Cuspidal unipotent Brauer characters^{∗}

Meinolf Geck, Gerhard Hiss and Gunter Malle

Lehrstuhl D für Mathematik, RWTH Aachen Templergraben 64, D–5100 Aachen, Germany and IWR, Universität Heidelberg Im Neuenheimer Feld 368, D–6900 Heidelberg, Germany

1 Introduction

Let q be a prime power and 𝔽̄_{q} be the algebraic closure of the finite field 𝔽_{q} with q elements. Let G = GL_{n}(𝔽̄_{q}) and F : G → G be a Frobenius map corresponding to some 𝔽_{q}-rational structure on G. Thus, either

G^{F} = GL_{n}(q) or G^{F} = GU_{n}(q),

the general linear group or the full unitary group over 𝔽_{q}. It is well-known that the ordinary irreducible characters of G^{F} are divided into Harish-Chandra series. Now let ℓ be a prime not dividing q. Recently, Hiss [25] has shown that there is a similar distribution into series of the ℓ-modular irreducible Brauer characters of G^{F}. As in the case of ordinary characters, one main step in classifying all ℓ-modular irreducible Brauer characters is the determination of those characters which lie in the discrete series, i.e., the cuspidal Brauer characters.

The groups GL_{n}(q) were studied extensively by Dipper and James (see [29] and the refences there). In particular, it turns out that the cuspidal ℓ-modular Brauer characters are precisely the ℓ-modular reductions of the cuspidal ordinary characters of GL_{n}(q). However, the situation is much more complicated for the groups GU_{n}(q) as can be seen already in very small examples like GU_{3}(q). It turns out that in general the reduction modulo ℓ of a cuspidal ordinary character of GU_{n}(q) need not be an irreducible Brauer character, and it is not clear whether every cuspidal Brauer character appears as a constituent in the ℓ-modular reduction of some cuspidal ordinary character. (We mention that there are examples of cuspidal Brauer characters in the finite Chevalley groups G_{2}(q) which are not constituents of the ℓ-modular reduction of any cuspidal ordinary character (see [23]).)

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

In the following, we only consider unipotent characters and unipotent ℓ-blocks. An ordinary irreducible character of G^{F} is called unipotent if it is a constituent of a Deligne–Lusztig character R_{T,1}^{G} for some F -stable maximal torus T of G. An ℓ-block is called unipotent if it is contained in the union of blocks E_{ℓ}(G^{F} , 1), which consists of all irreducible characters of G^{F} appearing in a Deligne–Lusztig character R_{T,θ}^{G} where T is an F -stable maximal torus of G and θ is a character of T^{F} of ℓ-power order. Finally, an ℓmodular Brauer character is called unipotent if it lies in a unipotent ℓ-block. It is known [23] that a Brauer character is unipotent if and only if it is a constituent of the ℓ-modular reduction of some ordinary unipotent character. In this paper, we begin a systematic investigation on the classification of the cuspidal unipotent Brauer characters of GU_{n}(q).

This paper is organised as follows. In Section 2, we study relations between the distribution of modules into series via Green’s theory of vertices and sources on the one hand, and via Harish-Chandra induction and restriction as initiated by Hiss [25] on the other hand. In [22] it was shown that the restriction of the ordinary Gelfand–Graev character to the unipotent blocks of G^{F} is the character of a projective indecomposable module, hence it corresponds to a unique unipotent ℓ-modular Brauer character which we denote by φ_{St}. In Section 3, we use the general results of Section 2 to give a sufficient and necessary criterion for φ_{St} to be a cuspidal Brauer character. In Section 4, we prove a property of the degrees of cuspidal unipotent Brauer characters by using an argument which was shown to us by M.Broué. The results mentioned so far are valid for all finite groups of Lie type.

In [18], Kawanaka’s theory of generalized Gelfand–Graev representations was used to show that the ℓ-modular decomposition matrix of G^{F} has a lower uni-triangular shape. In Section 5, we prove compatibility properties of generalized Gelfand–Graev characters with Harish-Chandra induction. In Section 6, the concept of cuspidal unipotent classes will be introduced. A unipotent class is called cuspidal if it has empty intersection with every proper F -stable split Levi subgroup. We are then able to prove that the number of cuspidal unipotent Brauer characters is bounded above by the number of cuspidal unipotent classes. Furthermore, we show that the reduction modulo ℓ of a cuspidal unipotent character (which in our situation only exists if G^{F} = GU_{n}(q) and n = s(s + 1)/2 for some s ≥ 1) is an irreducible Brauer character. This verifies a conjecture of Geck [17] in the case of the finite unitary groups.

We wish to emphasize that these main ideas of this paper, namely Harish-Chandra induction of generalized Gelfand–Graev representations and comparison of Green correspondence with Harish-Chandra philosophy, seem to be very promising concepts for investigating properties (like distribution into Harish-Chandra series, determination of decomposition numbers) of ℓmodular Brauer characters of other finite groups of Lie type as well. In particular, if it were possible to prove in general that the decomposition matrix of G^{F} is unitriangular and that similar compatibility properties with Harish-Chandra induction hold for a modified version of generalized Gelfand–Graev representations (whose construction is sketched in [31] and which were already used in [17] in the case of Steinberg’s triality groups), then it should be possible to give a unified proof of the above mentioned conjecture of Geck for all finite groups of Lie type (possibly with mild restrictions on q and ℓ). We have verified this, with some restriction on q and ℓ and using a case-by-case analysis, for the exceptional groups G_{2}(q), ^{3}D_{4}(q), E_{6}(q), and 2E_{6}(q).

