRs is a progenerator for the representations of G, since the GGGR corresponding to the trivial class is the regular representation, hence a progenerator itself. Our construction relies on the work of Geck and the second author [20] who showed that any representation Γ

can be replaced by the Harish-Chandra induction of a GGGR from a suitable Levi subgroup L of G, up to adding and removing GGGRs corresponding to unipotent classes larger than C for the closure ordering. This led them to the notion of cuspidal classes for which no such proper Levi subgroup exists. Following their observation, we consider the projective module

P = M

(L,C)

R

(Γ

)

where L runs over the set of 1-split Levi subgroups of G and C over the set of cuspidal unipotent classes of L.

Date: November 24, 2016.

2010 Mathematics Subject Classification. Primary 20C33; Secondary 20C08.

The first author gratefully acknowledges financial support by the ANR grant GeRepMod. The second author gratefully acknowledges financial support by ERC Advanced Grant 291512.

1

We show that P is a progenerator (see Corollary 2.6). To this end we show how to adapt the work of Geck–H´ezard [16] on a conjecture of Kawanaka to prove that the family of characters of Harish-Chandra induced GGGRs form a basis of the space of unipotently supported functions. As a consequence, we obtain that cuspidal modules must appear in the head of GGGRs associated to cuspidal classes. We believe that this should be of considerable interest for studying projective covers of cuspidal modules, as there exist very few cuspidal classes in general. For example, when G is a group of type A, cuspidal classes are regular classes and only usual Gelfand–Graev representations are needed to define in P . In this specific case, such a progenerator already appears for example in [4, 3].

The construction of P provides a solution to a long-standing conjecture of Geck (see [12, (6.6)]):

Theorem A. Assume that p and ℓ are good for G and that ℓ ∤ | Z (G)

/Z

(G)

| . Then any cuspidal unipotent character of G

remains irreducible under reduction modulo ℓ.

This property was shown to be very important in order to compare Harish-Chandra series in characteristic zero and ℓ, and for the study of supercuspidal representations, see Hiss [23].

As a second application of our construction, we show how the a-value of a unipotent representation can restrict the Harish-Chandra series in which it lies. More precisely, we deduce that unipotent representations with small a-value must lie in a Harish-Chandra series corresponding to a small Levi subgroup of G. This was already known at the level of unipotent characters from Lusztig’s classification. We give the example of Harish- Chandra series of Spin

(q) for unipotent representations of a-value at most 3, as well as approximations of the corresponding partial decomposition matrices.

This paper is organised as follows. Section 2 is devoted to the construction of the progenerator. We discuss the relation between wave front sets and unipotent supports and show how to derive a progenerator from Gelfand–Graev representations, see Corol- lary 2.6. Section 3 contains our main application on the ℓ-reduction of cuspidal unipotent characters, with the proof of Theorem A. In Section 4 we prove Theorem 4.2 on upper bounds for Harish-Chandra series in terms of a-values. Finally in Section 5 we show how our results can be used to bound Harish-Chandra series for groups of type D and derive approximations to decomposition matrices.

2. A progenerator

Let G be a connected reductive linear algebraic group over F

and F be a Frobenius endomorphism endowing G with an F

-structure. If H is any F -stable closed subgroup of G, we denote by H := H

the finite group of F

-points in H.

We let ℓ 6 = p be a prime and let (K, O , k) denote a splitting ℓ-modular system for G.

Any representation of G will be considered over one of the rings among K, O , or k.

Throughout this section, we will always assume that p is good for G. The results

on generalised Gelfand–Graev representations that we shall need were originally proved

by Kawanaka [25], Lusztig [29] and Achar–Aubert [1], under some restriction on p and

q. This restriction was recently removed by Taylor [33], so that we can work under the

assumption that p is good for G. Note that this is already required for the classification of unipotent classes to be independent from p.

2.1. Generalized Gelfand–Graev representations and wave front sets. Given an F -stable unipotent element u ∈ G, we denote by Γ

, or simply Γ

, the generalised Gelfand–Graev representation associated with u. It is an O G-lattice. The construc- tion is given for example in [25, § 3.1.2] (with some extra assumption on p) or in [33, § 5].

The first elementary properties that can be deduced are

• if ℓ 6 = p, then Γ

is a projective O G-module;

• if u and u

are conjugate under G then Γ

∼ = Γ

.

The character of KΓ

is the generalised Gelfand–Graev character associated with u. We denote it by γ

, or simply γ

. It depends only on the G-conjugacy class of u. When u is a regular unipotent element then γ

is a usual Gelfand–Graev character as in [9, § 14].

Let ρ be an irreducible character of G. A unipotent conjugacy class C of G is said to be a wave front set of ρ if the two following conditions are satisfied

• there exists u ∈ C

such that h γ

; ρ i 6 = 0;

• if C

is an F -stable unipotent conjugacy class of G such that dim C

> dim C then h γ

; ρ i = 0 for all u

∈ C

.

Here h ; i denotes the usual scalar product on K -characters. For example, the only wave front set of the Steinberg character is the regular unipotent class whereas the wave front set of the trivial character is the trivial class.

Theorem 2.1 (Lusztig, Taylor [29, 33]). Each irreducible character of G has a unique wave front set.

We will denote by Wf (ρ) the wave front set of ρ. By definition it is the F -stable unipotent class of maximal dimension for which h γ

; ρ i 6 = 0 for some F -stable element u in the class. It was shown by Achar and Aubert [1, Thm. 9.1] that it is also maximal for the closure relation on unipotent classes.

Theorem 2.2 (Achar-Aubert, Taylor [33]). Let ρ ∈ Irr(G) and let C be an F -stable unipotent class. If u ∈ C

is such that h γ

; ρ i 6 = 0 then C ⊆ Wf (ρ).

