Towards a classification of the irreducible representations in non-describing characteristic of a

finite group of Lie type ^{∗}

Meinolf Geck, Gerhard Hiss and Gunter Malle

Lehrstuhl D für Mathematik, RWTH Aachen Templergraben 64, D-52062 Aachen, Germany and IWR, Universität Heidelberg Im Neuenheimer Feld 368, D-69120 Heidelberg, Germany

1 Introduction

Let G be a finite group with a split BN-pair of characteristic p satisfying the commutator relations. Let k be a splitting field for all subgroups of G, and let us first assume that k has characteristic 0 so that all kG-modules are semisimple. It is well-known that the classification of the simple kG-modules (up to isomorphism) into Harish-Chandra series is achieved by solving the following two problems.

(a) Classification of the cuspidal simple modules.

(b) Determination of the structure and the simple modules of the endomorphism algebras of modules obtained by Harish-Chandra induction of cuspidal modules from proper split Levi subgroups.

Roughly speaking, each simple kG-module can be parametrized by a triple (L, X, ϕ) where L ⊆ G is a split Levi subgroup, X is a cuspidal simple kL-module, and ϕ is a simple module of the endomorphism algebra of the Harish-Chandra induction of X. This follows from Fitting’s lemma on the correspondence between indecomposable direct summands of a given module and the simple modules of its endomorphism algebra, and the fact that in the “ordinary” situation all modules considered are semisimple. If G arises as the fixed point set of a connected reductive group with connected centre under a Frobenius map then this parametrization of the simple kG-modules by triples as above is completely known (due to Howlett–Lehrer and the work of Deligne–Lusztig and Lusztig).

∗This paper is a contribution to the DFG research project “Algorithmic Number Theory and Algebra”

Here, we are interested in the case that the characteristic of the field k is a prime ℓ different from the defining characteristic p of the group G. In [22], the second author has initiated the study of Harish-Chandra series of ℓmodular Brauer characters of G. Thus, there is a notion of cuspidal modules and Harish-Chandra induction and one can try to determine the structure of the endomorphism algebras of induced cuspidal modules. However, since the notions of indecomposable and simple modules substantially differ in the “modular” situation, the first problem now is the question whether or not there exists a bijection between the simple modules in a given Harish-Chandra series and the simple modules of the corresponding endomorphism algebra. If this were true, then we again would have a parametrization of the simple kG-modules by tripels as above. Another basic problem is then the determination of the structure and the simple modules of the “modular” endomorphism algebras.

The only groups for which complete results are known (in addition to explicitly worked-out examples of small rank) are the general linear groups (due to the work of Dipper–James, Dipper and Dipper–Du) where the above parametrization is established. In [18], we made some steps towards the classification of the cuspidal ℓ-modular Brauer characters of the finite unitary groups. We also formulated precise conjectures on the distribution of the Brauer characters of these groups into Harish-Chandra series for certain special choices of q and ℓ.

This paper is a continuation of the work begun in [18]. In our first main result (Theorem 2.4) we establish the above mentioned parametrization of the simple kG-modules by triples (L, X, ϕ) in complete generality. We next investigate the structure of endomorphism algebras of induced cuspidal modules. Following the approach initiated by Dipper and Fleischmann [6], we show (Theorem 3.12) that these algebras can be described in a similar fashion as those arising in characteristic 0. This is done by adapting the theory of Howlett–Lehrer [24] to our more general situation.

In Section 4 we apply these results to some series of classical groups, namely the unitary, symplectic and odd dimensional orthogonal groups. Let G = G(q) be a group in one of the latter two series. We show (Theorem 4.11) that if the multiplicative order of q modulo ℓ is odd, then the only simple unipotent cuspidal kG-modules are those arising from the ordinary unipotent cuspidal modules, which are of ℓ-defect 0 under the above hypothesis. A similar result is proved for a unitary group GU_{n}(q), if the multiplicative order of −q modulo ℓ is even (Theorem 4.11). In contrast to this, if ℓ divides q + 1, and if ℓ > n, there are as many cuspidal unipotent kG-modules for the unitary groups G = GU_{n}(q) as there can possibly be by [18, Proposition 6.8] (Theorem 4.12). The latter two results partly prove the conjectures of [18, Section 9]. Conjecture 9.6 of [18] is completely proved in case ℓ > n (Theorem 4.12). The principal ingredients in establishing these conjectures are the results of Sections 2 and 3 of this paper, the work of Dipper and James [8] on the representation theory of Hecke algebras of type B_{n}, and counting arguments.

In Section 5 we use similar arguments to determine the number of simple cuspidal unipotent modules for the exceptional groups of Lie type in certain special cases.

2 On Harish-Chandra series

We fix a finite group G with split BN-pair of characteristic p satisfying the commutator relations (see [2, Chapter 2]). Let k be a splitting field for all subgroups of G such that char(k) = ℓ ≠ p. All modules considered will be finitely generated left modules. We fix a Levi subgroup L ⊆ G, and denote by R_{L}^{G} the usual operation of Harish-Chandra induction. Let X be a cuspidal simple kL-module. Then the Harish-Chandra series corresponding to the pair (L, X) consists of the simple kG-modules which are composition factors of the head of the module R_{L}^{G}(X). It is well-known and easy to prove that, if char(k) = 0, then the modules in this series (up to isomorphism) are in bijective correspondence with the simple modules (up to isomorphism) of the endomorphism ring of R_{L}^{G}(X). It is the purpose of this section to establish this result in complete generality (this is already known to be true, for example, if X is the trivial module of a Borel subgroup or, more generally, if X has a “reduction stable” lift to characteristic 0; see (3.1) for the exact definition). This will be achieved by proving that Dipper’s hypothesis [4, (2.6)], for quotients of Hom-functors is satisfied.

(2.1) Let β : Y → X be a projective cover of X and denote by Z its kernel. We write X′ = R_{L}^{G}(X), Y′ = R_{L}^{G}(Y) and Z′ = R_{L}^{G}(Z). Then Y′ is a projective kG-module, and we have a surjective map β′ : Y′ → X′ with kernel Z′. We consider the map

f : End_{kG}(X′) → Hom_{kG}(Y′, X′), φ ↦ φ ∘ β′

which is obviously injective. In particular, we have [X′, X′]_{G} ≤ [Y′, X′]_{G}, where we write

[M_{1}, M_{2}]_{G} := dim_{k} Hom_{kG}(M_{1}, M_{2}),

for any given kG-modules M_{1}, M_{2}. Furthermore, given an element g ∈ G, we denote by ^{g}X the k^{g}L-module conjugate to X. As vector spaces X equals gX. The action of an element h ∈ ^{g}L on ^{g}X is given by h.x = g^{−1}hgx, x ∈ ^{g}X = X.

Lemma 2.2 The map f is surjective.

Proof. We only need to prove that [X′, X′]_{G} = [Y′, X′]_{G}. Using the Mackey formula for the operations of Harish-Chandra induction R_{L}^{G} and restriction ∗R_{L}^{G} we find, for any kL-module V (cf. [6, Lemma 2.7]),

[R_{L}^{G}(V), X′]_{G} = ∑_{w}[^{w}V , X]_{L}

where w runs over a certain set of elements of G which normalize L.

Let us first put V := Y. Since Y is a projective cover of X and X is simple, we have [^{w}Y , X]_{G} ≠ 0 if and only if ^{w}Y is isomorphic to Y, and this holds if and only if ^{w}X is isomorphic to X. If these conditions are satisfied then we have [^{w}Y , X]_{G} = 1 = [^{w}X, X]_{G}. Hence we get the same result as for the case V := X which proves [Y′, X′]_{G} = [X′, X′]_{G}. □

Corollary 2.3 The kernel of β′ is invariant under each endomorphism of Y′.

Proof. Let ψ ∈ End_{kG}(Y′). By Lemma 2.2 we can find φ ∈ End_{kG}(X′) such that β′ ∘ ψ = φ ∘ β′. Now recall that Z′ = ker(β′). Then we can deduce

β′(ψ(Z′)) = φ(β′(Z′)) = φ(0) = 0,

and so ψ(Z′) ⊆ ker(β′) = Z′. □

Corollary 2.3 has very strong implications on the structure of X′. These were first observed in [4, (2.32)]. The following result summarizes the consequences in our special situation.

Theorem 2.4 Recall that L ⊆ G is a Levi subgroup and X is a cuspidal simple kL-module. Denote by H the endomorphism ring of R_{L}^{G}(X).

(i) Every indecomposable direct summand of R_{L}^{G}(X) has a simple head, and any two direct summands are isomorphic if and only if their heads are.

(ii) There exists a bijective correspondence between the isomorphism classes of simple modules of H and the simple kG-modules in the Harish-Chandra series determined by (L, X).

(iii) The Cartan matrix of the algebra H records the multiplicities of the simple components of the head of R_{L}^{G}(X) as composition factors of the various indecomposable direct summands of R_{L}^{G}(X).

Proof. For the convenience of the reader let us briefly recall the main arguments in [4, (2.32)]. As before, we put X′ = R_{L}^{G}(X). Let π : P_{X′} → X′ be a projective cover of X′. By Corollary 2.3 and [4, (2.7)], the kernel of π is also invariant under each endomorphism of P_{X′}. So we have an induced surjective homomorphism of k-algebras End_{kG}(P_{X′}) → H whose kernel lies in the radical of End_{kG}(P_{X′}). Hence we can lift idempotents, and we see that π defines a bijection between the indecomposable direct summands of P_{X′} and those of X′. Using this and Fitting’s lemma on the correspondence between indecomposable direct summands of a module and the projective indecomposable modules of its endomorphism algebra, the proof is completed. □

(2.5) In order to classify the simple kG-modules (up to isomorphism) we are therefore now in a completely analogous position as in the case of ordinary representations. Namely, we have to solve the following two problems.

(a) Classify the cuspidal simple modules.

(b) Determine the structure and classify the simple modules of the endomorphism algebra of an induced cuspidal module.

In the first paper in this series [18], we made some first small steps towards (a). In the present paper, we shall say more about (b). Theorem 2.4 together with a knowledge of the simple modules of the endomorphism algebras also allows us to use inductive arguments to count the number of simple modules in the various Harish-Chandra series, hence in particular the cuspidal modules. This will be entirely analogous to the arguments Lusztig used in [28] in order to classify the irreducible representations (in characteristic 0) of the finite classical groups.

As a prototype of this method, we briefly sketch a counting argument which gives an alternative proof of the fact that G = GL_{n}(q) has at most one cuspidal unipotent simple module, and that there exists one if and only if n = 1 or n = eℓ^{i} for some i ≥ 0 where e is the multiplicative order of q modulo ℓ (compare [18, § 7]).

The result is certainly true for n = 1, so let us assume n > 1 from now on. Let L ⊂ G be a proper Levi subgroup, X be a cuspidal unipotent module of L, and H the endomorphism algebra of R_{L}^{G}(X). By induction on n, there exist non-negative integers m_{−1}, m_{0}, m_{1}, m_{2}, … such that m_{−1} + em_{0}+eℓm_{1}+eℓ^{2}m_{2}+… = n and L is a direct product of groups GL_{1}(q) (m_{−1} factors) and GL_{eℓ^{j}} (m_{j} factors, j ≥ 0). Here, we understand that m_{−1} = 0 if e = 1. By Theorem 2.4, there exists a bijection between the simple modules in the Harish-Chandra series determined by (L, X) and the simple modules of H. So we only need to count the number of simple modules of H. By [3], we have H = H_{m_{−1}} ⊗ H_{m_{0}} ⊗ H_{m_{1}} ⊗ ⋯, where each H_{m_{j}} is an Iwahori-Hecke algebra of type A_{m_{j}−1} with parameters all equal to q̄ (the residue class of q modulo ℓ) for j = −1, and parameters all equal to 1 for j ≥ 0 (see [loc. cit., Part I, Lemma 3.5]). So, by using [7], we conclude that the number of simple modules of G in the Harish-Chandra series determined by (L, X) equals

p_{e}(m_{−1})p_{ℓ}(m_{0})p_{ℓ}(m_{1}) ⋯,

where, for any positive integer f, we denote by p_{f}(m) the number of fregular partitions of m. Varying over all such proper L ⊂ G as above and noting that, by induction, each such Levi subgroup has precisely one cuspidal unipotent module, we find that the total number of non-cuspidal unipotent simple modules of G is given by the expression

∑_{(m_{−1},m_{0},m_{1},...)}(p_{e}(m_{−1})p_{ℓ}(m_{0})p_{ℓ}(m_{1}) ⋯),

where (m_{−1}, m_{0}, m_{1}, …) runs over all sequences of non-negative integers such that m_{−1} + em_{0} + eℓm_{1} + … = n (and with ∑_{i} m_{i} > 1 since L is proper). To proceed, we need the following purely combinatorial result which will also be used later on.