In the last three sections, we give various explicit results on the groups G^{F} = GL_{n}(q) and G^{F} = GU_{n}(q). In Sections 7 and 8, we determine the precise conditions on ℓ and q such that φ_{St} is cuspidal. In Section 9, we give examples of decomposition matrices for the groups GU_{n}(q), where n ≤ 10, and formulate conjectures on the distribution of the irreducible Brauer characters into Harish-Chandra series.

2 Vertices

In this section we first recall the main results of [25, 11] introducing the notions of L-vertices and semisimple vertices. We then compare these with the usual Green vertex of an indecomposable module.

(2.1) Throughout the section G will denote a finite group with a split BNpair of characteristic p satisfying the commutator relations (see [6, Chapter 2]). We choose a prime ℓ different from p and a splitting ℓ-modular system (K, O, k) for G. This means that O is a local principal ideal domain, K is its field of fractions and k the residue class field of O modulo its maximal ideal. The characteristics of K and k are 0 and ℓ, respectively, and K is a splitting field for G and for all of its subgroups. Let θ ∈ {K, O, k}. A θG-lattice is, by definition, a θG-module which is free and finitely generated as θ-module. The restriction of a θG-lattice X to a subgroup U of G will be written as X↓_{U}, and Y ↑_{U}^{G} means the θG-lattice obtained by inducing Y from U up to G.

The set of standard generators of the Weyl group W of G will be denoted by S. Let P = {P_{J}^{n} | J ⊆ S, n ∈ N} and L = {L_{J}^{n} | J ⊆ S, n ∈ N} denote the set of all N-conjugates of standard parabolic subgroups respectively standard Levi subgroups of G.

(2.2) Let L ∈ L. Then there is some P ∈ P such that P = UL, where U is a normal p-subgroup of P and U ∩ L = {1}. Given any θL-lattice X, we denote its inflation to P by X̃. We then write

R_{L}^{G}(X) := X̃↑_{P}^{G} .

This functor is called “Harish-Chandra induction”. It was shown by Howlett and Lehrer [26] and independently by Dipper and Du [11] that Harish-Chandra induction is independent of the parabolic subgroup used to define it. There is also the adjoint functor ^{∗}R_{L}^{G} sending a θG-lattice X to the θL-lattice consisting of the fixed points of U on X.

A θG-lattice X is called “cuspidal” if ^{∗}R_{L}^{G}(X) = 0 for every proper Levi subgroup L of G. If θ ∈ {K, k} and M is a simple θG-module, then the Levi subgroups L which are minimal subject to the condition that ^{∗}R_{L}^{G}(M) ≠ 0 are conjugate under N. This was long known for θ = K and was proved for θ = k in [25, Corollary 5.2]. Following Dipper and Du [11], we call any such minimal Levi subgroup a “semisimple vertex” of M. Note that a semisimple vertex is only defined for a simple module.

The set of pairs (L, M), L ∈ L, and M a simple kL-module, can be given a partial order by putting (L′, M′) ≤ (L, M) if L′ ≤ L and M′ is a composition factor of ^{∗}R_{L′}^{L}(M). It is clear that (L, M) is a minimal pair in this partial order if and only if M is a cuspidal kL-module. The main result of [25] states that, given a simple kG-module M′, there is a unique (up to conjugation in N) minimal pair (L, M) ≤ (G, M′). This allows the definition of “ℓ-modular Harish-Chandra series”. These correspond to minimal pairs (L, M). Two simple kG-modules are said to lie in the (L, M) Harish-Chandra series, if they lie above (L, M). Since Harish-Chandra induction is independent of the parabolic subgroup, by [25, Theorem 5.8] the head and the socle of R_{L}^{G}(M) have the same set of composition factors, and this set consists exactly of the (L, M)-series. But unlike the case of characteristic 0, there may be composition factors of R_{L}^{G}(M) not lying in the (L, M)-series.

It was shown in [25, Lemma 5.7] that L is contained in the semisimple vertices of such composition factors, but the proof of that lemma actually gives the sharper result that L is strictly contained in their semisimple vertices. Let P_{M} denote the projective cover of the cuspidal irreducible kL-module M. Then R_{L}^{G}(P_{M}) is a projective kG-module and every indecomposable direct summand of it corresponds to a simple kG-module whose semisimple vertex is contained in L. This was proved in [25, Lemma 5.6], but again the proof of that lemma together with [25, Corollary 5.4] shows that a sharper result is true: If an indecomposable direct summand of R_{L}^{G}(P_{M}) corresponds to a simple kG-module M′′ which does not lie in the (L, M)-series, then the semisimple vertex of M′′ is strictly contained in L.