Here, C denotes the Zariski closure of a conjugacy class C.

2.2. Unipotent support and duality. Given ρ ∈ Irr(G) and C an F -stable unipotent class of G , we denote by AV(C, ρ) = | C

|

P

g∈C^F

ρ(g) the average value of ρ on C

. We say that C is a unipotent support of ρ if C has maximal dimension for the property that AV(C, ρ) 6 = 0. Geck [14, Thm. 1.4] showed that every irreducible character of G admits a unique unipotent support whenever p is good for G. We will denote it by C

. It turns out that it coincides with the wave front set of the Alvis-Curtis dual (see [9, § 8]) of the character.

Theorem 2.3 (Lusztig, Taylor [29, 33]). Let ρ ∈ Irr(G) and ρ

∈ Irr(G) be its Alvis- Curtis dual. Then

C

= Wf (ρ

).

In [32, III.1] Spaltenstein studied an order-reversing map d on the set of unipotent classes partially ordered by inclusion of closures. When p is good, the image of d consists of the so-called special unipotent classes, and d restricts to an involution on this subset of classes.

By [29, § 11], unipotent supports of unipotent characters are special classes. They can be computed as follows: any family F in the Weyl group of G contains a unique special representation, which is the image under the Springer correspondence of the trivial local system on a special unipotent class C

. Then this class is the common unipotent support of all the unipotent characters in F . Now, if φ is the image of (C, 1) via the Springer correspondence, then φ ⊗ sgn is the image of a local system on d(C) via the Springer correspondence (see [2, § 3]). Moreover, that local system is trivial except for the special characters of exceptional families in type E

and E

(see for example [6, § 11.3 and § 12.7]).

Consequently, for any unipotent character ρ of G we have

(1) d(C

) = C

= Wf (ρ).

This property does not hold in general for other series of characters.

2.3. Cuspidal unipotent classes. Following Geck and Malle [20] we say that an F - stable unipotent class C of G is non-cuspidal if there exists a 1-split proper Levi subgroup L of G such that

• C ∩ L 6 = ∅ ;

• for all u ∈ C ∩ L, the natural map C

(u)/C

(u) −→ C

(u)/C

(u) is an isomor- phism.

Here recall that a 1-split Levi subgroup of (G, F ) is by definition an F -stable Levi com- plement of an F -stable parabolic subgroup of G. If no such proper Levi subgroup exists, we say that the class C is cuspidal. Note that cuspidality is preserved under the quotient map G → G/Z(G)

. In particular, when G has connected centre, a unipotent class C is cuspidal if and only if its image in the adjoint quotient G

(with same root system as G ) is cuspidal.

2.4. A progenerator. Recall that G is connected reductive in characteristic p and that ℓ 6 = p. In particular every generalised Gelfand–Graev representation of G is projective for O G.

When G = GL

, it is known from [18, Thm. 7.8] that any cuspidal kG-module N lifts to characteristic zero in a (necessarily) cuspidal KG-module. The latter is a constituent of some Gelfand–Graev representation Γ of G so that N is in the head of kΓ. We prove an analogue of this result for G of arbitrary type.

Theorem 2.4. Assume that p is good for G and that ℓ 6 = p. Let N be a cuspidal kG- module. Then there exists an F -stable unipotent class C of G which is cuspidal for G

and u ∈ C

such that Hom

(kΓ

, N) 6 = 0.

This results from the following version of a conjecture by Kawanaka that was later proved by Geck and H´ezard [16, Thm. 4.5].

Proposition 2.5. Assume that the centre of G is connected. Then the space of Z-valued

unipotently supported class functions on G is generated by { R

(γ

) } where L runs over

1-split Levi subgroups of G, and u over F -stable unipotent elements of cuspidal unipotent classes of L.

Proof. Let C

, . . . , C

be the F -stable unipotent classes of G, ordered by increasing di- mension. For each C

, we choose a system of representatives u

, u

, . . . of the G-orbits in C

. In [16, § 4], it is shown, under the assumption that p is large, that to each u

one can associate an irreducible character ρ

of G such that

(1) ρ

has unipotent support C

; (2) h D

(ρ

); γ

i = ± δ

for all s.

Thus the { D

(ρ

) } span the space of Z-valued unipotently supported functions on G.

Here D

denotes the Alvis–Curtis duality on characters (so D

(ρ) = ± ρ

in our earlier notation). Let us explain why the arguments in [16, § 4] can be generalised to the case of good characteristic. For a given i, the irreducible characters { ρ

, ρ

, . . . } are obtained as characters lying in a family of a Lusztig series belonging to an isolated element, and whose associated unipotent support is C

. When the finite group attached by Lusztig to the family is abelian (resp. isomorphic to S

), these characters are constructed using [15, Prop. 6.6] (resp. [15, Prop. 6.7]). As mentioned in [15, § 2.4], this requires a generalisation of some of Lusztig’s results on generalised Gelfand–Graev representations [29] to the case of good characteristic. This was recently achieved by Taylor [33]. Finally, when the finite group attached to the family is isomorphic to S

or S

, then C is a specific special unipotent class of F

or E

. In that case one can use the results in [8, § 4] which hold whenever p is good or G, again thanks to [33].

Now let us choose the system of representatives u

∈ C

in a minimal Levi subgroup L

given by [20, Thm. 3.2]. Then property (2) is still satisfied if we replace γ

by R

(γ

) by [20, Cor. 2.7] and [9, Thm. 8.11]. Moreover, for j < i we have h D

(ρ

); R

(γ

) i = 0 since ρ

vanishes on the support of D

R

(γ

)

. Indeed, by [20, Prop. 2.3], this support is contained in the union of unipotent classes C

satisfying C

⊂ C

. But if C

⊂ C

with dim C

≤ dim C

we would have C

= C

. Therefore the matrix

D

D

(ρ

); R

(γ

) E

G

j,r;i,s

is upper unitriangular, with identity blocks on the diagonal, hence invertible. Therefore the { R

(γ

) } also span the space of Z-valued unipotently supported functions on G.