Lemma 2.6 Let m ≥ 0, r, s ≥ 1, and denote by p_{∞}(m) the number of partitions of m. Then

p_{∞}(m) = ∑_{(m_{−1},m_{0},m_{1},m_{2},...)}(p_{r}(m_{−1})p_{s}(m_{0})p_{s}(m_{1})p_{s}(m_{2}) ⋯)

where the summation is over all sequences (m_{−1}, m_{0}, m_{1}, m_{2}, …) of nonnegative integers such that m_{−1} + rm_{0} + rsm_{1} + rs^{2}m_{2} + … = m (and we understand that m_{−1} = 0 if r = 1, and that m_{1} = m_{2} = … = 0 if s = ∞).

Proof. This can be proved by using generating functions. We will work in the ring of formal power series in an indeterminate t over the rational numbers. The idea is to multiply both sides of the above equation by t^{m}, then sum over all m ≥ 0 and perform suitable manipulations on the resulting infinite sums. Thus, the left hand side becomes the well-known identity

∑_{m≥0} p_{∞}(m)t^{m} = ∏_{i≥1}(1 − t^{i})^{−1}.

The right hand side is the coefficient of t^{m} in the sum

∑_{(m_{−1},m_{0},m_{1},...)} (p_{r}(m_{−1})p_{s}(m_{0})p_{s}(m_{1}) ⋯) t^{m_{−1}+rm_{0}+rsm_{1}+...}

where (m_{−1}, m_{0}, m_{1}, …) runs over all sequences of non-negative integers with a finite number of non-zero terms. This expression can be transformed to the infinite product

( ∑_{m_{−1}≥0} p_{r}(m_{−1})t^{m_{−1}} ) ( ∑_{m_{0}≥0} p_{s}(m_{0})t^{rm_{0}} ) ( ∑_{m_{1}≥0} p_{s}(m_{1})t^{rsm_{1}} ) · · · .

Now we readily check that

∑_{m′≥0} p_{s′}(m′)t^{m′} = ∏_{i≥1}(1 + t^{i} + t^{2i} + … + t^{(s′−1)i}) = ∏_{i≥1} \frac{1 − t^{is′}}{1 − t^{i}} ,

for any s′ ≥ 1 and m′ ≥ 0. Substituting this into the above infinite product we obtain

∏_{i≥1} (\frac{1 − t^{ir}}{1 − t^{i}} · \frac{1 − t^{irs}}{1 − t^{ir}} · \frac{1 − t^{irs^{2}}}{1 − t^{irs}} ⋯) = ∏_{i≥1}(1 − t^{i})^{−1}

whence the result. □

By applying this with m := n, r := e, and s := ℓ to the above formula we see that our expression equals p_{∞}(n) − 1 if n = eℓ^{j} for some j ≥ 0, and p_{∞} otherwise. Since the number of both the unipotent ordinary and the unipotent Brauer characters of G equals p_{∞}(n), we see that there exists a cuspidal unipotent Brauer character of G if and only if n = eℓ^{j} for some j, as required.

In Section 4, we will use completely analogous arguments to derive results about cuspidal unipotent Brauer characters for some finite classical groups, under suitable conditions on q and ℓ.

We shall now discuss a further application of Theorem 2.4. Recall that the operation of Harish-Chandra induction is transitive, in the following sense. If M ⊆ G is a Levi subgroup such that L ⊆ M, then R_{L}^{G}(X) = R_{M}^{G}(R_{L}^{M}(X)). In some cases, this will allow us to reduce the problem of classifying modules in a given Harish-Chandra series to smaller groups.

Proposition 2.7 Recall that L ⊆ G is a Levi subgroup, X is a cuspidal simple kL-module, and H the endomorphism ring of R_{L}^{G}(X). Let M ⊆ G be a Levi subgroup such that L ⊆ M, and denote by E the endomorphism ring of R_{L}^{M}(X). If H ≅ E then the application of R_{M}^{G} defines a bijection between the indecomposable direct summands of R_{L}^{M}(X) and R_{L}^{G}(X), preserving composition series.

In particular, the application of R_{M}^{G} defines a bijection between the simple kM-modules and the simple kG-modules in the Harish-Chandra series corresponding to the pair (L, X).

Proof. Let R_{L}^{M}(X) = X_{1} ⊕ … ⊕ X_{r} where the X_{i} are indecomposable kMmodules. Transitivity of Harish-Chandra induction shows that R_{L}^{G}(X) = R_{M}^{G}(X_{1}) ⊕ . . . . . . ⊕ R_{M}^{G}(X_{r}). Since E ≅ H, the numbers of indecomposable direct summands of R_{L}^{M}(X) and R_{L}^{G}(X) are equal. Hence each R_{M}^{G}(X_{i}) must be indecomposable.

Now let 0 ⊂ X_{i1} ⊂ X_{i2} ⊂ … ⊂ X_{in_{i}} = X_{i} be a composition series of X_{i}. From the fact that induction of modules preserves exact sequences we deduce that 0 ⊂ R_{M}^{G}(X_{i1}) ⊂ R_{M}^{G}(X_{i2}) ⊂ … ⊂ R_{M}^{G}(X_{in_{i}}) = R_{M}^{G}(X_{i}) is a chain of submodules in R_{M}^{G}(X_{i}). Since E and H are isomorphic they have the same Cartan matrix. Using Theorem 2.4 we conclude that the number of terms in a composition series of R_{M}^{G}(X) and R_{L}^{G}(X) are the same. Hence the above chain of submodules in R_{L}^{G}(X_{i}) cannot be refined, i.e., it is a composition series.

The second statement follows from the fact that the X_{i} and the R_{M}^{G}(X_{i}) have simple heads. □

(2.8) We finally recall what happens in the case when k has characteristic 0. Let L ⊆ G be a Levi subgroup, X a cuspidal simple kL-module and H the endomorphism ring of R_{L}^{G}(X). By Fitting’s Lemma, there is a bijective correspondence between the simple kG-modules in the Harish-Chandra series determined by (L, X) and the simple H-modules. Being the endomorphism algebra of a semisimple module over a splitting field, the algebra H itself is semisimple and split. Howlett and Lehrer completely described the structure of H. It turns out that H is isomorphic to the group algebra of a certain finite group associated with L and X, twisted by a 2-cocycle which Howlett and Lehrer conjectured to be trivial. The conjecture is equivalent to the statement that R_{L}^{G}(X) has a simple constituent with multiplicity 1, and this is proved now in full generality if G arises as the fixed point set of a connected reductive group under a Frobenius map (see [15]).

An analogous question also arises in the general case when k has characteristic ℓ > 0:

Question 2.9 Assume that X is a cuspidal simple kL-module. Is it always true that there exists a simple kG-module with multiplicity 1 as a composition factor of the head of R_{L}^{G}(X)?

(2.10) We describe some cases in which (2.9) is known to have an affirmative answer. Let us assume that G arises as the set of fixed points of a connected reductive group under a Frobenius map.

(i) Let O be a discrete valuation ring in some large enough finite extension K of the field of ℓ-adic numbers such that the residue class field of O modulo its maximal ideal is k. Assume that X lifts to a reduction stable OG-lattice X̂ (see (3.1) below for the exact definition). It easily follows from the results in [10, Abschnitt 2.6], that Question 2.9 has an affirmative answer in this case. (We omit further technical details.) The assumptions are always satisfied, for example, if X is the trivial module of the Borel subgroup B ⊆ G.

(ii) Assume that G = GL_{n}(q). Then Dipper [3] has shown that Question 2.9 has an affirmative answer. Moreover, he explicitly describes the structure of the endomorphism algebra H as an Iwahori-Hecke algebra associated with a direct product of various finite Weyl groups of type A.

(iii) In Section 4 we shall show that this is also the case for cuspidal unipotent modules in the finite unitary, symplectic and odd-dimensional orthogonal groups.

3 Endomorphism rings

In this section we study the endomorphism ring of a Harish-Chandra induced cuspidal module. We continue with the notation of the previous section. As before, G is a finite group with a split BN-pair of characteristic p, satisfying the commutator relations. The Weyl group of G is denoted by W, the root system by Σ, and its fundamental system by Π. Let J ⊆ Π, and let L = L_{J} denote the corresponding standard Levi subgroup. We assume that k is a sufficiently large field of characteristic ℓ ≠ p. (In particular, all simple modules considered will be absolutely simple.) Finally, let X be a simple cuspidal kL-module.

Definition. Let N (L) := (N_{G}(L) ∩ N)L and W_{G}(L) := W (L) := N (L)/L. Then

W (L) ≅ {w ∈ W | w(J) = J} ≤ W,

and this is called the relative Weyl group of L in G. We also put N (L, X) := {g ∈ N (L) | ^{g}X ≅ X} and

W (L, X) := N (L, X)/L.

This is called the ramification group of R_{L}^{G}(X).

We are interested in the structure of the k-algebra

H(L, X) := End_{kG}(R_{L}^{G}(X)).

If k is the field of complex numbers, then the theory of Howlett and Lehrer [24], by giving a convenient presentation of H(L, X), completely determines its structure.

(3.1) In some special situations the Howlett–Lehrer theory can also be used in characteristic ℓ ≠ p. Let O denote a valuation ring of characteristic 0 with residue class field k. Suppose that X is liftable to an OL-lattice X̂ (i.e., an OL-module which is free as an O-module). Let K denote the field of fractions of O. Then K ⊗_{O} X̂ is a cuspidal KL-module. We may view R_{L}^{G}(X̂) as being a subspace of K ⊗_{O} R_{L}^{G}(X̂) ≅ R_{L}^{G}(K ⊗_{O} X̂). The Howlett–Lehrer basis B of End_{KG}(K ⊗_{O} R_{L}^{G}(X̂)) (see [24, §3] and (4) below) leaves the OG-lattice R_{L}^{G}(X̂) invariant. Following Dipper and Fleischmann [6], we call R_{L}^{G}(X̂) reduction stable with respect to B, if B is also a basis of End_{OG}(R_{L}^{G}(X̂)) and if k ⊗_{O} End_{OG}(R_{L}^{G}(X̂)) ≅ H(L, X). In this case the image B̄ of B is a k-basis of H(L, X), and the structure constants of H(L, X) with respect to B̄ are obtained by reducing the structure constants of End_{OG}(R_{L}^{G}(X̂)) (with respect to B) modulo the maximal ideal of O. In his PhD-thesis [10, Korollar 2.6.9], the first author has proved that R_{L}^{G}(X̂) is reduction stable if W (L, X) = W (L, K ⊗_{O} X̂).

It is easy to find examples of cuspidal kL-modules which are not liftable to OL-lattices. There are other examples of cuspidal kL-modules which are liftable, but do not have any reduction stable lift. An example for this has been discussed in [18, (8.6)]. Even worse, there exist cuspidal unipotent kL-modules which do not occur as a composition factor in the reduction modulo ℓ of any ordinary cuspidal module. An example is provided in the principal 5-block of the groups G_{2}(q), if 5 divides q + 1, but 5^{2} does not (see [21]).

It is for these reasons that we have to investigate H(L, X) in its own right. We wish to transfer those results and methods of [24] which only depend on the fact that p is invertible in k. For this purpose we rearrange the original proof of Howlett and Lehrer [24]. They introduce the parameters corresponding to certain distinguished basis elements at an early stage and use these parameters to derive structural properties of the endomorphism ring (for example [24, Lemmas (4.1),(4.6)]). To be able to introduce these parameters at this early stage it is necessary to work over a ground field of characteristic 0. In the following treatment we therefore delay the definition of the parameters to a later stage. We finally obtain a result similar to the main theorem of Howlett and Lehrer [24, Theorem 4.14]. We cannot, however, give a recipe to calculate the parameters in all cases. For reasons of simplicity we use the notations of [24]. (3.2) The structure of the ramification group W (L, X) is described in [24, §2]. Following [24], we put

Ω̂ := {a ∈ Σ \ J | w(J ∪ {a}) ⊆ Π for some w ∈ W },

and

v(a, J) := w_{J∪{a}}w_{J}, for a ∈ Ω̂,

where for J′ ⊆ Σ contained in some fundamental system we let W_{J′} denote the corresponding parabolic subgroup of W and w_{J′} the longest element in W_{J′}. Let

Ω := {a ∈ Ω̂ | v(a, J)^{2} = 1}.