Let θ ∈ {K, O, k} and let X be an indecomposable θG-lattice. Then Dipper and Du have shown in [11, Theorem 2.3] that the Levi subgroups L which are minimal subject to the condition that X is a direct summand of R_{L}^{G}(Y) for some θL-lattice Y are all conjugate under N. Any such minimal Levi subgroup is called an “L-vertex” of X. If θ ∈ {O, k}, then the Green vertices of X are contained in the L-vertices. Also, the semisimple vertex of an irreducible kG-module is contained in its L-vertex. Of course, if θ = K, semisimple vertices and L-vertices of irreducible modules coincide.

(2.3) Let θ ∈ {O, k}. Fix a parabolic subgroup P ∈ P with Levi decomposition P = UL, L ∈ L and an ℓ-subgroup D of G. We shall frequently assume that D satisfies

DC_{G}(D) ≤ L.  (1)

This condition implies C_{U}(D) = {1} and therefore

N_{P} (D) = N_{L}(D).  (2)

In particular, the normalizer of D in P is contained in L and Green correspondence between P and L is defined. We thus get the following generalization of [23, Lemma 6.1.6].

Lemma 2.4 Suppose that D satisfies (1). If X is an indecomposable θLlattice with Green vertex D, then X̃ has Green vertex D. Conversely, every indecomposable θP -lattice Y with Green vertex D is of the form Y = X̃ for some θL-lattice X.

Proof. Apply Green correspondence between P and L.

We can now relate the Green vertex with the L-vertex.

Proposition 2.5 Let X be an indecomposable θG-lattice with Green vertex D satisfying (1). Then there exists a θL-lattice Y such that X is a direct summand of R_{L}^{G}(Y). In other words, the L-vertex of X is contained in L.

Proof. Since the Green vertex of X equals D, there is an indecomposable θP -lattice Z with Green vertex D such that X is a direct summand of Z↑_{P}^{G}. By Lemma 2.4 we have Z = Ỹ for some θL-lattice Y.

We note two remarkable consequences.

Corollary 2.6 Suppose that D is a Sylow ℓ-subgroup of G satisfying (1) and that L ≠ G. If X is any indecomposable θG-lattice with L-vertex G, then the Green vertex of X is properly contained in D. In particular, the dimension of X is divisible by ℓ.

We shall discuss the existence of proper Levi subgroups containing a Sylow ℓ-subgroup for the general linear and unitary groups in Sections 7 and 8.

Corollary 2.7 Suppose D is abelian and satisfies (1). If X is an irreducible kG-module or an OG-lattice with irreducible ordinary character belonging to an ℓ-block with defect group D, then the L-vertex of X is contained in L.

Proof. By a result of Knörr [32, Corollary 3.7], X has Green vertex D.

(2.8) For later use in the unitary groups we generalize some results of [23, Kapitel 6]. First we have to introduce some more notation. We put

N (L) := (N_{G}(L) ∩ N)L.

In non-degenerate cases this will coincide with the normalizer of L in G. We also set

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

Now let θ ∈ {K, k} and let M be a simple θL-module. For x ∈ N (L) we denote by M^{x} the module conjugate to M by x. Then

N (L, M) := {x ∈ N (L) | M^{x} ≅ M}

is the “inertia group” of M and

W (L, M) := N (L, M)/L

the “ramification group” of R_{L}^{G}(M).

Proposition 2.9 Let M be a simple cuspidal kL-module with Green vertex D satisfying (1). Suppose that N_{G}(D) ≤ N (L, M) and that W (L, M) is an ℓ′-group. Then M ↑_{L}^{N (L,M)} is semisimple and there exists a multiplicity preserving bijection between the indecomposable direct summands of R_{L}^{G}(M) and the constituents of M↑_{L}^{N (L,M)}. If every composition factor of R_{L}^{G}(M) is self-dual, then R_{L}^{G}(M) is semisimple.

Proof. Put Z = M↑_{L}^{N (L,M)}. By Mackey’s theorem and the fact that M is invariant in N (L, M), we have

Z↓_{L}≅ |W (L, M)| M.  (3)

Using Frobenius reciprocity and (3), we see that the endomorphism ring of Z has dimension |W (L, M)|. Since W (L, M) is an ℓ′-group, Z is semisimple. Write

Z = ⨁_{i=1}^{r} m_{i}Z_{i},

where the Z_{i} are pairwise non-isomorphic simple kN (L, M)-modules. Then ∑_{i=1}^{r} m_{i}^{2} = |W (L, M)|. The fact that the Green vertex of M equals D together with equation (3) implies that every Z_{i} has Green vertex D. Applying Green correspondence between G and N (L, M) with respect to D, we find

M↑_{L}^{G}= Z↑_{N (L,M)}^{G}= ⨁_{i=1}^{r} m_{i}Y_{i} ⊕ Y_{0},

where each Y_{i} is an indecomposable kG-module with Green vertex D, and each indecomposable direct summand of Y_{0} has a Green vertex properly contained in D.