Proof of Theorem 2.4. Let G ֒ → G e be a regular embedding, compatible with F , that is e

G has connected centre and same derived subgroup as G . Note that this restricts to an isomorphism on the variety of unipotent elements. To avoid any confusion, given a unipotent element u in G e we shall denote by e Γ

(resp. Γ

) the corresponding generalised Gelfand–Graev representation of G e (resp. G) and by e γ

(resp. γ

) its character. By construction we have Γ e

= Ind

Γ

. Fix a system of coset representatives g

, . . . , g

of G/G. Since e G E G, the Mackey formula yields an isomorphism of functors e

(2) Res

◦ Ind

≃

M

ad(g

).

In particular Res

e Γ

≃ L

Γ

.

Let N be a cuspidal simple kG-module. From (2) we deduce that Ind

N is a cuspidal k G-module. Let e M be a simple constituent of the socle of Ind

N and let ψ be its Brauer character. We denote by ψ

the unipotently supported class function which coincides with ψ on the set of unipotent elements. Then for every unipotent element u contained in a Levi subgroup L e of G e we have

h R

( e γ

); ψ

i

= h R

( e γ

); ψ i

,

which is zero whenever L e 6 = G e since ψ is cuspidal. Together with Proposition 2.5, we deduce that there exists an F -stable cuspidal unipotent class C of G e and u ∈ C

such that he γ

; ψ i

6 = 0. By construction of the generalised Gelfand–Graev characters there exist a unipotent subgroup U

of G e and a linear character χ

of U

such that e γ

= Ind

χ

. If k

denotes the 1-dimensional kU

-module on which U

acts by χ

then since U

is a p-group, hence an ℓ

-group, we get

dim Hom

(k Γ e

, M ) = dim Hom

(k

, Res

M )

= h χ

; Res

ψ i

= he γ

; ψ i

which is non-zero. This proves that there is a surjective map k Γ e

։ M . By re- striction to G we get surjective maps Res

k Γ e

։ Res

M ։ N . It follows that Hom

(kΓ

i

, N ) 6 = 0 for at least one i ∈ { 1, . . . , r } , thus proving the claim.

Given C an F -stable cuspidal unipotent class of G we will denote by Γ

the sum of all the generalised Gelfand–Graev representations corresponding to representatives of G- orbits in C

. If N is a simple kG-module then there exist a 1-split Levi subgroup L of G and a cuspidal kL-module M such that Hom

(R

(M ), N ) 6 = 0. Consequently, we can build a progenerator of kG using Theorem 2.4 for all 1-split Levi subgroups. For this, define L to be the set of pairs (L, C ) such that L is a 1-split Levi subgroup of G, and C is an F -stable unipotent class of L which is cuspidal for L

Corollary 2.6. Let G be connected reductive with Frobenius map F and assume that p is good for G. Then the module

P = M

(L,C)∈L

R

(kΓ

) is a progenerator of kG.

Remark 2.7. Except for groups of type A, even the multiplicities of unipotent characters in the character of P depend on q (e.g. the multiplicity of the Steinberg character).

Remark 2.8. One can use Theorem 2.4 to reprove that there is at most one unipotent

cuspidal module in GL

(q) (see for example [7, Thm. 5.21 and Cor. 5.23]). Indeed, the

only cuspidal class in PGL

(F

) is the regular class, therefore any cuspidal module must

appear in the head of the usual Gelfand–Graev representation Γ

for some (any) regular

unipotent element u. Since γ

has only the Steinberg character as a unipotent constituent,

it follows that Γ

has only one unipotent projective indecomposable summand, therefore at most one unipotent cuspidal module in its head.

For general linear groups again, the progenerator given in Corollary 2.6 involves only parabolic induction of usual Gelfand–Graev representations. This progenerator was al- ready studied in [4] and [3]. Our construction is a natural generalisation to arbitrary finite reductive groups.

3. ℓ-reduction of cuspidal unipotent characters

Recall that G is a connected reductive group defined over F

, with corresponding Frobe- nius endomorphism F . Throughout this section we will assume that p, the characteristic of F

, is good for G. In that case we can use the result of the previous section to show that under some mild assumptions on ℓ, cuspidal unipotent characters remain irreducible after ℓ-reduction.

3.1. Multiplicities in generalized Gelfand–Graev characters. Recall from § 2 that given a unipotent character ρ with unipotent support C

, any generalized Gelfand–Graev character γ

with u / ∈ d(C

) satisfies h γ

; ρ i = 0. We combine here results from [20, 16]

to compute h γ

; ρ i when ρ is cuspidal and u ∈ d(C

).

Given an F -stable unipotent class, recall that Γ

is the sum of all the Γ

’s where u runs over a set of representatives of G-orbits in C

. We will denote by γ

the character of K Γ

.

Proposition 3.1. Assume that G is simple of adjoint type. Let ρ be a cuspidal unipotent character with unipotent support C

. Then h γ

; ρ i = 1.

Proof. Let F be the family of unipotent characters containing ρ. Since the Alvis–Curtis dual ρ

of ρ is cuspidal, it must also lie in F , therefore d(C

) = C

by (1). By [20, Thm.

3.3] C

is a cuspidal class, and for u ∈ C

the finite group A

(u) is isomorphic to the small finite group associated to F as in [28, § 4]. In particular, the condition ( ∗ ) in [16, Prop. 2.3] holds for C

(and for d(C

)).