For each a ∈ Ω, the corresponding element v(a, J) is contained in W (L). This follows from [23, Lemma 4]. Moreover, for w ∈ W (L) we have

wv(a, J)w^{−1} = v(w(a), w(J)),  (1)

and so the subgroup R(L) := ⟨v(a, J) | a ∈ Ω⟩ of W (L) is normal (see [23, Corollary 7]). Let V denote the Euclidean vector space affording the reflection representation of W, and V_{J} the subspace spanned by the roots corresponding to the reflections in J. Write V_{J}^{⊥} for the orthogonal complement of V_{J} in V. For root systems of finite reflection groups we follow the notion of Humphreys [26, 1.2]. In particular, the roots need not be unit vectors, neither do they need to span the underlying vector space. Then the image of Ω under the projection V → V_{J}^{⊥} is a root system in the Euclidean space V_{J}^{⊥} with Weyl group R(L) [23, Theorem 6]. Furthermore, if Σ^{+} is the positive system of roots determined by Π, then Ω ∩ Σ^{+} is projected onto a positive system. To make this last statement more precise, let for the moment v̄ denote the projection of v ∈ V to V_{J}^{⊥}. Then {ā | a ∈ Π \ J} is an ℝ-basis of V_{J}^{⊥}. Order this basis and consider the corresponding lexicographic ordering on V_{J}^{⊥}. Let a ∈ Ω and w ∈ W (L). Then a > 0 if and only if ā > 0. Also, w(a) > 0 if and only if w(ā) > 0. Finally, we put

Γ′ := {a ∈ Ω | v(a, J) ∈ W (L, X)},  (2)

and let Γ be any subset of Γ′ which is invariant under W (L, X). Let R_{Γ}(L, X) := ⟨v(a, J) | a ∈ Γ⟩, and C_{Γ}(L, X) := {w ∈ W (L, X) | w(Γ^{+}) = Γ^{+}}, where Γ^{+} := Γ ∩ Σ^{+}. From (1) it follows that R_{Γ}(L, X) is normal in W (L, X). Furthermore, R_{Γ}(L, X) is a reflection group in V_{J}^{⊥} whose root system is the projection of Γ to V_{J}^{⊥}. Its positive system (with respect to the ordering of V_{J}^{⊥} discussed above) is the projection of Γ^{+} = Γ ∩ Σ^{+}. The corresponding fundamental system is the projection to V_{J}^{⊥} of

Δ = {a ∈ Γ^{+} | v(a, J)(b) ∈ Γ^{+} for all b ∈ Γ^{+} \ {a}}.  (3)

(see [24, Lemma (2.7)] or [2, Lemma 10.7.6]). Some of the results in the following sections are proved for a general W (L, X)-invariant subset Γ ⊆ Γ′, others only for Γ′. If Γ = Γ′ we denote the set (3) of simple roots by Δ′.

We have the following semi-direct product decomposition.

Proposition 3.3 W (L, X) = R_{Γ}(L, X)C_{Γ}(L, X) is a semi-direct product, i.e., R_{Γ}(L, X) ⊴ W (L, X) and R_{Γ}(L, X) ∩ C_{Γ}(L, X) = 1.

Proof. See [23, Lemma 2] and [24, Lemma (2.7)]. □

It is convenient to note the following connection between the length function l in W, and the length function in R_{Γ}(L, X).

Lemma 3.4 For a_{1}, a_{2}, …, a_{n} ∈ Δ write v_{i} = v(a_{i}, J). If v_{1}v_{2} ⋯ v_{n} is a reduced expression in R_{Γ}(L, X), then l(v_{1}v_{2} ⋯ v_{n}) = l(v_{1})+l(v_{2})+…+l(v_{n}).

Proof. This follows from [24, Lemma (2.11)(i)] by induction on n. Let w = v_{2} ⋯ v_{n}. Since v_{1}v_{2} ⋯ v_{n} is reduced, we must have w^{−1}(a_{1}) ∈ Γ^{+}. But then w^{−1}(a_{1}) ∈ Σ^{+}, and so l(v_{1}w) = l(v_{1}) + l(w). □

(3.5) We are now going to introduce the basis B of the endomorphism ring H(L, X). Let X denote a kL-representation afforded by X. Since X is invariant in N (L, X), there is a projective representation X̃ of N (L, X) extending X. Let λ : W (L, X) × W (L, X) → k^{∗} denote the corresponding cocycle. We assume that X̃ is chosen in such a way that λ satisfies the conditions of [24, (3.4)]. (In order to achieve this normalization, we may need some additional square roots in k.) If X is extendible to N (L, X), we take X̃ to be one such extension. In this case, λ = 1.

For the following considerations it is convenient to realize R_{L}^{G}(X) as the set Hom_{kP} (kG, X), where P = P_{J} is the standard parabolic subgroup corresponding to J (see [6, (1.3)]). The action of G on R_{L}^{G}(X) is then given by

(gf)(a) = f(ag), f ∈ Hom_{kP} (kG, X), g ∈ G, a ∈ kG.

For w ∈ W (L, X) choose an element ẇ ∈ N mapping to w under the canonical epimorphism. If P = U_{J}L is the Levi decomposition of the parabolic subgroup P, write [J] := 1/|U_{J}| ∑_{u∈U_{J}} u (note that [J] ∈ kG since U_{J} is a p-group and ℓ, the characteristic of k, is different from p), and define the map

B_{w} : Hom_{kP} (kG, X) → Hom_{kP} (kG, X)

by

(B_{w}f)(a) := X̃(ẇ)f(ẇ^{−1}[J]a), f ∈ Hom_{kP} (kG, X), a ∈ kG. (4)

Then B_{w} is an element of H(L, X), it is independent of the choice of the representative ẇ ∈ N (see [24, Remark 2, p. 45]), and the set

{B_{w} | w ∈ W (L, X)}

is a k-basis of H(L, X) (see [6, Theorem 2.9]). Note that B_{1} is the identity.

Many of the results of [24] on the multiplication of the B_{w}’s carry over to our more general situation, since the calculations involve only denominators which are powers of p, and hence non-zero elements of k. For example, we have the following two results.

Proposition 3.6 Let w_{1}, w_{2} ∈ W (L, X) such that l(w_{1}w_{2}) = l(w_{1}) + l(w_{2}). Then B_{w_{1}}B_{w_{2}} = λ(w_{1}, w_{2})B_{w_{1}w_{2}}.

Proof. This follows from [24, Proposition (3.12)]. □

Proposition 3.7 Let a ∈ Π \ J such that v := v(a, J) ∈ W (L, X). Then there are α, β ∈ k, α ≠ 0 such that

B_{v}^{2} = αB_{1} + βB_{v}.

Proof. This is proved as in [24, (3.13-16)]. □

To recall the definition of the set of roots Γ of [24] we assume for the moment that the characteristic of k is 0. For a ∈ Γ′ let Σ_{a} denote the subsystem of Σ generated by J ∪ {a}. Then we define L_{a} := ⟨B ∩ N, U_{b} | b ∈ Σ_{a}⟩, where U_{b} is the root subgroup of G corresponding to b (L_{a} is denoted by M_{H} in [24, p. 49]). It is shown in [loc. cit.] that R_{L}^{L_{a}}(X) = X_{1}^{a} ⊕ X_{2}^{a} with two simple, non-isomorphic kL_{a}-modules X_{1}^{a} and X_{2}^{a}. Howlett and Lehrer define Γ in [loc. cit., (3.21)] by

Γ := {a ∈ Γ′ | dim_{k}(X_{1}^{a}) ≠ dim_{k}(X_{2}^{a})}.  (5)

Note that their definition of Γ′ agrees with the one in (2). They show that Γ is invariant under W (L, X) [loc. cit., (3.23)]. Also, a ∈ Γ′ \ Γ implies that B_{v(a,J)}^{2} = αB_{1} [loc. cit., (3.18) and p. 49]. This in turn simplifies the multiplication rules involving B_{v(a,J)} [loc. cit., (4.3)]. The statements of Lemmas (4.1) and (4.2) of [24] remain true if the set of roots Γ of [loc. cit.] is replaced by the larger set Γ′. Note that under this stronger hypothesis the last case in the proof of Lemma (4.1) does not occur. We return to the more general case that the characteristic of k is different from p. It is of course possible to define a W (L, X)-invariant set of roots Γ as in (5), but it is not clear whether we could also derive the consequence on the square of B_{v(a,J)} (cf. Lemma 3.17 below). We therefore continue with Γ′, observing that the (simpler) proofs of [24, (4.1), (4.2)] carry over to our situation. These two lemmas have the following important consequence.

Proposition 3.8 Let a ∈ Δ′. Then there is an element u ∈ W such that u(J) ⊆ Π, u(a) = b ∈ Π. Furthermore, there exists a kG-isomorphism

θu̇ : Hom_{kP} (kG, X) → Hom_{kP_{u(J)}}(kG, u̇X)

and an element γ ∈ k^{∗} such that the following diagram commutes:

[figure: Hom_{kP} (kG, X); Hom_{kP} (kG, X); Hom_{kP_{u(J)}}(kG, u̇X); Hom_{kP_{u(J)}}(kG, u̇X); B_{v(a,J)}; γB_{v(b,u(J))}; θu̇; θu̇]

Here, u̇ denotes an element of N mapping to u under the canonical homomorphism. The endomorphism B_{v(b,u(J))} of Hom_{kP_{u(J)}}(kG, u̇X) is defined by (4) with respect to the projective representation X̃^{u̇} of N (u̇L, u̇X) = u̇N (L, X).

Proof. The assertion follows from Equation (4.3.3) in the proof of [24, Proposition (4.3)(v)] (with θu̇ := B_{D,u̇}). This proof appeals to Lemma (4.1), but its weaker form suffices to obtain our result. □

Proposition 3.8 shows that every B_{v} with v = v(a, J), a ∈ Δ′, satisfies a quadratic relation as in Proposition 3.7. It is easy to replace the element B_{v} by a scalar multiple T_{v}, which satisfies a quadratic relation with β = α − 1. We write T_{1} = B_{1} in the following. In order to ensure that the quadratic equations below have solutions, we may have to pass to a finite extension of k (but see Lemma 3.20).

Lemma 3.9 Let a ∈ Δ′ and put v := v(a, J). Then there exists ξ ∈ k^{∗} and p_{a} ∈ k^{∗} such that T_{v} := ξB_{v} satisfies

T_{v}^{2} = p_{a}T_{1} + (p_{a} − 1)T_{v}.  (6)

If a multiple of B_{v} satisfies relation (6) with some number p′_{a}, then p′_{a} ∈ {p_{a}, 1/p_{a}}. Furthermore, p_{a} = 1 if and only if B_{v}^{2} is a multiple of the identity.

Proof. Let ξ be a solution of the quadratic equation

x^{2}α − xβ − 1 = 0.  (7)

It is easily checked that T_{v} = ξB_{v} satisfies (6) with p_{a} = ξ^{2}α. On the other hand, if κB_{v} satisfies (6) with some p′_{a}, then κ also is a solution of (7). Finally, −1/p_{a}T_{v} satisfies (6) with p_{a} replaced by 1/p_{a}. □

Lemma 3.10 Let the assumptions be as in Proposition 3.8. Suppose that some multiple of B_{v(b,u(J))} satisfies a quadratic relation (6) with parameter p_{b}. Then some multiple of B_{v(a,J)} satisfies (6) with the same parameter.

Proof. Let T_{v} be a scalar multiple of B_{v(b,u(J))} such that

T_{v}^{2} = p_{b}T_{1} + (p_{b} − 1)T_{v}.

Then the same quadratic relation holds for θ−1 ; u̇T_{v}θu̇ which, by the diagram in Proposition 3.8, is a scalar multiple of B_{v(a,J)}. The result now follows from Lemma 3.9. □

For the next result we let Γ ⊆ Γ′ be W (L, X)-invariant, and use the notation of (3.2).

Lemma 3.11 Let a, a′ ∈ Δ and put v := v(a, J), v′ := v(a′, J). Suppose that v and v′ are conjugate in R_{Γ}(L, X), and that some multiple T_{v} of B_{v} satisfies a quadratic relation (6) with parameter p_{a}. Then there is a multiple T_{v′} of B_{v′} which satisfies a quadratic relation (6) with the same parameter.

Proof. The two fundamental reflections v and v′ of the finite Coxeter group R_{Γ}(L, X) are connected on its Coxeter diagram by a finite path whose edge labels are odd integers. Suppose first that v and v′ lie on adjacent nodes and that the order of vv′ equals 2m + 1. Then (vv′)^{m}v = v′(vv′)^{m} and the elements on either side of this equation are reduced (in R_{Γ}(L, X)). Let w = (vv′)^{m}. Then, by Lemma 3.4, l(wv) = l(w) + l(v) = l(v′w). By Proposition 3.6, B_{w}B_{v} = γB_{v′}B_{w}, and so B_{w}T_{v} = δB_{v′}B_{w}, for some γ, δ ∈ k^{∗}. Lemma 3.9 now implies the assertion, since B_{w} is invertible by [24, Corollary (3.17)]. For the general case use induction on the length of the path connecting v and v′ on the Dynkin diagram. □

We shall show below how to determine the parameters p_{a} in some special cases.