Now M ↑_{L}^{G}= (M ↑_{L}^{P}) ↑_{P}^{G}. Applying Green correspondence between P and L, which we may by condition (1), we find that M ↑_{L}^{P}= M̃ ⊕ M′, where each indecomposable direct summand of M′ has Green vertex properly contained in D. Therefore

R_{L}^{G}(M) = ⨁_{i=1}^{r} m_{i}Y_{i} ⊕ Y′_{0},

where Y′_{0} is a direct summand of Y_{0}. The endomorphism ring of R_{L}^{G}(M) has dimension |W (L, M)| by [12, Theorem 2.9]. A dimension count yields that Y′_{0} = 0, that the Y_{i} have one-dimensional endomorphism rings and that there are no non-trivial homomorphisms between Y_{i} and Y_{j} for i ≠ j.

Suppose now that every composition factor of R_{L}^{G}(M) is self-dual. Then there is a non-trivial homomorphism from the head of each Y_{i} to the socle of its dual. Hence each Y_{i} is self-dual and, since it has a one-dimensional endomorphism ring, the non-trivial homomorphisms from its head to its socle are isomorphisms. This implies that Y_{i} is simple.

(2.10) Let θ ∈ {O, k}. We investigate a connection between the semisimple vertex of an irreducible kG-module M and the L-vertices of certain θGlattices contained in the projective cover of M.

Proposition 2.11 Let Y be a projective indecomposable θG-lattice with corresponding irreducible kG-module M. Let X be an indecomposable θG-lattice with projective cover Y. Then the semisimple vertex of M is contained in the L-vertex of X.

Proof. We must show that the Harish-Chandra restriction ^{∗}R_{L}^{G}(M) is not the zero module. By the adjointness of Harish-Chandra induction and restriction, this is equivalent to showing that Y is a direct summand of some R_{L}^{G}(Y_{0}), where Y_{0} is a projective θL-lattice. By assumption, X is a direct summand of R_{L}^{G}(X_{0}) for some θL-lattice X_{0}. If we choose Y_{0} to be the projective cover of X_{0}, then R_{L}^{G}(Y_{0}) is projective and maps onto X. Hence Y, the projective cover of X, must be a direct summand of R_{L}^{G}(Y_{0}).

3 Degrees of cuspidal characters

Let p be a prime and q a power of p. Let G be a connected reductive algebraic group with some 𝔽_{∥}-rational structure and corresponding Frobenius morphism F. We choose an F -stable maximal torus T_{0} of G and an F -stable Borel subgroup B_{0} containing T_{0}. Then the group G^{F} of F -fixed points of G is a finite group with a split BN-pair of characteristic p with B = B_{0}^{F} and N = N_{G}(T_{0})^{F}. Furthermore, G^{F} satisfies the commutator relations. The set L consists of the groups of the form L^{F}, where L is an F -stable Levi subgroup containing T_{0} of an F -stable parabolic subgroup. The centre of G will be written as Z(G). We keep to the notation of the preceding section. In particular, ℓ is a prime different from p.

(3.1) We briefly want to discuss a result on the degrees of cuspidal modules of G^{F}. If χ is an ordinary cuspidal irreducible character of G^{F}, then the highest power of (q − 1), which divides the order polynomial (see [3]) of G/Z(G)^{0}, also divides the degree of χ. A similar result is true for cuspidal ℓ-modular characters. The proof we give is due to Broué.

Let W denote the Weyl group of G. We write T_{w} for the F -stable maximal torus of G obtained from T_{0} by twisting with w ∈ W. An F - stable maximal torus T of G is minisotropic, if its 𝔽_{∥}-rank is as small as possible. This is the same as saying that T does not lie in any proper F - stable parabolic subgroup of G. The 𝔽_{∥}-rank of a minisotropic torus is the 𝔽_{∥}-rank of Z(G). We write rank_{q}(G) for the 𝔽_{∥}-rank of G.

It will be more convenient to state and prove the following result in terms of characters. We think of a Brauer character of a kG^{F} -module as being defined on all of G^{F}, by letting its values be 0 on ℓ-singular classes. The usual scalar product on the set of K-valued class functions of G^{F} will be written as ( , ). We write reg_{G} for the Brauer character of the regular kG^{F} -module. The p′-part of the integer z will be denoted by z_{p′}.

Proposition 3.2 (Broué) Let φ be the character (respectively Brauer character) of a simple KG^{F} - or kG^{F} -module M. If L^{F} ∈ L is the semisimple vertex of M, then (q − 1)^{rank_{q}(G)−rank_{q}(Z(L))} divides |W |φ(1). In particular, if M is cuspidal, then (q − 1)^{rank_{q}(G)−rank_{q}(Z(G))} divides |W |φ(1).

Proof. We have (see [8, Corollary 12.14])

reg_{G} = \frac{1}{|W |} ∑_{w∈W} |G^{F}: T_{w}^{F} |_{p′}R_{T_{w}}^{G}(reg_{T_{w}}).

Now φ(1) = (φ, reg_{G}), and so

|W |φ(1) = ∑_{w∈W} |G^{F}: T_{w}^{F} |_{p′}(φ, R_{T_{w}}^{G}(reg_{T_{w}}))
= ∑_{w∈W} |G^{F}: T_{w}^{F} |_{p′}(^{∗}R_{T_{w}}^{G}(φ), reg_{T_{w}}),

where ^{∗}R_{T_{w}}^{G} denotes the adjoint of the Deligne-Lusztig map.