If G is of classical type or of type E

, then the claim follows from [16, Prop. 4.3] since A

(u) is abelian in that case.

Let u ∈ C

. If G is of exceptional type different from E

, then A

(u) equals S

(for type G

and E

), S

(for type F

) or S

(for type E

). In that case, by [26, Lemma 1.3.1], there is at most one local system on C

that is not in the image of the Springer correspondence and the multiplicity of ρ in γ

was first computed by Kawanaka [26]. An explicit formula can be also found in [8, § 6]. The assumption (6.1) in loc. cit. is satisfied and the projection of γ

on F can be explicitly computed from Lusztig’s parametrisa- tion. Namely, let now u ∈ C

be such that F acts trivially on A = A

(u). Then the conjugacy classes of A are in bijection with the G-orbits in C

. Given a ∈ A, we fix u(a) a representative of the corresponding orbit. Then the unipotent characters in F are parametrized by A-conjugacy classes of pairs (a, χ) where χ ∈ Irr(C

(a)) and γ

is given by [8, Thm. 6.5(ii)]

γ

= X

χ∈Irr(CA(a))

χ(1) ρ

.

Now the proposition follows from the fact that a cuspidal character always corresponds to a pair (a, χ) with χ(1) = 1. This can be checked case-by-case using for example [6,

§ 13.9] (thanks to [27, Thm. 1.15, § 1.16], the parametrization for

E

can be deduced from

the one for E

by Ennola duality).

3.2. ℓ-reduction. We can now state and prove the main application of the construction of our progenerator, viz. Theorem A, which we restate.

Theorem 3.2. Assume that p and ℓ are good for G and that ℓ ∤ | Z (G)

/Z

(G)

| . Then any cuspidal unipotent character of G

remains irreducible under reduction modulo ℓ..

Proof. We start with some reductions. Let G ֒ → G e be a regular embedding. Under the assumptions on ℓ, the unipotent characters form a basic set of the union of the unipotent blocks [13]. In addition, the restrictions of unipotent characters of G e to G remain irre- ducible (see [5, Prop. 17.4]). Therefore the unipotent parts of the decomposition matrix of G e and G are equal, so that we can assume without loss of generality that the centre of G is connected. Now, Z(G)

⊂ Z(G) and the unipotent characters are trivial on the centre. Consequently the unipotent part of the decomposition matrix of G and of G/Z (G)

are equal, and we can assume that G is semisimple of adjoint type. In that case G is a direct product G = G

× · · · × G

where each G

is simple of adjoint type, with F cyclically permuting the copies of G

in G

. Now the projection onto the first component G

→ G

induces a group isomorphism (G

)

≃ (G

)

mapping unipotent characters to unipotent characters. Therefore we can and we will assume without loss of generality that G is simple of adjoint type.

Let ρ be a cuspidal unipotent character of G, and C

be its unipotent support. It is a special and self-dual cuspidal unipotent class. Since ρ is cuspidal, it does not occur in any R

(γ

) for proper 1-split Levi subgroups L of G. In addition, if C is a cuspidal class different from C

then by [20, Thm. 3.3] we have d(C

) = C

⊂ C which forces h γ

; ρ i = 0 for every u ∈ C

. Finally, with the notation introduced before Corollary 2.6, it follows from Proposition 3.1 that h γ

; ρ i = 1. Consequently

X

(L,C)∈L

R

(γ

); ρ

= 1

which by Corollary 2.6 proves that ρ appears in the character of exactly one projective indecomposable module. In other words, the ℓ-reduction of ρ is irreducible.

Remark 3.3. If G is simple of adjoint type, then the previous proof shows that cuspidal unipotent characters remain irreducible after ℓ-reduction without any assumption on ℓ.

By a result of Hiss [23, Prop. 3.3], this has the following consequence which might be of interest for applications to representations of p-adic groups:

Corollary 3.4. Assume that p and ℓ are good for G and that ℓ ∤ | Z(G)

/Z

(G)

| . Then any unipotent supercuspidal simple kG-module is liftable to an O G-lattice.

4. Bounding Harish-Chandra series

Recall that G is a connected reductive group defined over F

, with corresponding Frobe-

nius endomorphism F . Throughout this section we assume that p, the characteristic of

F

, is good for G. The irreducible representations of G = G

are partitioned into Harish- Chandra series, but this partition is not known in general. In this section we give a necessary condition for a unipotent module to lie in a given series. It involves a numerical invariant coming from Lusztig’s a-function.

4.1. Lusztig’s a-function. Let ρ ∈ Irr(G). By [28, 4.26.3], there exist nonnegative integers n

, N

, a

with n

≥ 1 and N

≡ ± 1 (mod q) such that

dim ρ = 1 n

q

N

.

Moreover, by [28, 13.1.1], the integer a

is equal to the dimension of the Springer fibre at any element of the unipotent support C

of ρ. More precisely, for any u ∈ C

we have

(3) a

= 1

2 (dim C

(u) − rk(G)) = dim B − 1

2 dim C

where B is the variety of all Borel subgroups of G.

Let χ ∈ ZIrr(G). We define a

to be the minimum over the a-values of the irreducible unipotent constituents of χ (and ∞ if there is none). If ϕ ∈ IBr(G) is unipotent we define a

to be the a-value of its projective cover. By extension we set a

= a

if S is a simple kG-module with Brauer character ϕ.

4.2. Cuspidal classes and a -value. Cuspidal unipotent classes for simple groups of adjoint type in good characteristic were classified by Geck and Malle [20, Prop. 3.6].

From the classification we can see that they are all special classes of rather large dimension (compared to the rank of the group). We give here, for every classical type, a lower bound for the dimension of the Springer fibre over the dual of any cuspidal unipotent class.