The main theorem of Howlett and Lehrer [24, Theorem 4.14] can be reformulated for our more general situation as follows. Remember that H = H(L, X) denotes the kG-endomorphism ring of R_{L}^{G}(X). The ramification group W (L, X) has a decomposition as a semi-direct product W (L, X) = R_{Γ′}(L, X)C_{Γ′}(L, X) where R_{Γ′}(L, X) is a normal subgroup of W (L, X) and is a reflection group. We omit the subscript Γ′ in the following.

Theorem 3.12 (1) There is a 2-cocycle λ′ of W (L, X), cohomologous to λ, such that the following multiplication rules are satisfied for all w ∈ W (L, X) and x ∈ C(L, X):

(i) B_{w}B_{x} = λ′(w, x)B_{wx},

(ii) B_{x}B_{w} = λ′(x, w)B_{xw},

(2) For all a ∈ Δ′ there is a scalar multiple T_{v} = T_{v(a,J)} of B_{v(a,J)} such that the following conditions are satisfied:

(iii) T_{v}^{2} = p_{a}T_{1} + (p_{a} − 1)T_{v}, for some 0 ≠ p_{a} ∈ k.

(iv) Let v = v(a, J), v′ = v(a′, J) for a, a′ ∈ Δ′ and write m(v, v′) for the order of vv′. Then there is a sign ε(v, v′) such that

T_{v}T_{v′}T_{v} ⋯ = ε(v, v′)T_{v′}T_{v}T_{v′} ⋯

(m(v, v′) factors on each side). If v, v′ ∈ Δ′ are conjugate by an element in W (L, X), then p_{a} = p_{a′}. If p_{a} ≠ 1, or if m(v, v′) is odd, then ε(v, v′) = 1.

Proof. The existence of λ′ and the first two formulae follow from [24, Proposition (4.3)(i)].

By Proposition 3.8 and Lemmas 3.9 and 3.10 we can find the elements T_{v} and parameters p_{a} such that the quadratic relations (iii) are satisfied. Furthermore, by Lemma 3.11, we may choose these elements in such a way that p_{a} = p_{b}, whenever v(a, J) and v(b, J) are conjugate by an element in W (L, X).

From Lemma 3.4 and Proposition 3.6, it follows that

B_{v}B_{v′}B_{v} ⋯ = ε′(v, v′)B_{v′}B_{v}B_{v′} ⋯ (m(v, v′) factors on each side)

for some ε′(v, v′) ∈ k. Hence an analogous relation holds for the product T_{v}T_{v′}T_{v} ⋯ with some ε(v, v′) ∈ k. Suppose first that m(v, v′) = 2m is even. Then (T_{v}T_{v′})^{m} = ε(v, v′)(T_{v′}T_{v})^{m}. This implies that

T_{v}^{2}T_{v′}(T_{v}T_{v′})^{m−1} = ε(v, v′)^{2}(T_{v′}T_{v})^{m−1}T_{v′}T_{v}^{2}.  (8)

If m(v, v′) = 2m+1 is odd, then T_{v}(T_{v′}T_{v})^{m} = ε(v, v′)(T_{v′}T_{v})^{m}T_{v′}. It follows that

T_{v}^{2}(T_{v′}T_{v})^{m} = ε(v, v′)^{2}(T_{v′}T_{v})^{m}T_{v′}^{2}.  (9)

If m(v, v′) is odd, then v and v′ are conjugate by an element in W (L, X) and so p_{a} = p_{a′} in this case. Since the parameters are non-zero, and the elements (T_{v}T_{v′})^{m−1} and (T_{v′}T_{v})^{m} are scalar multiples of the basis elements B_{(vv′)^{m−1}} respectively B_{(v′v)^{m}}, it follows from the equations (8) respectively (9) and the quadratic relations that ε(v, v′)^{2} = 1. If p_{a} ≠ 1, these equations and the quadratic relations show that ε(v, v′) = 1. We finally show that we can choose the elements T_{v} in such a way that ε(v, v′) = 1, whenever m(v, v′) is odd. Suppose then that m(v, v′) is odd. By what we have shown so far, we may assume that p_{a} = p_{a′} = 1. Consider the set S′ of all nodes of the Coxeter graph of R(L, X) which are connected to v by a path whose edge labels are all odd. Then v′ ∈ S′ and the parameters corresponding to the elements of S′ are all equal to 1. Thus changing T_{s} to −T_{s} for s ∈ S′ does not affect the quadratic relation T_{s}^{2} = T_{1}. The subgraph spanned by S′ is a connected tree. Starting from a leaf of this tree, we substitute, if necessary, the basis elements T_{s} by their negatives, to ensure that ε(s, s′) = 1 for all s, s′ ∈ S which lie on adjacent nodes. This does not affect the signs ε(s, t) for s ∈ S′ and t ∉ S′, since for such a pair m(s, t) is even. □

With additional assumptions we can force the signs ε(v, v′) to be 1.

Corollary 3.13 Suppose that X is extendible to N (L, X) or that R_{L}^{G}(X) has an indecomposable direct summand with multiplicity 1. Then the restriction of the 2-cocycle λ′ to C(L, X) is cohomologous to the trivial 2-cocycle. Furthermore, ε(v(a, J), v(a′, J)) = 1 for all a, a′ ∈ Δ′, and the subalgebra of H = H(L, X) spanned by the elements B_{w}, w ∈ R(L, X), is isomorphic to the Iwahori-Hecke algebra (over k) corresponding to R(L, X) and parameters p_{a}, a ∈ Δ′.

Proof. If X is extendible to N (L, X), then, by our choice, λ is the trivial 2-cocycle, and the first assertion follows. Let v = v(a, J), v′ = v(a′, J) for a, a′ ∈ Δ′. There is nothing to prove if m(v, v′) is odd. Suppose therefore, that m(v, v′) = 2m is even. Thus (vv′)^{m} = (v′v)^{m}, and so (B_{v}B_{v′})^{m} = B_{(vv′)^{m}} = (B_{v′}B_{v})^{m} by Proposition 3.6. Since T_{v} is a scalar multiple of B_{v}, the second assertion follows. If R_{L}^{G}(X) has an indecomposable direct summand with multiplicity 1, then H has a 1-dimensional k-representation. Applying this 1-dimensional representation to the equations in Theorem 3.12(1), the first assertion follows. Take v and v′ as above and apply this representation to the equation in Theorem 3.12(iv). Since the elements T_{v} are mapped to non-zero elements of k, this forces ε(v, v′) to be 1.

The subalgebra spanned by the elements B_{w}, w ∈ R(L, X), has the same dimension and satisfies the same relations as the Iwahori-Hecke algebra. This completes the proof. □

(3.14) In this paragraph we show how to calculate the parameters p_{a} in some special cases. We continue with the notation of the previous paragraph. Let a ∈ Δ′, and write v := v(a, J). By Lemma 3.10 we may assume that a ∈ Π \ J. Let I := J ∪ {a}. Consider Hom_{kP} (kP_{I}, X) as subspace of Hom_{kP} (kG, X) = R_{L}^{G}(X) by identifying it with the set of elements of Hom_{kP} (kG, X) which vanish on the k-span of G \ P_{I}. Since v ∈ L_{I} ≤ P_{I}, the operator B_{v} leaves Hom_{kP} (kP_{I}, X) invariant. Clearly (cf. [24, Lemma (3.13)]), B_{v} and hence B_{v}^{2} are determined by their action on Hom_{kP} (kP_{I}, X). Since the unipotent radical of P_{I} is contained in the unipotent radical of P, which acts trivially on X, the latter space is isomorphic, as kP_{I}-module, to Hom_{k(P ∩L_{I})}(kL_{I}, X), which is the inflation of R_{L}^{L_{I}} (X) to P_{I}. To calculate p_{a}, it suffices therefore to consider the situation where G = L_{I} and W_{G}(L, X) = W_{G}(L) = ⟨v(a, J)⟩ has just two elements. Then H := H(L, X) is 2-dimensional with basis T_{1}, T_{v}. For the remainder of this paragraph we assume that G = L_{I}.

Lemma 3.15 Either R_{L}^{G}(X) is indecomposable, or else the direct sum of two non-isomorphic simple kG-modules.

Proof. Suppose that R_{L}^{G}(X) = X_{1} ⊕ X_{2} is decomposable. Since H = End_{kG}(X_{1} ⊕ X_{2}) is 2-dimensional, each X_{i} has a 1-dimensional endomorphism ring, and there are no non-trivial homomorphisms from X_{1} to X_{2} nor the other way. The set of composition factors of the head of R_{L}^{G}(X) equals the set of simple kG-modules in the Harish-Chandra series corresponding to (L, X), and this in turn equals the set of composition factors of the socle of R_{L}^{G}(X) ([22, Theorem 5.8], [25, Theorem 1.1] and [5, 5.2]). If Y is a simple module in the head of X_{1}, then it must also be in the socle of X_{1}, since there are no non-trivial homomorphisms from X_{1} to X_{2}. Since the endomorphism ring of X_{1} is 1-dimensional, this implies that Y = X_{1}. The same argument applies to X_{2}. □

Lemma 3.16 R_{L}^{G}(X) is indecomposable if and only if p_{a} = −1. In this case, ℓ divides the dimension of R_{L}^{G}(X).

Proof. It is readily checked that H has no non-trivial idempotent if p_{a} = −1. Suppose that p_{a} ≠ −1. Then

\frac{1}{p_{a} + 1}T_{1} + \frac{1}{p_{a} + 1}T_{v}

is a non-trivial idempotent of H. If R_{L}^{G}(X) is indecomposable, the only eigenvalue of T_{v} is −1. The trace of T_{v} as operator on R_{L}^{G}(X) equals 0 [2, Proposition 10.9.4], which proves the last assertion. □

Lemma 3.17 Suppose that R_{L}^{G}(X) = X_{1} ⊕X_{2} is semisimple. Then we may choose notation such that

p_{a}dim_{k}(X_{1}) = dim_{k}(X_{2}).  (10)

If ℓ does not divide dim_{k}(R_{L}^{G}(X)), then ℓ does not divide dim_{k}(X_{1}), and we have p_{a} = dim_{k}(X_{2})/dim_{k}(X_{1}).

Proof. Let ξ and η denote the two eigenvalues of T_{v} acting on R_{L}^{G}(X). The quadratic relation for T_{v} implies the relations

ξ(ξ + 1) = p_{a}(ξ + 1),  and  η(η + 1) = p_{a}(η + 1).

Since T_{v} is not a multiple of the identity, ξ ≠ η. We may therefore assume that ξ ≠ −1. This implies p_{a} = ξ ≠ −1, and also η = −1. The fact that T_{v} has trace 0 now implies (10).

Since p_{a} ≠ 0, we have that ℓ divides dim_{k}(X_{1}) if and only if ℓ divides dim_{k}(X_{2}). In this case ℓ divides dim_{k}(R_{L}^{G}(X)). □

If k has characteristic 0, the above Lemma reduces the determination of the parameters of H(L, X) to the problem of finding the dimensions of the irreducible constituents of R_{L}^{G}(X).

(3.18) There is another instant where one can reduce the calculation to a simpler situation. Suppose that M is a finite group with a split BN-pair of characteristic p. We consider the situation where G is replaced by the direct product G × M and L by L × M. If P is a parabolic subgroup of G with Levi complement L, then P ×M is a parabolic subgroup of G×M with Levi complement L×M. We still assume that W (L, X) = ⟨v⟩ has order 2. Let Y denote a cuspidal irreducible kM-module. Then W_{G×M}(L × M) = W (L) and W_{G×M}(L × M, X ⊗ Y) = W (L, X). Write b_{v} for the element of the endomorphism ring of R_{L×M}^{G×M}(X ⊗ Y) corresponding to v defined by (4) with respect to the obvious projective representation of N (L×M, X ⊗Y) = N (L, X) × M.