Fix w ∈ W such that the summand corresponding to w in the above formula is non-zero. Let L′ be an F -stable split Levi subgroup of G minimal with respect to the condition that it contains T_{w}. Then T_{w} is a minisotropic torus of L′ and so rank_{q}(T_{w}) = rank_{q}(Z(L′)). Now ^{∗}R_{T_{w}}^{G}(φ) is non-zero, and so ^{∗}R_{L′}^{G}(φ) is non-zero, by the transitivity of the adjoint Harish-Chandra map. This implies that L, the semisimple vertex of M, is conjugate to a subgroup of L′ (see [25, Lemma 5.1]). Hence rank_{q}(Z(L)) ≥ rank_{q}(Z(L′)) and so (q − 1)^{rank_{q}(G)−rank_{q}(Z(L))} divides |G^{F}: T_{w}^{F} |_{p′}. The result follows.

4 The modular Steinberg character

Let the notation be as in the preceding section. We assume moreover that G has connected centre. An irreducible ℓ-modular character of G^{F} is called unipotent if it lies in E_{ℓ}(G^{F} , 1), which is a union of ℓ-blocks of G^{F} [5]. The Gelfand-Graev module of G^{F} (see [6, Chapter 8]) is a projective kG^{F} - module, and its component lying in E_{ℓ}(G^{F} , 1) is indecomposable (see [22]).

(4.1) Let us denote by φ_{St} the unique irreducible unipotent Brauer character of G^{F} corresponding to the E_{ℓ}(G^{F} , 1)-component of the Gelfand-Graev character. We are now able to prove the following extension of [25, Lemma 6.2].

Theorem 4.2 If ℓ divides |G^{F}: L^{F} | for all L^{F} ∈ L \ {G^{F} }, then φ_{St} is cuspidal.

Let L^{F} ∈ L\{G^{F} } such that ℓ does not divide |G^{F}: L^{F} |. If D ∈ Syl_{ℓ}(L^{F}) satisfies (1), then the semisimple vertex of φ_{St} is contained in L^{F}. In particular, φ_{St} is not cuspidal.

Proof. The first part of the theorem has already been proved in [25, Lemma 6.2]. To prove the second, let Y denote the component of the Gelfand-Graev character of G^{F} containing the ordinary Steinberg character. For X take the quotient of Y whose Brauer character is the Steinberg character. Then the Green vertex of X is D, since ℓ does not divide the dimension of X. By Proposition 2.5, the L-vertex of X is contained in L. The assertion now follows from Proposition 2.11.

5 Harish-Chandra induction of GGGR’s

Let G be a connected reductive group defined over 𝔽_{q} with corresponding Frobenius map F. We assume that the centre of G is connected and that the root system of G has only simple components of type A. In this section, we prove compatibility properties of generalized Gelfand–Graev representations, which were introduced by N.Kawanaka in [30], with Harish-Chandra induction (see Proposition 5.5 below). The exposition and the notation are independent of the remaining parts of this paper.

Given a unipotent element u ∈ G^{F} we can form the character γ_{u}^{G} of the corresponding generalized Gelfand–Graev representation (GGGR for short) of G^{F}. It is obtained by inducing a certain representation from a unipotent subgroup of G^{F} and only depends on the G^{F} -conjugacy class of u. In particular, we have that γ_{u}^{G}(g) = 0 unless the element g ∈ G^{F} is unipotent. A special feature of the group G is that the centralizer of every element is connected (this implies that, if two elements in G^{F} are conjugate in G, then they are already conjugate in G^{F}).

For any element g ∈ G, we denote by (g)_{G} its conjugacy class in G. For any subset S ⊆ G, denote by \overline{S} its Zariski-closure in G. Given unipotent classes C, C′ of G we write C ≤ C′ if and only if C ⊆ \overline{C}′. This defines a partial order relation on the set of unipotent classes.

(5.1) Following Lusztig [34] we define a map σ_{G} from the irreducible characters of G^{F} to the F -stable unipotent classes of G as follows. Let χ be an irreducible character of G^{F}. Then there exists a unique F -stable unipotent class C of G with maximal dimension subject to the condition that ∑_{x∈C^{F}} χ(x) ≠ 0. The class σ_{G}(χ) := C is called the unipotent support of χ. In addition, we have χ(g) = 0 unless the unipotent part of the element g ∈ G^{F} lies in C or in a class of dimension strictly smaller than dim C. Denote by D_{G} the Curtis–Alvis duality operation on the character ring of G^{F}. It is also shown in [loc. cit.] that

(i) σ_{G}(χ) = ξ_{G}(±D_{G}(χ)),

where ξ_{G} is the map from irreducible characters to unipotent classes defined in [33, Chapter 13]. Kawanaka [31] has suggested an alternative description of the map ξ_{G} in terms of GGGR’s. Indeed, by [34, Theorem 11.2], we have, for any unipotent element u ∈ G^{F} and C = σ_{G}(χ):

(ii) (D_{G}(χ), γ_{u}^{G}) = 0 if dim \overline{C} < dim \overline{(u)}_{G} or dim \overline{C} = dim \overline{(u)}_{G} and C ≠ (u)_{G}, (iii) (D_{G}(χ), γ_{u}^{G}) ≠ 0 and ”independent of q”, if u ∈ C^{F} is chosen suitably.

and these two properties characterize C uniquely. (See also the note at the end of this section).