Lemma 4.1. Assume that G is adjoint simple of classical type and rank m. Let C be an F -stable cuspidal unipotent class of G and d(C) be the dual special class. For u ∈ d(C) we set

a

:= 1

2 (dim C

(u) − m).

Then lower bounds for a

are as given in Table 1.

Proof. For type A

the regular class is the only cuspidal class by [20, Prop. 3.6]. Its dual class is the trivial class (see Section 2.2), for which a

=

(dim PGL

− m) = m(m + 1)/2.

For the other types, the cases with m ≤ 8 can easily be worked out explicitly, using the C hevie system [31] for example. Now assume that m ≥ 9. Let us first consider a partition λ = (λ

≥ · · · ≥ λ

> 0) of n. We denote by λ

= (1

, 2

, . . .) the conjugate partition in exponential notation, so that r

= λ

− λ

. Let r be the minimum of the non-zero r

’s and h be the number of non-zero r

’s (equivalently, the number of distinct parts in λ).

Then

(4) n = X

i≥1

ir

≥ X

i=1

ir ≥ h

2 r.

Table 1. Lower bounds for a

m 1 2 3 4 5 6 7 8 ≥ 9

A

1 3 6 10 15 21 28 36

m(m + 1)

2

A

− 1 3 4 4 7 9 10

(m + 1) q

m+1 2

− 1 B

− 1 3 6 7 7 11 13

m( √

2m + 1 − 4) C

− 1 4 5 7 7 12 15

m( √

2m − 2) D

,

D

− − − 3 7 9 12 13

m( √

2m − 2) −

√ 2m

Let R be the maximum number of non-zero (consecutive) equal parts in λ. For example R = 1 if λ has distinct parts. Then the number of non-zero parts s in λ satisfies s ≤ Rh ≤ R p

2n/r. Therefore by the Cauchy–Schwarz inequality we get

(5) X

i≥0

(r

+ r

+ · · · )

≥ 1 s

X

i≥0

(r

+ r

+ · · · )

= n

s ≥ n √ nr R √

2 .

Now to any conjugacy class C corresponds a partition λ coming from the elementary divisors of any element in C in the natural matrix representation of a classical group isogenous to G. If C is cuspidal, then depending on the type of G, the partition λ satisfies the following properties (see [20, Prop. 3.6]):

• type

A

: λ is a partition of m + 1 into distinct parts;

• type B

: λ is a partition of 2m + 1 into odd parts, and each part occurs at most twice;

• type C

: λ is a partition of 2m into even parts, and each part occurs at most twice;

• type D

and

D

: λ is a partition of 2m into odd parts, and each part occurs at most twice, and λ 6 = (m, m) in type D

.

(It was claimed erroneously in loc. cit. that for type D

, m odd, classes with label (m, m) are cuspidal, but in fact they do lie in a Levi subgroup of type A

.) Except in types D

and

D

, the dual of a special class corresponding to λ is the special class corresponding to the conjugate of λ (see [6, § 12.7 and § 13.4]). With the same notation as above, if C is cuspidal then R = 1 and r ≥ 1 for type

A

; R ≤ 2 and r ≥ 2 for type C

. For type B

we have R ≤ 2 and all nonzero r

are at least 2, except possibly that for the largest i (if λ

= 1), but it is easy to check that (4) continues to hold in this case with r = 2. Using the formula for dim C

(u) given in [6, § 13.1] together with (5) we deduce, for type

A

,

a

= 1 2

X

i≥0

(r

+ r

+ · · · )

− 1 − m

≥ 1

2 (m + 1)

r m + 1 2 − 1

;

for type B

a

= 1 4

X

i≥0

(r

+ r

+ · · · )

− X

iodd

r

− 2m

≥ 1 4

(2m + 1) √

2m + 1

2 − 4m − 1

≥ 1 4 m( √

2m + 1 − 4);

and for type C

a

= 1 4

X

i≥0

(r

+ r

+ · · · )

+ X

iodd

r

− 2m

≥ 1 4

2m √ 2m

2 − 2m

= 1 4 m( √

2m − 2).

Finally, we consider type D

. There are no elementary operation to deduce the effect of Lusztig–Spaltenstein duality on unipotent classes of groups of type D. However one can use (1) to compute a

for u ∈ d(C) from the a-value of the Alvis–Curtis dual of the unique special unipotent character with unipotent support C. Let

S =

a

. . . a

a

. . . a

be the symbol of an irreducible character of W (D

). Then by [30, (5.15)], for example (noting that the last “ − ” sign in that formula should be a “+”), the A-value of the corresponding unipotent principal series character ρ

is given by

A( S ) = X

i<j

max { a

, a

} − X

i=1

2i 2

− 2 X

i=1

a

+ 1 2

+ m

, which is easily seen to equal

A( S ) = X

i=1

(i − 1)a

− a

(a

+ 1) − i(i − 4)/4

− 7s/4 + m

= − 1 4

X

(2a

− i)

+ 4(2a

− i)

− 7s/4 + m

. Thus the a-value of the Alvis–Curtis dual ρ

= D

(ρ

) of ρ

is

a( S

) = m(m − 1) − A( S ) = − m + 1 4

X

(2a

− i)

+ 4(2a

− i)

+ 7s/4.

Now let λ be a partition of 2m labelling a cuspidal unipotent class C for D

. Note that by the description of cuspidal unipotent classes, λ has an even number of non-zero parts, say λ = (λ

≥ λ

≥ . . . ≥ λ

> 0). Also note that since all parts λ

are odd and each occurs at most twice, we have s ≤ √

2m. The Springer correspondent of C is obtained as follows, see [21, § 2D]: it is labelled by a symbol S as above with entries a

= (λ

− 1)/2 + i − 1 − ⌊

⌋ , 1 ≤ i ≤ 2s. Using that

X

(2a

− i)

≥ X

i=1

(2a

− i)

/2s and X

i=1

a

= m + s(s − 1)

we obtain

a( S

) ≥ − m + 1

8s (2m − 3s)

+ 2m − 3s + 7s/4

≥ m +

√ 2m 8 ( √

2m − 3)

− 5 4

√ 2m ≥ m( √

2m − 2)/4 −

√ 2m 8 .