Lemma 3.19 There exists a k(G × M)-isomorphism

θ : Hom_{kP} (kG, X) ⊗_{k} Y → Hom_{k(P ×M)}(k(G × M), X ⊗_{k} Y)

such that the following diagram commutes:

[figure: Hom_{kP} (kG, X) ⊗_{k} Y; Hom_{kP} (kG, X) ⊗_{k} Y; Hom_{k(P ×M)}(k(G × M), X ⊗_{k} Y); Hom_{k(P ×M)}(k(G × M), X ⊗_{k} Y); B_{v} ⊗ id_{Y}; b_{v}; θ; θ]

Proof. Define θ by f ⊗ y ↦ f_{y} ∈ Hom_{k(P ×M)}(k(G × M), X ⊗_{k} Y) with f_{y}(g, m) = f(g) ⊗ my, for f ∈ Hom_{kP} (kG, X), y ∈ Y, g ∈ G, m ∈ M. It is straightforward to check all the assertions. □

We conclude this section with some remarks on the smallest field containing the parameter p_{a} of the k-algebra H.

Lemma 3.20 Let k_{0} ⊆ k be a subfield such that the projective k-represen-tation X̃ and all simple kG-modules in the head of R_{L}^{G}(X) can be realized over k_{0}. We assume that k_{0} ⊆ k is a separable extension. Then p_{a} ∈ k_{0}.

Proof. Let X_{0} be a k_{0}G-module such that X = k ⊗_{k_{0}} X_{0}, and H_{0} the endomorphism ring of R_{L}^{G}(X_{0}) so that H = k ⊗_{k_{0}} H_{0}. By our assumption on k_{0}, the elements B_{1}, B_{v} lie in H_{0}. Hence we have that

H_{0} ≅ k_{0}[t]/(t^{2} − βt − α)

where t is an indeterminate and α, β ∈ k_{0} are as in Proposition 3.7. Assume, if possible, that the polynomial t^{2} − βt − α is irreducible in k_{0}[t]. Then H_{0} is a semisimple algebra (even a field). Since k ⊇ k_{0} is a separable extension we conclude that H is a semisimple algebra. Using Lemma 3.15 we see that R_{L}^{G}(X) is a direct sum of two non-isomorphic simple kG-modules. Since these can be realized over k_{0} we deduce that R_{L}^{G}(X_{0}) is a direct sum of two non-isomorphic, absolutely simple k_{0}G-modules. Hence H_{0} ≅ k_{0} ⊕ k_{0}, a contradiction.

So the polynomial t^{2} − βt − α has a root in k_{0}. Then the equation x^{2}α − xβ − 1 = 0 also has a solution in k_{0}. The proof of Lemma 3.9 shows that p_{a} equals the product of α and the square of such a solution, hence p_{a} ∈ k_{0}. The proof is complete. □

We note that, from the beginning, we could take k to be an algebraic closure of the prime field of characteristic ℓ so that k is a separable extension of any of its subfields.

4 Application to some classical groups

In this section we consider the finite symplectic, unitary, and odd dimensional special orthogonal groups, and study the Harish-Chandra series of their unipotent ℓ-modular Brauer characters. We are not able to give complete answers but we can at least determine the number of cuspidal unipotent Brauer characters for special choices of q and ℓ. The counting argument that we will use is similar to the one which Lusztig used in [28] in order to classify the unipotent representations of finite classical groups.

(4.1) Throughout this section, we let q be a power of a prime p, and denote by G = G_{n}(q) one of the following groups.

(a)  GU_{n}(q) (any q, n ≥ 0),
(b)  Sp_{n}(q) (q a power of 2, n ≥ 0 even),
(c)  CSp_{n}(q) (symplectic similitudes, q odd, n ≥ 0 even),
(d)  SO_{n}(q) (q odd, n ≥ 1 odd),

In each case, we use the convention that G_{0}(q) is the trivial group, except for (c) in which case G_{0}(q) is 𝔽_{q}^{∗}. In all cases, G_{n}(q) is the group of rational points of a connected reductive group defined over 𝔽_{q} and with connected centre.

Our notation differs from Lusztig’s [28]. He writes Sp_{2n}(q), CSp_{2n}(q) and SO_{2n+1}(q). The reason for this change is that in some statements and proofs below we explicitly work with the natural matrix representations of these groups. Let ℓ be a prime, and (K, O, k) a splitting ℓ-modular system for all subgroups of G_{n}(q). In case (a) we let d denote the multiplicative order of −q modulo ℓ, and set the parameter δ = 2. In cases (b), (c), (d) we let d denote the multiplicative order of q modulo ℓ, and set the parameter δ = 1. Since we will also deal with general linear groups over the field with q^{δ} elements (which occur as direct factors in split Levi subgroups of G_{n}(q)) we let e be the multiplicative order of q^{δ} modulo ℓ. Thus d = e in the cases (b), (c), (d) whereas in case (a) we have

e = \{ \begin{matrix}d/2 & , if d is even, \\ d & , if d is odd. \end{matrix}

(4.2) (i) Let us write n = m′ + 2m, for some non-negative integers m′, m, and choose a partition λ ⊢ m. We then have a corresponding (split) Levi subgroup L = L_{m′,λ} ⊆ G_{n}(q) which is isomorphic to the direct product

G_{m′}(q) × GL_{1}(q^{δ})^{n_{1}} × GL_{2}(q^{δ})^{n_{2}} × ⋯

where λ = (1^{n_{1}}, 2^{n_{2}}, …). Every unipotent Brauer character φ of L can be written as a tensor product of unipotent Brauer characters of the various factors, and φ is cuspidal if and only if each factor in the tensor product decomposition is. (ii) In [18] we have defined the modular Steinberg character φ_{St} for every group G_{n}(q) as above. By [18, (6.9) and Theorem 7.6], only φ_{St} can possibly be a cuspidal unipotent Brauer character of GL_{r}(q^{δ}) (for r ≥ 2), and this happens if and only if r = eℓ^{i} for some i ≥ 0. If r = 1, then all characters of GL_{1}(q^{δ}) are cuspidal but only the trivial character (which we will also denote by φ_{St} in this case) is unipotent. Let us assume from now on that φ is a cuspidal unipotent Brauer character of L. Then we must have

λ = (1^{m_{−1}}, e^{m_{0}}, (eℓ)^{m_{1}}, (eℓ^{2})^{m_{2}}, …)

and φ has a tensor product decomposition

φ = φ′ ⊗ φ_{λ,St}, φ_{λ,St} := φ_{St}^{m_{−1}} ⊗ φ_{St}^{m_{0}} ⊗ φ_{St}^{m_{1}} ⊗ ⋯  (11)

where φ′ is a cuspidal unipotent Brauer character of G_{m′}(q) and the factors φ_{St}^{m_{i}} correspond to the various factors GL_{eℓ^{i}}(q^{δ})^{m_{i}}. In particular, the number of cuspidal unipotent Brauer characters of L = L_{m′,λ} is the same as for the factor G_{m′}(q). Note that, if e = 1 then we understand that m_{−1} = 0. (iii) We denote by W (L) the relative Weyl group of L in G = G_{n}(q) (see the definition in the introduction of Section 3). By [23] we know that W (L) is a direct product of finite Weyl groups of type B_{m_{i}}, namely

B_{m_{−1}} × B_{m_{0}} × B_{m_{1}} × B_{m_{2}} × · · · .

Let X be a kL-module affording φ. We denote by H the endomorphism algebra of R_{L}^{G}(X).

We first investigate the ramification group of R_{L}^{G}(X).

Proposition 4.3 The stabilizer of X in N (L) is the whole of N (L), and X is extendible to N (L).

Proof. The group W (L) is generated by the permutations of the factors GL_{r}(q^{δ}) of the same size, and an automorphism of order 2 of each factor GL_{r}(q^{δ}). Since all these groups have a unique cuspidal unipotent Brauer character, their tensor product is fixed under this action. Since W (L) induces the identity on the factor G_{m′}(q), the first assertion follows. To prove the second, remember that L is isomorphic to a direct product

G_{m′}(q) × GL_{1}(q^{δ})^{m_{−1}} × GL_{e}(q^{δ})^{m_{0}} × GL_{eℓ}(q^{δ})^{m_{1}} · · · .

Fix r ∈ {1, e, eℓ, …} and write s for the multiplicity of GL_{r}(q^{δ}) as a direct factor in the above decomposition of L. Thus s = m_{−1} for r = 1 and s = m_{j} for r = eℓ^{j}. Write L_{i} for the direct factors of GL_{r}(q^{δ})^{s}, 1 ≤ i ≤ s. Then for each i, there is an extension L̂_{i} := L_{i}.2 of L_{i} by a cyclic group of order 2 such that the groups L̂_{i} generate the direct product L̂_{1} × L̂_{2} × ⋯ × L̂_{s}. Furthermore, there is a subgroup S_{s} of N (L), isomorphic to the symmetric group on s letters, which permutes the groups L̂_{i}, 1 ≤ i ≤ s. Any element of S_{s} which fixes one of the L̂_{i} centralizes it. Thus the group generated by the L̂_{i} together with S_{s} is the wreath product ĜL_{r}(q^{δ}) ≀ S_{s}, where ĜL_{r}(q^{δ}) denotes the extension of the direct factor GL_{r}(q^{δ}) of L by the automorphism of order 2 introduced above. Then N (L) is isomorphic to a direct product

G_{m′}(q) × ĜL_{1}(q^{δ}) ≀ S_{m_{−1}} × ĜL_{e}(q^{δ}) ≀ S_{m_{0}} × ĜL_{eℓ}(q^{δ}) ≀ S_{m_{1}} ⋯

Let Y_{i} denote the Steinberg module of L_{i}, and put X_{r} := Y_{1} ⊗ Y_{2} ⊗ ⋯ ⊗ Y_{s}. Since the Steinberg character of L_{i} is invariant under the automorphisms of L_{i}, it follows that Y_{i} is invariant in L̂_{i}.

Choose an extension Y′_{1} of Y_{1} to L̂_{1}. Then for each i, there is a unique extension Y′_{i} of Y_{i} to L̂_{i} which is conjugate to Y′_{1} under S_{s}. Let X′_{r} = Y′_{1} ⊗ Y′_{2} ⊗ ⋯ ⊗ Y′_{s}. Then X′_{r} is invariant under S_{s}, and thus extends to ĜL_{r}(q^{δ}) ≀ S_{s} by a well known construction for modules of wreath products. It follows that X extends to N (L). □

Proposition 4.4 The algebra H = H(L, X) is isomorphic to an Iwahori-Hecke algebra associated with the Weyl group W (L). It is therefore a tensor product of Iwahori-Hecke algebras H_{i} of type B_{m_{i}}, for i = −1, 0, 1, 2, …, with parameters given by

\begin{matrix}B_{m_{−1}} : [figure: …; p_{1}; q̄^{δ}; q̄^{δ}; q̄^{δ}; q̄^{δ}; 1; 2; 3; m_{−1}] \\ B_{m_{i}} : [figure: …; p′_{1}; 1; 1; 1; 1; 1; 2; 3; m_{i}] \end{matrix}

where p_{1} and p′_{1} are elements in k^{∗}, and z̄ denotes the residue class of the integer z modulo ℓ. If ℓ > 2 then p_{1}, p′_{1} ∈ 𝔽_{ℓ^{2}}.

Proof. Since X is extendable to N (L, X) = N (L) we know by Theorem 3.12 that H is isomorphic to the tensor product of the Iwahori-Hecke algebras corresponding to the various direct factors of W (L). We consider the direct summand B_{m_{j}} for some −1 ≤ j. Let r = 1 if j = −1 and r = eℓ^{j} if 0 ≤ j. We first calculate the parameter corresponding to a node different from the first (in the numbering of the diagram in 4.4). By the considerations in Paragraph (3.21) and by Lemma 3.19, it suffices to assume that G = GL_{2r}(q^{δ}), that L = GL_{r}(q^{δ}) × GL_{r}(q^{δ}), and that X = X_{St} ⊗ X_{St} (where X_{St} denotes a simple module with Brauer character φ_{St}). Then R_{L}^{G}(X) has a reduction stable lift to a cuspidal OG-lattice (see [6, § 5]), so we can calculate the parameter as in characteristic 0, and reduce it modulo ℓ. The parameter in characteristic 0 equals (q^{δ})^{r}; in k this equals q̄^{δ} if r = 1, and 1 if r = eℓ^{j}.

To prove the last assertion we first observe that X and all simple modules in the head of R_{L}^{G}(X) can be realized over the prime field 𝔽_{ℓ}. This will follow from a well known result in modular representation theory, if we can show that the values of the corresponding Brauer characters are integers. By [16, Theorem 5.1], the ordinary unipotent characters form a basic set in the union of all unipotent ℓ-blocks. Since the values of the ordinary unipotent characters in a classical group are integers, this implies that the values of the unipotent Brauer characters are integers. An extension of X to a group in which L has index 2 and X is invariant, can be realized over 𝔽_{ℓ^{2}}. The result follows from Lemma 3.20. □

(4.5) The parameters corresponding to node 1 of the Dynkin diagrams in 4.4 are not so easy to determine. We find them only in special cases. First we remark that, in order to calculate these parameters, we may assume that n = m′ + 2r and G = G_{m′+2r}(q), that L = G_{m′}(q) × GL_{r}(q^{δ}), and that φ = φ_{0} ⊗φ_{St}, where φ_{0} is a cuspidal irreducible Brauer character of G_{m′}(q). Here, r = 1 or r = eℓ^{j} for some 0 ≤ j.