From now on, we fix an F -stable parabolic subgroup P of G with rational Levi decomposition P = UL. Suppose that we have a unipotent element u ∈ L^{F}. Then we can form the characters γ_{u}^{L} and R_{L}^{G}(γ_{u}^{L}) where R_{L}^{G} denotes Harish-Chandra induction from L^{F} to G^{F}. We wish to show that the formulae in (5.1)(ii) and (iii) hold with γ_{u}^{G} replaced by R_{L}^{G}(γ_{u}^{L}).

The following Lemma is certainly well-known. We shall give a direct and elementary proof here.

Lemma 5.2

(i) Let u, v ∈ L be unipotent elements such that (u)_{L} ≤ (v)_{L}. Then (u)_{G} ≤ (v)_{G}.

(ii) Let u ∈ L be unipotent and x ∈ U. Then (u)_{G} ≤ (xu)_{G}.

Proof. (i) Note that u lies in the closure of (v)_{L}, hence also in the closure of (v)_{G}.

(ii) If L = G then U = {1}, and there is nothing to prove. Let us assume that L ≠ G. Choose an F -stable maximal torus T and an F -stable Borel subgroup B of G such that T ⊂ B ⊆ P and T ⊆ L. Let Φ be the root system of G with respect to T, and I ⊂ Φ the system of simple roots determined by B. Then P = P_{J}, U = U_{J} and L = L_{J} for some proper subset J ⊂ I (for the notation we refer to [6, Chapter 2]). The element x ∈ U can now be written as

x = ∏_{α∈Φ^{+}\Φ_{J}} x_{α}(t_{α}) where t_{α} ∈ 𝔽̄_{q}.

We claim that there exist non-zero integers c_{α} (α ∈ Φ^{+} \ Φ_{J}) such that, if we define

x(λ) = ∏_{α∈Φ^{+}\Φ_{J}} x_{α}(λ^{c_{α}}t_{α}) ∈ U_{J}, for λ ∈ 𝔽̄_{q},

then (xu)_{G} contains all elements of the form x(λ)u where λ ∈ 𝔽̄_{q}^{×}. As in [1, E II 3.7], this then implies that the closure of (xu)_{G} contains the element u = x(0)u proving the assertion.

Our claim can be proved as follows. Let X and Y be the character and cocharacter group of T, and ⟨ , ⟩ : X × Y → ℤ the natural pairing. Since the matrix (⟨γ, β^{∨}⟩)_{γ,β∈I} is invertible, there exist integers n_{β} (β ∈ I) such that

∑_{β∈I} n_{β}⟨γ, β^{∨}⟩ = \{ \begin{matrix}= 0 & for γ ∈ J, \\ > 0 & for γ ∉ J. \end{matrix}  (4)

For α ∈ Φ we define c_{α} := ∑_{β∈I} n_{β}⟨α, β^{∨}⟩. Then (4) implies that c_{α} = 0 for all α ∈ Φ_{J}, and c_{α} ≠ 0 for all α ∈ Φ^{+} \ Φ_{J}. We identify T with Y ⊗ 𝔽̄_{q}^{×} and consider the element t(λ) := ∑_{β∈I} β^{∨} ⊗ λ^{n_{β}} ∈ T for λ ∈ 𝔽̄_{q}^{×}. The action of t(λ) on an element x_{α}(μ) (μ ∈ 𝔽̄_{q}) of a root subgroup is then given by

t(λ)x_{α}(μ)t(λ)^{−1} = x_{α}(α(t(λ))μ)
= x_{α}((∏_{β∈I} λ^{n_{β}⟨α,β^{∨}⟩})μ)
= x_{α}(λ^{c_{α}}μ)

for all μ ∈ 𝔽̄_{q}, λ ∈ 𝔽̄_{q}^{×}. Since c_{α} = 0 for α ∈ Φ_{J}, the element t(λ) commutes with u. So we obtain

t(λ)xut(λ)^{−1} = x(λ)u for λ ∈ 𝔽̄_{q}^{×},

and our claim is established.

(5.3) Let u, v ∈ G^{F} be unipotent elements. It follows from the definition of the GGGR that

γ_{u}^{G}(v) ≠ 0 implies (v)_{G} ≤ (u)_{G}.

By [30, (3.2.20)], there exists a polynomial f_{u,v} ∈ ℤ[X] and an integer a_{u,v} such that

γ_{u}^{G}(v) = f_{u,v}(q) and D_{G}(γ_{u}^{G})(v) = ±q^{a_{u,v}}f_{v,u}(q^{−1})

(note that also D_{G}(γ_{u}^{G})(g) is zero unless g ∈ G^{F} is unipotent). Then, in fact, we have the stronger statement that f_{u,v} ≠ 0 implies (v)_{G} ≤ (u)_{G}. In particular, we see that

D_{G}(γ_{u}^{G})(v) ≠ 0 implies (u)_{G} ≤ (v)_{G}.