4.3. Harish-Chandra series and a-value. It results from Lusztig’s classification of unipotent characters that there is at most one family containing a cuspidal unipotent character. In addition, the unipotent class attached to such a family is self-dual and has a large a-value compared to the rank of the group. We give a generalisation of this second statement to positive characteristic using the progenerator constructed in § 2.4 and the definition of the a-function in § 4.1.

Theorem 4.2. Let (L, X ) be a cuspidal pair of G. Let S be a simple kG-module lying in a Harish-Chandra series above (L, X ). Then there exists an F -stable unipotent class C of L which is cuspidal for L

and such that a

≤ a

.

Proof. By Theorem 2.4, there exists an F -stable unipotent class C of L which is cuspidal for L

and u ∈ C

such that the generalised Gelfand–Graev module Γ

maps onto X.

Let P

be the projective cover of X. Since Γ

is projective, it must contain P

as a direct summand. In particular, any irreducible unipotent constituent χ of a lift to O of P

is also a constituent of γ

. For such a character χ we have by definition C ⊆ Wf(χ). Recall from Theorem 2.3 that Wf (χ) coincides with the unipotent support of χ

, the Alvis–Curtis dual of χ. Since χ (and χ

) is unipotent, this support is a special unipotent class. By applying Spaltenstein’s duality d (see § 2.2), we deduce from (1) that d(C) ⊃ d(C

) = C

. Consequently a

≤ a

for every unipotent constituent χ of a lift to O of P

, therefore a

≤ a

.

It remains to see that a

≤ a

. This follows from the fact that P

is a direct summand of R

(P

) and that Harish-Chandra induction can not decrease the a-value (see for example

[28, Cor. 8.7]).

Consequently, we see from Table 1 that unipotent modules with “small” a-value must lie in “small” Harish-Chandra series, that is in Harish-Chandra series corresponding to Levi subgroups of small semisimple rank. In particular, cuspidal modules have a large a-value compared to the rank of the group.

5. Small Harish-Chandra series in even-dimensional orthogonal groups We give an application of Theorem 4.2 to the determination of the Harish-Chandra series of unipotent characters with small a-value in the finite orthogonal groups.

5.1. Small cuspidal Brauer characters. We first determine the cuspidal unipotent Brauer characters with small a-value.

Lemma 5.1. Let G be of type A

(q) (m ≥ 1), D

(q) (m ≥ 4) or

D

(q) (m ≥ 2), and

let ℓ be a prime not dividing 6q. If ϕ ∈ IBr(G) is a cuspidal unipotent Brauer character

with a

≤ 3 then ϕ occurs in Table 2. Any such ϕ is liftable to characteristic 0.

Table 2. Some cuspidal unipotent Brauer characters in types A

, D

and

D

G ϕ a

condition on ℓ A

(q) 1

1 ℓ | (q + 1)

A

(q) 1

3 ℓ | (q

+ q + 1), ℓ 6 = 3

2

D

(q) -.1 2 ℓ | (q

+ 1)

2

D

(q) -.2 3 ℓ | (q + 1) D

(q) D

3 always

The Brauer characters are labelled, via the triangularity of the decomposition matrix, by the Harish-Chandra labels of the ordinary unipotent characters.

Proof. Let ϕ ∈ IBr(G) be as in the statement. By Theorem 4.2, there exists a cuspidal unipotent class C of G with a

≤ 3. By Lemma 4.1, G must then be of one of the types A

, A

,

D

,

D

or D

. Since ℓ > 3 by assumption, for all of these groups it is known that the unipotent characters form a basic set for the union of unipotent ℓ-blocks of G [17], and that the decomposition matrix with respect to these is uni-triangular (see [24, 10, 22]). Moreover, from the explicit knowledge of these decomposition matrices, it follows that ϕ must be as in Table 2. The cuspidal Steinberg character of A

(q), ℓ | Φ

, is liftable to an ordinary cuspidal character by [18, Thm. 7.8], the cuspidal character -.2 of

D

(q) =

A

(q) is liftable by [10, Tab. 1] (there denoted 21

) when ℓ > 3, and the cuspidal unipotent character of D

(q) by [22, Prop. 5.1(iii)].

5.2. Small Harish-Chandra series. We can now determine those unipotent Harish- Chandra series that may contain Brauer characters of small a-value.

Lemma 5.2. Let G be of type A

(q) and ℓ | (q + 1).

(a) If m ≥ 2 then a

≥ 3 for all unipotent Brauer characters ϕ ∈ IBr(G) in the (A

, 1

)-Harish-Chandra series.

(b) If m ≥ 4 then a

≥ 4 for all unipotent Brauer characters ϕ ∈ IBr(G) in the (A

, 1

⊗ 1

)-Harish-Chandra series.

Proof. It follows from the known unipotent decomposition matrices (see [24, App. 1]), or from the known distribution into modular Harish-Chandra series, that a

= 3 for m = 2 in (a), and a

≥ 4 for m = 4 in (b) respectively. Since Harish-Chandra induction does not diminish the a-value (see for example [28, Cor. 8.7]), the claim also holds for all

larger m.

Proposition 5.3. Assume that n ≥ 4 and ℓ 6 | 6q. Then the unipotent Brauer characters ϕ of Spin

(q) with a

≤ 3 lie in a Harish-Chandra series as given in Table 3.