Lemma 4.6 Let G = G_{n}(q) = GU_{n}(q) as in (4.1a). Suppose that d = 1, and also that ℓ > n. Then the parameters corresponding to node 1 equal −1.

Proof. We place ourselves in the minimal situation discussed before. By Lemma 3.16 we have to show that R_{L}^{G}(X) is indecomposable. Suppose this is not the case. Then, by Lemma 3.15, R_{L}^{G}(X) is a direct sum of two simple kG-modules X_{1} and X_{2}. In particular, the Green vertex of X_{1} is contained in L.

On the other hand, by the classification of ℓ-blocks by Fong and Srinivasan [9], X_{1} is contained in the principal ℓ-block of G, since d = 1. Since ℓ > n, a Sylow ℓ-subgroup of G, and hence the defect group of the principal ℓ-block of G is abelian. By Knörr’s theorem [27], the Green vertex of X_{1} is equal to the defect group of the principal block. This is a contradiction, since ℓ divides the index of L in G [18, Lemma 8.2]. □

(4.7) We next aim at getting some information about the parameter p′_{1} in another special situation. Recall from [28, (8.11)], that G = G_{n}(q) in (4.1) has at most one ordinary cuspidal unipotent character, and that there exists one if and only if (note the notational difference to [28])

n = t(t + 1)/2 for some t ≥ 0  in case (4.1a) ,
n = 2t(t + 1) for some t ≥ 0  in cases (4.1b,c) ,
n = 2t(t + 1) + 1 for some t ≥ 0 in case (4.1d).

We shall study the situation in which this cuspidal unipotent character (if it exists) has ℓ-defect 0 and hence, remains irreducible under reduction modulo ℓ. The degree formulae in [loc. cit.] show that this holds under the following conditions.

d = 2e even (hence ℓ ≠ 2)  for G_{n}(q) as in (4.1a),
d = e odd, ℓ ≠ 2,  for G_{n}(q) as in (4.1b,c,d).

As in (4.5) write n = m′ + 2r. Let r′ = ℓ^{j} for some j ≥ 0 and put r = er′. Consider the Levi subgroup

L = G_{m′}(q) × GL_{r}(q^{δ})

of G = G_{m′+2r}(q).

The proof of the next result is inspired by the proof of [9, Proposition (3E)].

Lemma 4.8 Let the notation be as above. Then there is a Sylow ℓ-subgroup D of GL_{r}(q^{δ}) such that C_{G}(D) ≤ L and N_{G}(D) ≤ N (L).

Proof. Write L′ = GL_{r}(q^{δ}) ≤ L and L_{0} = G_{m′}(q) ≤ L. There is a standard Levi subgroup L_{1} × L_{2} × ⋯ × L_{r′} of L′ with each L_{i} isomorphic to GL_{e}(q^{δ}). Let S ≤ L′ be the set of (e × e)-block permutation matrices. Then S is isomorphic to the relative Weyl group in L′ of L_{1} × L_{2} × ⋯ × L_{r′}; it permutes the direct factors L_{i}.

Let T_{1} denote a Coxeter torus of L_{1}. Then each L_{i} contains a unique Coxeter torus T_{i} conjugate to T_{1} under the action of S. Let A_{i} denote the Sylow ℓ-subgroup of T_{i} and write A = A_{1}×A_{2}×⋯×A_{r′} and T = T_{1}×T_{2}×⋯×T_{r′}. Finally, choose a Sylow ℓ-subgroup S_{ℓ} of S. Then, by [30], D = AS_{ℓ} is a Sylow ℓ-subgroup of L′.

Since ℓ is odd, it follows from a result of Alperin [1, Theorem 2], that A is the unique maximal abelian normal subgroup of D. Hence A, and then also C_{G}(A) are normalized by N_{G}(D). By using an argument similar to the one in the last part of the proof of [18, Theorem 8.3], it follows that

C_{G}(A) = L_{0} × T.  (12)

In the symplectic and orthogonal case, one has to observe that for an ℓelement x in the Coxeter torus T_{1}, the sets of eigenvalues of x respectively x^{−1} are disjoint. But this follows from the assumption that d is odd. Equation (12) shows in particular that C_{G}(D) ≤ L. It also shows that C_{G}(A) = C_{G}(T) and so N_{G}(A) = N_{G}(T). From general results on the structure of the normalizer of a torus in a finite group of Lie type it follows that N_{G}(T)/C_{G}(T) is isomorphic to a wreath product C_{2e} ≀ S_{r′}, where C_{2e} denotes a cyclic group of order 2e and S_{r′} a symmetric group on r′ letters. For each 1 ≤ i ≤ r′, there is an element v_{i} in N (L_{0} × L_{1} × ⋯ × L_{r′}), centralizing L_{j} for j ≠ i and inducing an outer automorphism of order 2 on L_{i}. Let L̂_{i} = ⟨L_{i}, v_{i}⟩, and put N_{i} := NL̂_{i}(T_{i}), 1 ≤ i ≤ r′. Then N_{i}/T_{i} is a cyclic group of order 2e. Since S permutes the L̂_{i}, it also permutes the groups N_{i}, and so N_{G}(T) = L_{0} × (N_{1} × ⋯ × N_{r′})S. In particular, N_{G}(T) ≤ N (L_{0} × L_{1} × ⋯ × L_{r′}) = L_{0} × (L̂_{1} ⋯ × L̂_{r′})S. Let ls ∈ N_{G}(D), with l ∈ L_{0} × L̂_{1} ⋯ × L̂_{r′} and s ∈ S. Let us write x̄ for the image of an element x of N (L_{0} × L_{1} × ⋯ × L_{r′}) in W_{G}(L_{0} × L_{1} × ⋯ × L_{r′}). Then l̄ ∈ ⟨v̄_{1}, …, v̄_{r′}⟩ and l̄s normalizes the image S_{ℓ} of D. It follows that Sl̄ ; ℓ ≤ S. Since r′ is a power of ℓ, the Sylow ℓ-subgroup S_{ℓ} of S is transitive (in the natural permutation representation of S on r′ points). This implies that either l̄ = 1 or else that l̄ equals the product v̄_{1}v̄_{2} ⋯ v̄_{r′}. Since the latter generates W (L), the second assertion follows. □

Lemma 4.9 Assume that d and ℓ are as in (4.7). Suppose also that

m′ = t(t + 1)/2 for some t ≥ 0  in case (4.1a) ,
m′ = 2t(t + 1) for some t ≥ 0  in cases (4.1b,c) ,
m′ = 2t(t + 1) + 1 for some t ≥ 0 in case (4.1d),

and that φ′ in (11) is the reduction modulo ℓ of the ordinary irreducible cuspidal unipotent character (which is a defect 0 character, cf. (4.7)). Then we have p_{1} = q̄^{2t+1} and p′_{1} ≠ −1.

Proof. To prove the assertion for p_{1}, we may assume that G = G_{m′+2}(q), L = G_{m′}(q) × GL_{1}(q^{δ}). Also, X is liftable to an O-lattice X̂ such that R_{L}^{G}(X̂) is reduction stable. Then R_{L}^{G}(X̂) has two distinct ordinary irreducible constituents; the quotient of their degrees equals q^{2t+1}. We now turn to the proof that p′_{1} ≠ −1. We may assume that G = G_{m′+2r}(q), L = G_{m′}(q) × GL_{r}(q^{δ}), for r = eℓ^{j}, j ≥ 0. Furthermore, φ = φ′ ⊗ φ_{St}. Let X′ and X denote simple modules with Brauer characters φ′ respectively φ_{St}. Then X′ ⊗ X is the simple kL-module with character φ. To prove that p′_{1} ≠ −1, it suffices, by Lemma 3.16, to show that R_{L}^{G}(X′ ⊗ X) decomposes.

Let Ẑ denote the indecomposable direct summand of the Gelfand-Graev OGL_{r}(q^{δ})-lattice which has the ordinary Steinberg character as constituent. Let Ŷ denote the unipotent quotient of Ẑ (see [22, Section 6]). Thus Ŷ is the unique factor of Ẑ which has the sum of the unipotent constituents of Ẑ (which in this case is just the Steinberg character) as character. Let Z and Y denote the reductions modulo ℓ of Ẑ respectively Ŷ.

Let D be a Sylow ℓ-subgroup of the direct factor GL_{r}(q^{δ}) of L satisfying the conclusions of Lemma 4.8. Then D is a Green vertex of Ŷ, and thus also one of the OL-lattice X̂′ ⊗ Ŷ, where X̂′ denotes the lift of the defect zero module X′. This follows since the rank of Ŷ is not divisible by ℓ, and since X′ is a defect zero module of G_{m′}(q).

By Lemma 4.8, C_{G}(D) ≤ L and N_{G}(D) ≤ N (L). The index of L in N (L) is 2, and X′⊗X is invariant in N (L). Hence X′⊗X has 2 distinct extensions to N (L). (Remember that ℓ ≠ 2.) It follows that the projective cover X̂′ ⊗ Ẑ of X̂′ ⊗ X̂ also has 2 distinct extensions to N (L). Each of these extensions contains, with multiplicity 1, one of the two extensions of the ordinary character of X̂′ ⊗Ŷ. This implies that Ind_{L}^{N (L)}(X′ ⊗Y) has exactly two direct summands. Furthermore, for reasons of degrees, the Green vertex of each of the two direct summands of Ind_{L}^{N (L)}(X′ ⊗ Y) equals D.

We now apply Green correspondence between N (L) and G. It follows that Ind_{L}^{G}(X′ ⊗ Y) has exactly two direct summands with Green vertex D. The remaining direct summands (if any) have Green vertex strictly contained in D. On the other hand, Ind_{L}^{G}(X′ ⊗ Y) = R_{L}^{G}(X′ ⊗ Y) ⊕ Ind_{P}^{G}(S), where P is a parabolic subgroup of G which has L as Levi complement, and where S is a direct sum of kP -modules with Green vertex strictly contained in D (see [18, Lemma 2.4]). We conclude that R_{L}^{G}(X′ ⊗ Y) has two direct summands Y_{1} and Y_{2} with Green vertex D.

For i = 1, 2, let Z_{i} denote that direct summand of the projective module R_{L}^{G}(X′ ⊗ Z), which is a projective cover of Y_{i}. To prove that Z_{i} is indecomposable, we first observe that X′ ⊗ Y is the unipotent quotient of X′ ⊗ Z. Taking unipotent quotients commutes with Harish-Chandra induction [22, Lemma 6.1] and direct sums. Therefore, Y_{i} is the unipotent quotient of an indecomposable direct summand of R_{L}^{G}(X′ ⊗ Z). This direct summand is a projective cover of Y_{i}.

Suppose that R_{L}^{G}(X′ ⊗ X) is indecomposable. This means that the Harish-Chandra series of the pair (L, X′ ⊗ X) consists of a unique simple kGmodule which occurs in the head and in the socle of R_{L}^{G}(X′ ⊗ X). The space of homomorphisms from R_{L}^{G}(X′ ⊗ Z) to R_{L}^{G}(X′ ⊗ X) has dimension 2 (see the proof of Lemma 2.2) and the latter module is not simple (having a 2-dimensional endomorphism ring). Therefore, the projective cover of R_{L}^{G}(X′ ⊗ X) has multiplicity 1 as a direct summand of R_{L}^{G}(X′ ⊗ Z). It follows that the head of Z_{1}, say, does not belong to the Harish-Chandra series of (L, X′ ⊗ X).

Let M denote the head of Z_{1}. Then the semisimple vertex L_{0} of M is properly contained in L (see [18, Section 2]). There is a projective indecomposable module Z_{0} of L_{0} such that Z_{1} is a direct summand of R_{L_{0}}^{G}(Z_{0}). The ordinary irreducible constituents of the unipotent quotient Ŷ_{1} of Ẑ_{1} all lie in the Harish-Chandra series corresponding to (G_{m′}(q)×GL_{1}(q^{δ})^{r}, X̂′ ⊗1^{r}). Hence we must have L_{0} = G_{m′}(q) × S_{0}, where S_{0} is a proper Levi subgroup of GL_{r}(q^{δ}).