We now show that this property behaves well with respect to Harish-Chandra induction.

Lemma 5.4 Assume that a unipotent element u ∈ L^{F} is chosen such that dim \overline{(u)}_{L} ≥ dim \overline{(u′)}_{L} for all u′ ∈ (u)_{G} ∩ L^{F}. Let v ∈ G^{F} be unipotent.

(i) If R_{L}^{G}(D_{L}(γ_{u}^{L}))(v) ≠ 0, then (u)_{G} ≤ (v)_{G}.

(ii) If D_{L}(γ_{u}^{L})(u) ≠ 0, then R_{L}^{G}(D_{L}(γ_{u}^{L}))(u) ≠ 0.

Proof. We write P^{F} ∩ (v)_{G^{F}} = C_{1} ∪ … ∪ C_{r} where the C_{i} are conjugacy classes of P^{F}. Choose v_{i} ∈ C_{i} and write v_{i} = x_{i}u_{i} where x_{i} ∈ U^{F} and u_{i} ∈ L^{F}. Using the induction formula for characters, we then have

R_{L}^{G}(D_{L}(γ_{u}^{L}))(v) = ∑_{i=1}^{r} \frac{|C_{G}(v)^{F} |}{|C_{P} (v_{i})^{F} |}D_{L}(γ_{u}^{L})_{P^{F}} (v_{i})

= ∑_{i=1}^{r} \frac{|C_{G}(v)^{F} |}{|C_{P} (v_{i})^{F} |}D_{L}(γ_{u}^{L})(u_{i}).

(i) Assume that R_{L}^{G}(D_{L}(γ_{u}^{L}))(v) ≠ 0. Then P^{F} ∩ (v)_{G^{F}} must be nonempty, and the above formula implies D_{L}(γ_{u}^{L})(u_{i}) ≠ 0 for some i ∈ {1, …, r}. Hence (u)_{L} ≤ (u_{i})_{L} (see the remarks in (5.3)) and, by Lemma 5.2, we then conclude (u)_{G} ≤ (u_{i})_{G} ≤ (x_{i}u_{i})_{G} = (v)_{G}.

(ii) Let v = u = u_{1}. Suppose that D_{L}(γ_{u}^{L})(u_{i}) ≠ 0 for some i > 1. Again, this implies (u)_{L} ≤ (u_{i})_{L}, and hence dim \overline{(u)}_{L} ≤ dim \overline{(u_{i})}_{L}. By Lemma 5.2, we then have (u)_{G} ≤ (u_{i})_{G} ≤ (x_{i}u_{i})_{G} = (v_{i})_{G} = (u)_{G}, so we conclude that u and u_{i} are conjugate in G. Our assumption on the choice of u now implies that we must have dim \overline{(u)}_{L} = dim \overline{(u_{i})}_{L}. So the closure of (u)_{L} equals the closure of (u_{i})_{L}, hence the classes themselves must be equal (since the relation ≤ is a partial order on the set of unipotent classes of G). Thus u_{i} lies in (u)_{L}^{F}. The fact that C_{L}(u) is connected implies that u_{i} is conjugate to u in L^{F}. So the values of D_{L}(γ_{u}^{L}) on u and u_{i} are equal. Hence, if D_{L}(γ_{u}^{L})(u) ≠ 0, the non-trivial summands in the above induction formula do not cancel out, and the result follows.

With this information we are now able to obtain results on the irreducible characters of G^{F} which can possibly occur as constituents of R_{L}^{G}(γ_{u}^{L}).

Proposition 5.5 Let u ∈ L^{F} be a unipotent element chosen as in Lemma 5.4. Let χ be an irreducible character of G^{F} and C := σ_{G}(χ). Then

(i) (D_{G}(χ), R_{L}^{G}(γ_{u}^{L})) = 0 if dim \overline{C} < dim \overline{(u)}_{G} or dim \overline{C} = dim \overline{(u)}_{G} and C ≠ (u)_{G}.

(ii) (D_{G}(χ), R_{L}^{G}(γ_{u}^{L})) ≠ 0 if u ∈ C^{F}.

Proof. Let C_{1}, …, C_{m} be the F -stable unipotent classes of G, where the labelling is chosen such that dim \overline{C}_{i} < dim \overline{C}_{j} implies i < j. Assume (u)_{G} = C_{r} and C = C_{s} for r, s ∈ {1, …, m}. Then we obtain, by using the selfadjointness of D_{G} and the equation R_{L}^{G} ∘D_{L} = D_{G} ∘R_{L}^{G} (see [8, Chapter 8]), the following formula.