Proof. (a) First assume that G = Spin

(q). Let ϕ ∈ IBr(G) be an ℓ-modular Brauer

character of a-value at most 3. It lies in the Harish-Chandra series associated to a cuspidal

pair (L, λ), with λ ∈ IBr(L). By Theorem 4.2, there exists a unipotent class C of L,

cuspidal for L

, such that a(C) ≤ a

≤ a

≤ 3. Thus, since the a-value is additive over

outer tensor products of characters, by Lemma 5.1 and the shape of parabolic subgroups

Table 3. Harish-Chandra series in

D

(q) with small a-value

(L, λ) conditions

D

(q) ( ∅ , 1), (D

, D

) always (A

, 1

), (D

, .2) ℓ | (q + 1)

(A

, 1

⊗ 1

), (D

A

, .2 ⊗ 1

) ℓ | (q + 1), n = 4

2

D

(q) ( ∅ , 1) always

(A

, 1

), (

D

, -.2) ℓ | (q + 1) (

D

, -.1) ℓ | (q

+ 1) (A

, 1

⊗ 1

) ℓ | (q + 1), n = 5

(A

, 1

) ℓ | (q

+ q + 1), n = 4

of a Weyl group of type D

, L is of one of the types

A

(m ≤ 3), D

, D

A

, A

, or D

,

and λ is a (liftable) cuspidal unipotent Brauer character of L as in Table 2. We consider these Harish-Chandra series (L, λ) in turn.

Assume that 3 < ℓ | (q

+ q + 1). For n = 4 it follows from the known Brauer tree that the Brauer characters in the Harish-Chandra series of the ℓ-modular Steinberg character 1

of a Levi subgroup of type A

have a-value at least 7. Now for n ≥ 5, G contains a Levi subgroup of type D

, and so by the previous observation, the (A

, 1

)-series again only contains Brauer characters of a-value bigger than 6.

It remains to consider the case that ℓ | (q + 1). By Lemma 5.2 neither the A

-series nor the D

A

-series can contribute if n ≥ 5, since their Harish-Chandra vertex is contained in a Levi subgroup of G of type A

, resp. of type D

A

. Similarly, the Harish-Chandra vertex of the A

-series is contained in a Levi subgroup of type A

A

when n ≥ 6 and so the a-values are at least 4.

(b) Now let G = Spin

(q) and ϕ as in the statement. So by Theorem 4.2, ϕ lies in a Harish-Chandra series (L, λ) with a

≤ 3. Thus, by Lemma 5.1, L is of one of the types

A

(m ≤ 3),

D

, A

, or

D

,

and λ ∈ IBr(L) is a (liftable) cuspidal Brauer character of L as in Table 2. We consider the contributions by these Harish-Chandra series in turn. For 3 < ℓ | (q

+ q + 1) the known Brauer trees show that there is one Brauer character in the (A

, 1

)-series with a-value 3 for n = 4, but all a-values are at least 6 for n = 5 (and hence for all larger n).

Now consider the case when ℓ | (q + 1). Since

D

contains a Levi subgroup of type A

if n ≥ 6 we see by Lemma 5.2 that the Harish-Chandra series of type A

cannot contain our Brauer character in that case. Similarly the A

-vertex lies in a Levi subgroup of type

A

A

and so does not contain ϕ.

5.3. A triangular subpart of the decomposition matrix. We give an approximation

to the ℓ-modular decomposition matrix of Spin

(q) for certain unipotent Brauer charac-

ters of a-value at most 3. For simplicity, in view of Table 3 we only consider primes ℓ

not dividing q + 1. The induction basis is given by the case n = 5. Recall that unipo- tent characters of Spin

(q) are labelled by symbols of rank n and defect congruent to 0 modulo 4.

Proposition 5.4. The first eight rows of the decomposition matrix of the unipotent ℓ- blocks of Spin

(q) for primes ℓ > 5 with (ℓ, q(q + 1)) = 1 are as given in Table 4, where

(a, b, c, d, e) =

 

 

 

 

 

 

 

 

 



(0, 0, 0, 0, 0) when ℓ | (q − 1), (1, 0, 0, 0, 0) when ℓ | (q

+ q + 1), (0, 1, 0, 0, 0) when ℓ | (q

+ 1),

(0, 0, 1, 0, 0) when ℓ | (q

+ q

+ q

+ q + 1), (0, 0, 0, 1, 0) when ℓ | (q

− q + 1),

(0, 0, 0, 0, 1) when ℓ | (q

+ 1).

5 0

1

4 1

e 1

1,5 0,1

c . 1

3 2

b+d b . 1

0,1,2,4

−

. . . . 1

0,4 1,2

. e . . . 1

1,4 0,2

. b d b . . 1

2,4 0,1

b . a . . . . 1

Table 4. Partial unipotent decomposition matrices for Spin

(q)

Proof. For primes ℓ > 3 dividing q

+ q + 1, q

+ q

+ q

+ q + 1, q

− q + 1 or q

+ 1 the Sylow ℓ-subgroups of G = Spin

(q) are cyclic, and the Brauer trees of unipotent blocks are easily determined. The unipotent decomposition matrix for primes 3 < ℓ | (q − 1) is the identity matrix by a result of Puig, see [5, Thm. 23.12]. Finally, for (q

+ 1)

> 5 the relevant part of the decomposition matrix of G can be obtained by Harish-Chandra inducing PIMs from a Levi subgroup of type Spin

(q) determined in [11, Thm. 3.3].

Theorem 5.5. Let G = Spin

(q) with q odd and n ≥ 5, and ℓ > 5 a prime not dividing q(q + 1). Then the first eight rows of the decomposition matrix of the unipotent ℓ-blocks of G are approximated from above by Table 5, where k = n − 5 and a, b, c, d, e are as in Proposition 5.4.