Now, if Y_{0} denotes the unipotent quotient of Z_{0}, then the unipotent quotient Y_{1} of Z_{1} is a direct summand of R_{L_{0}}^{G}(Y_{0}). However, the index of S_{0} in GL_{r}(q^{δ}) is divisible by ℓ (see [18, Lemma 7.2]), and so the Green vertex of Y_{1} is properly contained in D, a contradiction. □

(4.10) Let us return to the setting in (4.1) and (4.2). By Theorem 2.4, the number of ℓ-modular Brauer characters of G in the Harish-Chandra series determined by (L, φ) equals the number of simple modules (up to isomorphism) of H. We have seen that H = H_{−1} ⊗ H_{0} ⊗ H_{1} ⊗ H_{2} ⊗ ⋯ where H_{i} is the Iwahori-Hecke algebra associated with the Weyl group of type B_{m_{i}} and parameters given as in Proposition 4.4. We conclude that the number of simple modules of H is the product of the respective numbers for the algebras H_{i}. These numbers are not known in general, but from the results of Dipper and James in [8] we get at least complete information in some special cases.

To describe this, recall from (2.5) the notation p_{f}(m) for the number of fregular partitions of m. By convention, p_{f}(0) = 1 and p_{∞}(m) is the number of all partitions of m (which are, therefore, ∞-regular). Furthermore, we denote by q_{f}(m) the cardinality of the set

⋃_{0≤a≤m}{(μ, ν) | μ ⊢ a, ν ⊢ m − a and μ, ν are f-regular},

i.e., we have

q_{f}(m) = ∑_{(a,b)} p_{f}(a)p_{f}(b),

where a, b run over all non-negative integers such that a+b = m. The known results can now be described as follows.

(i) Let i ≥ 0 so that, by Proposition 4.4, all parameters except possibly p′_{1} equal 1.

If p′_{1} = −1 then, by [8, Remark 5.4], the simple H_{i}-modules are obtained by extending (in a unique way) the simple modules of the parabolic subalgebra of type A_{m_{i}−1}. Since this algebra is isomorphic to the group algebra of the symmetric group S_{m_{i}} we finally see that the number of simple H_{i}-modules equals p_{ℓ}(m_{i}), the number of ℓ-regular partitions of m_{i} (cf. [7]).

If p′_{1} ≠ −1 then, again by [8, Remark 5.4], the number of simple H_{i}-modules equals q_{ℓ}(m_{i}).

Thus, we have complete answers for i ≥ 0. Consequently, we know the number of simple modules of H in the case that e = 1 (since then m_{−1} = 0 by our conventions in (4.2)(ii)).

(ii) Now let i = −1 (and hence e > 1). In this case, complete results are known under the additional assumption that the element

f_{m_{−1}}(p_{1}, q̄^{δ}) = ∏_{j=−(m_{−1}−1)}^{m_{−1}−1}(p_{1} + q̄^{δj})

of [8, Definition 4.4], is non-zero. If this is the case then, by [8, Theorem 5.3], the number of simple H_{m_{−1}}-modules equals q_{e}(m_{−1}). (Note that e is the smallest integer s ≥ 0 such that 1 + q^{δ} + q^{2δ} + … + q^{δ(s−1)} ≡ 0 mod ℓ, since e > 1.)

Now we can prove the main results in this section. First let us recall that, if ℓ is a good prime for G then the number of unipotent Brauer characters equals the number of (ordinary) unipotent characters, by [16, Theorem 5.1]. By [28, Theorem 8.2], these numbers are explicitly known.

Theorem 4.11 (Cf. [18, Conjecture 9.2]). Assume that d and ℓ are as in (4.7). Then G_{n}(q) has a at most one cuspidal unipotent Brauer character, and there exists one if and only if n is as in (4.7) in which case it is the ℓ-modular reduction of the unique (ordinary) cuspidal unipotent character.

Proof. This will be done by induction on n. Step 1. Let L = L_{m′,λ} be a proper Levi subgroup of G and φ = φ′ ⊗ φ_{λ,St} as in (4.2). By induction, φ′ is the reduction modulo ℓ of an (ordinary) cuspidal unipotent character, hence m′ must have a form as in Lemma 4.9. We now claim that

x := f_{m_{−1}}(p_{1}, q^{δ}) ∈ k

is non-zero. This can be seen as follows. The element x is a product of terms of the form (q^{2t+1} + q^{δj}) for some t ≥ 0. Assume, if possible, that ℓ divides such a term. If δ = 2 then ℓ divides q^{a} + 1 for some odd a hence −q has odd order, contrary to our assumptions. If δ = 1 then ℓ divides q^{2t+1} + 1 or q^{a} + 1 for some even a. But then q has even order, again a contradiction. We conclude that x is non-zero. Using (4.10) we see that the total number of simple H-modules is given by

q_{e}(m_{−1})q_{ℓ}(m_{0})q_{ℓ}(m_{1}) ⋯

(with m_{−1} = 0 if e = 1). Step 2. We can now count the number of unipotent Brauer characters in the Harish-Chandra series of G determined by pairs (L_{m′,λ}, φ′ ⊗ φ_{λ,St}), with fixed m′ and φ′ but varying λ; it is given by the following expression.

∑_{(m_{−1},m_{0},m_{1},...)}(q_{e}(m_{−1})q_{ℓ}(m_{0})q_{ℓ}(m_{1}) ⋯)

where the summation is over all sequences (m_{−1}, m_{0}, m_{1}, …) such that m_{−1} + em_{0} + eℓm_{1} + … = m (and m_{−1} = 0 if e = 1). Now substituting the formula q_{f}(m_{i}) = ∑_{(a_{i},b_{i})} p_{f}(a_{i})p_{f}(b_{i}) (cf. (4.10)) and separating terms with a′_{i}s and b′_{i}s yields

∑_{a,b} ( ∑_{(a_{−1},a_{0},a_{1},...)} p_{e}(a_{−1})p_{ℓ}(a_{0})p_{ℓ}(a_{1}) ⋯) ( ∑_{(b_{−1},b_{0},b_{1},...)} p_{e}(b_{−1})p_{ℓ}(b_{0})p_{ℓ}(b_{1}) ⋯)

where the sum is over all a, b such that a + b = m and the sequences satisfy a_{−1} + ea_{0} + eℓa_{1} + … = a and b_{−1} + eb_{0} + eℓb_{1} + … = b (with a_{−1} = b_{−1} = 0 if e = 1). Applying Lemma 2.6 for m = a (resp. m = b) and with r = e and s = ℓ shows that our expression equals

λ(m) := ∑_{a,b} p_{∞}(a)p_{∞}(b)

where the sum is over all a, b such that a + b = m.

Step 3. Finally, varying m′ and noting that, by induction and (4.2(i)), L has only one cuspidal unipotent Brauer character, we see that the number of non-cuspidal unipotent Brauer characters of G is given by

∑_{m′,m} λ(m)

where the sum is over all pairs m′, m such that n = m′ + 2m and m > m′. By [28, (3.3), (8.10)] (for the symplectic and orthogonal groups) and [loc. cit., § 9] (for the unitary groups), this is the same formula which also counts the ordinary non-cuspidal unipotent characters of G. Hence the number of cuspidal unipotent (ordinary and Brauer) characters is 1, if n has the desired form, and 0 otherwise.

The proof is complete. □

Let us recall some facts and notation from [18, Sections 6, 9]. Due to the triangular shape of the decomposition matrix of G = GU_{n}(q), there is bijection between the unipotent ordinary and ℓ–modular characters of G. In order to obtain the triangular shape itself, a basic set of characters of projective modules is constructed by Harish-Chandra induction of the characters of generalized Gelfand–Graev representations (GGGR’s for short) of suitable Levi subgroups. This will be used several times in the arguments to follow without further reference.

We use the notation φ_{μ}, μ ⊢ n, to denote the unipotent Brauer characters of G. Let μ′ denote the partition conjugate to μ, and put |μ| := n if μ ⊢ n. The ℓ-adic expansion of a partition is defined in [18, Section 9(15)]. If μ ⊢ n we write μ = μ_{1} + 2μ_{2} where μ_{1} and μ_{2} are uniquely determined by the fact that μ′_{1} is 2-regular.

Theorem 4.12 (Cf. [18, Conjecture 9.6]). Assume that G = GU_{n}(q) and d = 1. Suppose that we always have p′_{1} = −1 in Proposition 4.4 (by Lemma 4.6, this holds for example if ℓ > n).

Then the unipotent character φ_{μ} of GU_{n}(q) lies in the ℓ-modular Harish-Chandra series (L, λ) with

L = GU_{m′}(q) × GL_{1}(q^{2})^{m_{0}} × GL_{ℓ}(q^{2})^{m_{1}} × ⋯ × GL_{ℓ^{r}}(q^{2})^{m_{r}},
λ = φ_{μ_{1}} ⊗ φ_{St} ⊗ φ_{St} ⊗ ⋯ ⊗ φ_{St},

where m′ = |μ_{1}| and the m_{i} are obtained from the ℓ-adic expansion of μ_{2} as m_{i} = |μ_{2}^{(i)}|. In particular, G has precisely p_{2}(n) cuspidal unipotent Brauer characters.

Proof. The idea is roughly the following: First, we find an upper bound for the semisimple vertex and source of φ_{μ}, using the results of [18] on GGGR’s. Secondly, Proposition 4.4 together with (4.10)(i) enables us to compute the precise number of characters in each series, and comparison then shows that the bound obtained previously is sharp.

Step 1. Let φ_{μ} be a unipotent ℓ-modular irreducible character of GU_{n}(q) and χ_{μ} the corresponding ordinary unipotent character. Let C be the class of unipotent elements u indexed by the partition μ′. Write μ = μ_{1} + 2μ_{2}, let m′ := |μ_{1}| and (k_{1}, …, k_{s}) be the parts of μ′_{2}. Then the class C is cuspidal in the Levi subgroup

M := GU_{m′}(q) × M′ where M′ := GL_{k_{1}}(q^{2}) × ⋯ × GL_{k_{s}}(q^{2}).

We assume that the (labels of the) unipotent characters of GU_{n}(q) and GU_{m′}(q) are ordered reversed lexicographically so that the trivial character is the first and the Steinberg character the last. The unipotent characters of M will be written in the form ψ = ψ_{ν} ⊗ψ′ where ψ_{ν} is a character of GU_{m′}(q) corresponding to the partition ν ⊢ m′ and ψ′ is a unipotent character of M′. These conventions refer both to ordinary and Brauer characters. We now claim that

(G, φ_{μ}) ≥ (GU_{m′}(q) × M′, ψ_{ν} ⊗ ψ′) with ν ≥ μ_{1},

where the above order relation is the one introduced in [22, Section 5] on pairs of Levi subgroups and irreducible Brauer characters thereof. This can be seen as follows. Denote the character of the GGGR of M associated with u by γ_{u}^{M}. Using the results in [18, (6.7)], we see that the projective cover of φ_{μ} occurs as a direct summand of R_{M}^{G}(γ_{u}^{M}). This shows that (G, φ_{μ}) ≥ (M, ψ) for some ψ, by [22, Proposition 4.1]. To derive the result about the form of ψ, we note that all ordinary constituents of R_{M}^{G}(γ_{u}^{M}) have labels λ with λ ≥ μ.

Step 2. Let φ_{μ} and M be as in Step 1. Since e = 1, the group GL_{f}(q^{2}) (where f ≥ 1) has a cuspidal unipotent Brauer character if and only if f = ℓ^{j} for some j ≥ 0. It follows that the set of Levi subgroups of GL_{k_{i}}(q^{2}) which have a cuspidal unipotent Brauer character contains a maximal element: the one corresponding to the ℓ-adic expansion of k_{i}. Now we consider the ℓadic expansions of the numbers k_{1}, …, k_{s}, collect together the corresponding factors GL_{ℓ^{j}}(q^{2}) of the same size, and conclude that the semisimple vertex of φ_{μ} is contained in the Levi subgroup

L = GU_{m′}(q) × L′ with L′ := GL_{1}(q^{2})^{m_{0}} × GL_{ℓ}(q^{2})^{m_{1}} × ⋯ × GL_{ℓ^{r}}(q^{2})^{m_{r}},

where the numbers m_{i} obtained in this way are precisely those in the statement of the theorem. Furthermore, any Levi subgroup of G, which possibly has a cuspidal unipotent Brauer character, is of this form and arises in this way (cf. [18, Sections 7, 8]). Using the result of Step 1 we also see that

(G, φ_{μ}) ≥ (GU_{m′}(q) × L′, ψ_{ν} ⊗ ψ_{St}) with ν ≥ μ_{1},  (13)

where ψ_{St} denotes the tensor product of the modular Steinberg characters of the various factors GL_{ℓ^{j}}(q^{2}).