Proof. By Proposition 5.4 we may assume that n ≥ 6. Let ϕ ∈ IBr(G) be an ℓ-modular constituent of one of the eight unipotent characters ρ

, . . . , ρ

listed in Table 5 (in that order). Then by Brauer reciprocity, ρ

is a constituent of the projective cover of ϕ. The degree formula [6, § 13.8] shows that a

≤ 3 for all i. Thus ϕ lies in one of the Harish- Chandra series (L, λ) given in Table 3. We consider these in turn.

The Hecke algebra for the principal series is the equal parameter Iwahori–Hecke algebra

of type D

, whose ℓ-decomposition matrix is known to be uni-triangular with respect to a

ρ a_ρ

n 0

0 1

n−1 1

1 e+k 1

1,n 0,1

2 c+k . 1

3 2

2 b+d+ke+^k₂

b+k . 1

0,1,2,n−1

−

3 . . . . 1

0,n−1 1,2

3 ke+^k₂

e+k . . . 1

1,n−1 0,2

3 k(c+e+k−1) b+k d+k b . . 1

2,n−1 0,1

3 b+kc+^k₂

. a+k . . . . 1

ps ps ps ps D₄ ps ps ps

Table 5. Approximate decomposition matrices for Spin

(q)

canonical basic set given by FLOTW bipartitions, see [19, Thm. 5.8.19]. All of the ρ

apart from ρ

are contained in this basic set, for all relevant primes ℓ. Thus the corresponding part of the decomposition matrix is indeed lower triangular. The given upper bounds on the entries in this part of the decomposition matrix are obtained by Harish-Chandra inducing the PIMs from our induction base of Spin

(q) in Proposition 5.4.

Next consider the Harish-Chandra series of the ordinary cuspidal unipotent character λ of a Levi subgroup L of type D

. Then R

(λ) only contains unipotent characters in the ordinary Harish-Chandra series of type D

, hence only ρ

, just once. So there is exactly one PIM in this series involving one of the ρ

, namely the projective cover of ρ

.

The following consequence seems relevant in view of determining the low-dimensional representations of orthogonal groups:

Corollary 5.6. Under the hypotheses of Theorem 5.5 assume that ϕ ∈ IBr(G) has pro- jective cover corresponding to the ith column of Table 5, 1 ≤ i ≤ 8. If ϕ(1) < q

− q

then i ≤ 3.

Proof. This is easily shown by inverting the given approximation of the decomposition

matrix in Table 5 and using the known degrees ρ

(1).

We now consider the same question for the non-split orthogonal groups Spin

(q). Re-

call that its unipotent characters are labelled by symbols of rank n and defect congruent

to 2 modulo 4. Since all considered characters lie in the principal series, they can also be

indexed by suitable bipartitions of n − 1.

Proposition 5.7. The first eight rows of the decomposition matrix of the unipotent ℓ- blocks of Spin

(q) for primes ℓ > 5 with (ℓ, q(q + 1)) = 1 are as given in Table 6, where

(a, b, c, d, e) =

 

 

 

 

 

 

 

 

 



(0, 0, 0, 0, 0) when ℓ | (q − 1), (1, 0, 0, 0, 0) when ℓ | (q

+ q + 1), (0, 1, 0, 0, 0) when ℓ | (q

+ 1), (0, 0, 1, 0, 0) when ℓ | (q

− q + 1), (0, 0, 0, 1, 0) when ℓ | (q

+ 1),

(0, 0, 0, 0, 1) when ℓ | (q

− q

+ q

− q + 1).

(4; -) 1 (31; -) b 1 (3; 1) e . 1 (2²; -) a . . 1 (21²; -) . b . . 1 (21,1) . d a b . 1 (2; 2) d . c . . . 1 (-; 4) b . . . 1

Table 6. Partial unipotent decomposition matrices for Spin

(q)

Proof. This is completely analogous to the proof of Proposition 5.4. For primes ℓ > 5 dividing q

+1 it follows from the known Brauer trees that there are two PIMs of

D

(q) in the

D

Harish-Chandra series, with unipotent parts (-; 3) and (1; 2) respectively. Harish- Chandra induction to

D

(q) shows that there is exactly one PIM in this series containing one of the ρ

, namely ρ

with label (-; 4) (see [11, Thm. 7.1]).

Theorem 5.8. Let G = Spin

(q) with q odd and n ≥ 5, and ℓ > 5 a prime not dividing q(q + 1). Then the first eight rows of the decomposition matrix of the unipotent ℓ-blocks of G are approximated from above by Table 7, where k = n − 5 and a, b, c, d, e are as in Proposition 5.7.

Proof. The eight unipotent characters ρ

, . . . , ρ

displayed in Table 7 all have a-value at most 3. Thus, arguing as in the proof of Theorem 5.5 and using Table 3 we see that any ℓ-modular constituent ϕ of the ρ

either lies in the principal series, or ℓ | (q

+ 1) and ϕ lies in the

D

-series.

First assume that ℓ | (q

+ 1) and consider the series of the cuspidal Brauer character (-; 1) of a Levi subgroup of type

D

. The Harish-Chandra induction of (-; 4) from

D

(q) to

D

(q) only contains ρ

among our ρ

, so there is at most one PIM in this series that contributes to the first eight rows of the decomposition matrix.

By [19, Rem. 6.7.12] the Iwahori-Hecke algebra H (B

; q

; q) for the principal series

has a canonical basic set indexed by suitable Uglov bipartitions. Now for n ≥ 6 the

bipartitions indexing ρ

, . . . , ρ

are Uglov, so these characters lie in the basic set. Hence,

there are at most eight PIMs involving one of the ρ

; since the unipotent characters form

a basic set for the unipotent blocks by [17], it must be exactly eight.