Step 3. We now use some counting arguments to derive sharper results than the estimates in Step 2. We fix a Levi subgroup L as above. The number of unipotent Brauer characters φ_{μ} giving rise to the Levi subgroup L equals, by construction, the number of partitions μ ⊢ n with μ = μ_{1} + 2μ_{2}, μ′_{1} ⊢ m′ 2-regular, μ_{2} = (μ_{2}^{(0)}, μ_{2}^{(1)}, …), μ_{2}^{(i)} ⊢ m_{i} ℓ-regular. Thus, this number is given by the expression

p_{2}(m′)p_{ℓ}(m_{0})p_{ℓ}(m_{1}) · · · .  (14)

In particular, the number of unipotent Brauer characters of G with semisimple vertex equal to L is bounded above by this expression. Summation over all such L yields the upper bound

∑_{(m′,m_{0},m_{1},...)}(p_{2}(m′)p_{ℓ}(m_{0})p_{ℓ}(m_{1}) ⋯)

for the total number of unipotent Brauer characters of G, where the sum is over all sequences (m′, m_{0}, m_{1}, …) of non-negative integers such that m′ + 2m_{0} + 2ℓm_{1} + … = n. By Lemma 2.6, this expression equals p_{∞}(n). We conclude that the above estimates are sharp. This means that all unipotent Brauer characters φ_{μ} of G giving rise to L by the above construction have semisimple vertex L, and their number is given by (14).

Step 4. Let us fix L as above. Combining the results obtained so far we see that, given a unipotent character φ_{μ} with semisimple vertex L, the partition ν in (13) must be 2-regular, by [18, Proposition 6.8]. So the proof of the theorem will be complete if we can show that in fact ν = μ_{1}. This can be seen as follows.

First we consider those φ_{μ} with μ_{1} = (1)^{m′}. Since this is the largest index for a cuspidal character of L, we trivially see that ν = μ_{1}. Moreover, there are precisely p_{ℓ}(m_{0})p_{ℓ}(m_{1}) ⋯ characters of this form. Hence, using Theorem 2.4 and (4.10)(i), we conclude that we have found all unipotent Brauer characters φ_{μ} with semisimple vertex L and source ψ_{(1)^{m′}} ⊗ ψ_{St}.

Assume now that the claim is already proved for all φ_{μ} with μ_{1} (lexicographically) smaller than a certain 2-regular partition ν ⊢ m′. The φ_{μ} with μ_{1} = ν all have a source of the form ψ_{ν′} ⊗ ψ_{St} with ν′ ≥ ν. But those for which ν′ > ν are already accounted for by induction. Moreover, again by Theorem 2.4 and (4.10)(i), we have found the correct number of characters with semisimple vertex L and source ψ_{ν} ⊗ ψ_{St}.

This completes the proof. □

5 Application to exceptional groups

We demonstrate the applicability of the previous results by determining the number of ℓ-modular cuspidal unipotent characters of exceptional groups (with the exception of E_{8}, d = 3 or 5) for primes ℓ dividing a cyclotomic polynomial Φ_{d}(q) for d odd. The reason for not treating the cases E_{8}, d = 3, 5 lies in the fact that firstly there are cases where C(L, X) ≠ 1 (notation of Theorem 3.12), and secondly for some of the occurring endomorphism algebras we are not able to determine the precise values of the parameters. If that could be achieved, we could complete our results for d odd.

(5.1) The following table gives for each type of group G and each odd d with Φ_{d} dividing the order polynomial of G the different ℓ-modular Harish-Chandra series for ℓ dividing Φ_{d}(q) in terms of the pair (L, λ), where L is a (split) Levi subgroup of G and λ the label for a cuspidal unipotent simple kL-module X. In the following, we denote X by its label λ. For L ≠ G of classical type the label is the name of an ordinary unipotent character of L. This makes sense since for these Levi subgroups L ≠ G it is known that the ℓ-modular decomposition matrix is unitriangular, hence providing a natural bijection between ordinary and modular irreducible unipotent characters, see [3, 11, 20]. For these pairs (L, λ) we give the relative Weyl group W (L, λ) (see [23]) in terms of R(L, λ) and C(L, λ), the structure of the endomorphism ring of R_{L}^{G}(λ) and its number of irreducible representations. (Here it is understood that H(W; u) is the Hecke algebra of type W with parameters u over a suitable field of characteristic ℓ for ℓ dividing Φ_{d}(q).

(5.2) The results can be obtained as follows: By induction we may assume that the number of ℓ-cuspidal unipotent characters λ of every proper Levi subgroup L < G is known. It then remains to count the number of irreducible characters of the endomorphism ring of R_{L}^{G}(λ) to find the number of cuspidals. The structure of End(R_{L}^{G}(λ)) is almost given in Corollary 3.13 if R_{L}^{G}(λ) contains an indecomposable direct summand with multiplicity one (this condition is always easily checked in the examples). In fact, there is a subalgebra isomorphic to the Iwahori-Hecke algebra of type R(L, λ), acted upon by the finite group C(L, λ). The latter turns out to be either trivial or the cyclic group of order 2, and its action on the irreducible characters of the Iwahori-Hecke subalgebra then follows from the structure of W (L, λ) with Theorem 3.12. Finally, the parameters of H(R(L, λ)) can be computed by looking at minimal situations as described in (3.16). In most cases, we then have cyclic defect group, and the powerful theory of Brauer trees determines a big enough part of the decomposition matrix to compute the parameters from the degree formula in Lemma 3.19 (here we also use the explicit results on Brauer trees of exceptional Hecke algebras in [12]).

(5.3) The number of simple modules of a Hecke algebra is either already known or easy to compute (like the rank 2 cases and the types G_{2}, F_{4}, E_{6}, see [19, 13, 14]), or is determined by using the results of Dipper and James for the classical types A_{n} and B_{n} (see [7, 8] and also (4.10)), or has to be calculated by explicit computation using GAP [29] and the character tables of the Hecke algebras contained in the library of CHEVIE [17]. The latter cases involve Hecke algebras of type D_{n} and E_{7}, E_{8}. Let us briefly indicate how this is done. If H has type D_{n} (with n ≤ 7) or E_{7}, the character table of the corresponding generic Hecke algebra is explicitly known and contained in the library of CHEVIE. The number of simple modules of H is then obtained by computing the rank of the specialized character table. Here, we first specialize to a cyclotomic field, find a maximally linearly independent set of characters, compute the determinant of the matrix of values on suitably chosen basis elements, and consider then specializations to finite fields with a characteristic ℓ which divides this determinant. Thus, we can deduce results for all choices of q and ℓ such that q has fixed multiplicative order modulo ℓ.

We give some more indications on the hardest case, viz. G = E_{7}, d = 3, L of type E_{6}, where there are six cuspidal unipotent Brauer characters to deal with. The relative Weyl group has type A_{1}, and we have to decide whether the Harish-Chandra induced modules are semisimple or indecomposable (see Lemma 3.15). A Sylow ℓ-subgroup D of G lies in a uniquely determined maximal torus T of L of order q^{2} + q + 1. Thus the centralizer respectively normalizer of D coincides with the centralizer respectively normalizer of T. These are contained in L respectively N (L) and so the assumptions of Proposition 2.9 of [18] are satisfied. This shows that the Harish-Chandra induced modules have two irreducible constituents. So by Lemma 3.16 the parameter in the Hecke algebra must be different from −1.

We remark that the result for G_{2} is already implicit in [21], while the one for ^{3}D_{4} is explicitly given in [10].

In the tables occur some parameters p_{i} ∈ 𝔽_{ℓ}^{∗} \ {−1}, i = 1, …, 8, which we could not determine explicitly. Also, note that H(G_{2}; 1, q) ≅ H(A_{2}; q) : 2.

Table 5.4. Modular d-Harish-Chandra series for exceptional groups, d odd, ℓ |̸|W |.

G d  L  λ  R(L, λ) C(L, λ) H  #irr.
G_{2} 3  1  1  G_{2}  1  H(G_{2}; q, q)  5

+ 5 cuspidals

3D_{4} 3 1 1 G_{2} 1 H(G_{2}; q, 1) 4

+ 4 cuspidals

F_{4} 3  1  1  F_{4}  1  H(F_{4}; q, q)  15
A_{2}  St  G_{2}  1  H(G_{2}; q, 1)  4
Ã_{2}  St  G_{2}  1  H(G_{2}; q, 1)  4
B_{2}  c  B_{2}  1  H(B_{2}; 1, 1)  5

+ 9 cuspidals

E_{6} 3  1  1  E_{6}  1  H(E_{6}; q)  13
A_{2}  St  A_{2} × A_{2} 2  H(A_{2}; q)^{⊗2} : 2  5
A_{2} × A_{2} St × St G_{2}  1  H(G_{2}; q, 1)  4
D_{4}  c  A_{2}  1  H(A_{2}; q)  2

+ 6 cuspidals

5  1  1  E_{6}  1  H(E_{6}; q)  23
A_{4}  St  A_{1}  1  H(A_{1}; q)  2
D_{4}  c  A_{2}  1  H(A_{2}; q)  3

+ 2 cuspidals

9  1  1  E_{6}  1  H(E_{6}; q)  24
D_{4}  c  A_{2}  1  H(A_{2}; q)  3

+ 3 cuspidals

2E_{6} 3  1  1  F_{4}  1  H(F_{4}; q, q^{2})  15
A_{2}  St  G_{2}  1  H(G_{2}; q, 1)  4
Ã_{2}  St  G_{2}  1  H(G_{2}; q^{2}, 1)  4
2A_{5}  c  A_{1}  1  H(A_{1}; 1)  2

+ 5 cuspidals

E_{7} 3  1  1  E_{7}  1  H(E_{7}; q)  32
A_{2}  St  A_{5}  2  H(A_{5}; q) : 2  14
A_{2} × A_{2} St × St G_{2} × A_{1} 1  H(G_{2}; q, 1) ⊗ H(A_{1}; 1) 8
D_{4}  c  C_{3}  1  H(C_{3}; q, q)  4
E_{6}  c_{1}, c_{2} A_{1}  1  H(A_{1}; 1)  2 + 2
E_{6}  c_{3}, …, c_{6} A_{1}  1  H(A_{1}; p_{i}), i = 1, …, 4 8

+ 6 cuspidals

G d L  λ R(L, λ) C(L, λ) H  #irr.
5 1  1 E_{7}  1  H(E_{7}; q)  54
A_{4} St A_{2}  2  H(A_{2}; q) : 2  6
D_{4} c C_{3}  1  H(C_{3}; q, q^{4})  10
E_{6} c_{1}, c_{2} A_{1}  1  H(A_{1}; q^{4})  2 + 2

+ 2 cuspidals

7 1  1 E_{7}  1  H(E_{7}; q)  58
D_{4} c C_{3}  1  H(C_{3}; q, q^{4})  10
A_{6} St A_{1}  1  H(A_{1}; p_{5})  2
E_{6} c_{1}, c_{2} A_{1}  1  H(A_{1}; q^{2})  2 + 2

+ 2 cuspidals

9 1  1 E_{7}  1  H(E_{7}; q)  58
D_{4} c C_{3}  1  H(C_{3}; q, q^{4})  10
E_{6} c_{1}, c_{2} A_{1}  1  H(A_{1}; 1)  2 + 2
E_{6} St A_{1}  1  H(A_{1}; p_{6})  2

+ 2 cuspidals

E_{8} 7 1  1 E_{8}  1  H(E_{8}; q)  108
D_{4} c F_{4}  1  H(F_{4}; q, q^{4})  25
A_{6} St A_{1} × A_{1} 1  H(A_{1}; q) ⊗ H(A_{1}; p_{7}) 4
E_{6} c_{1}, c_{2} G_{2}  1  H(G_{2}; q, q^{9})  6 + 6
E_{7} c_{1}, c_{2} A_{1}  1  H(A_{1}; q^{15})  2 + 2

+ 13 cuspidals

9 1  1 E_{8}  1  H(E_{8}; q)  106
D_{4} c F_{4}  1  H(F_{4}; q, q^{4})  25
E_{6} c_{1}, c_{2} G_{2}  1  H(G_{2}; q, q^{9})  6 + 6
E_{6} St G_{2}  1  H(G_{2}; q, p_{8})  6
E_{7} c_{1}, c_{2} A_{1}  1  H(A_{1}; q^{15})  2 + 2

+ 13 cuspidals

15 1  1 E_{8}  1  H(E_{8}; q)  110
D_{4} c F_{4}  1  H(F_{4}; q, q^{4})  25
E_{6} c_{1}, c_{2} G_{2}  1  H(G_{2}; q, q^{9})  5 + 5
E_{7} c_{1}, c_{2} A_{1}  1  H(A_{1}, q^{15})  2 + 2

+ 17 cuspidals

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