(D_{G}(χ), R_{L}^{G}(γ_{u}^{L})) = (χ, D_{G}(R_{L}^{G}(γ_{u}^{L}))) = (χ, R_{L}^{G}(D_{L}(γ_{u}^{L})))
= \frac{1}{|G^{F} |} ∑_{i=1}^{m} ∑_{x∈C_{i}^{F}} χ̄(x)R_{L}^{G}(D_{L}(γ_{u}^{L}))(x)

= \frac{1}{|G^{F} |} ∑_{i} ∑_{x∈C_{i}^{F}} χ̄(x)R_{L}^{G}(D_{L}(γ_{u}^{L}))(x),

where i runs over those indices such that dim \overline{C}_{i} ≤ dim \overline{C}_{s} (by the definition and properties of σ_{G}(χ) = C_{s}), and C_{r} ≤ C_{i} (by Lemma 5.4(i)).

Now, if dim \overline{C}_{s} < dim \overline{C}_{r} or dim \overline{C}_{s} = dim \overline{C}_{r} and C_{s} ≠ C_{r}, then C_{r} ≰ C_{s} and the summation in the above formula is over the empty set. This proves (i).

Now assume u ∈ C^{F}. Then C_{r} = C = C_{s}, and there is only one summand in the above formula left, viz. for i = r = s. The fact that C_{G}(u) is connected implies that C^{F} is a single G^{F} -conjugacy class. Hence the right hand side of the above formula becomes

\frac{1}{|G^{F} |}|C^{F} |χ̄(u)R_{L}^{G}(D_{L}(γ_{u}^{L}))(u).

In order to prove (ii) we therefore must show that (a) χ(u) ≠ 0 and

(b) R_{L}^{G}(D_{L}(γ_{u}^{L}))(u) ≠ 0. To prove (a) we simply apply the above argument with L = G, use (5.1)(iii), and obtain:

0 ≠ (D_{G}(χ), γ_{u}^{G}) = \frac{1}{|G^{F} |}|C^{F} |χ̄(u)D_{G}(γ_{u}^{G})(u).

We also see from this that

D_{G}(γ_{u}^{G})(u) ≠ 0.  (5)

By [33, (13.4.2)], the map ξ_{G}, and hence also σ_{G}, is surjective. Thus, (5) holds for every unipotent element u ∈ G^{F}. To prove (b) we now apply this argument to the group L which is possible since L has connected centre and only simple components of type A. Thus D_{L}(γ_{u}^{L})(u) ≠ 0 and the result follows from Lemma 5.4(ii).

(5.6) A final note on the proofs of the results in (5.1) is in order. The fact that the map σ_{G} is well-defined and that σ_{G} and ξ_{G} are related by the formula (5.1)(i) is proved in [34] under the hypothesis that q is a power of a sufficiently large prime and that G is split over 𝔽_{q}. Since we are dealing with groups G such that every simple component is of type A, the above mentioned results are true without any assumption on q and with G being split or non-split. This follows easily from the facts (a) that character values of GL_{n}(q) at unipotent elements are given by evaluating certain polynomials at q (this is already due to Green [21]), (b) that there is an Ennola duality between GL_{n}(q) and GU_{n}(q), both for the irreducible characters and for the characters of the GGGR’s (see [30, (3.2.10) and (3.2.14)]), and (c) that the scalar product in (5.4)(ii) is non-zero and bounded above independently of q (see [34, Theorem 11.2(ii)]). We omit further details. Similar arguments can also be found in [18, Step 2 and 3 in the proof of Theorem A].

6 Cuspidal classes and cuspidal Brauer characters

In this section, we let again G = GL_{n}(𝔽̄_{q}) and F : G → G be a Frobenius map corresponding to some 𝔽_{q}-rational structure on G. We will apply the results of the previous section to prove results on cuspidal unipotent Brauer characters of G^{F} (see Proposition 6.8 and Theorem 6.10 below).

Recall that all unipotent classes of G are F -stable, and that the centralizer of every element of G is connected. Both the unipotent classes and the unipotent characters of G^{F} have a natural parametrization by partitions of n: for the classes, this comes from the Jordan canonical form of matrices; for the characters, this comes from the parametrization of the irreducible characters of the symmetric group S_{n} and the definition of “almost characters” in [8, Section 15.4]. For example, the trivial class and the Steinberg character of G^{F} have label (1^{n}) while the class of regular elements and the trivial character have label (n). These parametrizations are compatible in the sense that, if χ is a unipotent character and C := ξ_{G}(χ) is the associated unipotent class (see (5.1)), then ±D_{G}(χ) and C are labelled by the same partition. This follows immediately from the original definition of ξ_{G} in [33, Chapter 13], and the description of Springer’s map from unipotent classes to characters of the Weyl group in [6, Section 13.3]. The partial order relation on the unipotent classes is given by the usual dominance order on partitions of n (see [6, Section 13.4]).

(6.1) Let P = UL be an F -stable parabolic subgroup of G with unipotent radical U and F -stable Levi subgroup L. In this case, we call L a split Levi subgroup of G. We say that the unipotent class C is cuspidal if C ∩ L^{F} = ∅ for all proper split Levi subgroups L of G. (Of course, this definition applies to every connected reductive group defined over a finite field.)

Proposition 6.2 Let λ be a partition of n and C_{λ} be the corresponding unipotent class.

(i) Assume that G^{F} = GL_{n}(q). Then C_{λ} is cuspidal if and only if λ = (n), i.e., C_{λ} is the class of regular unipotent elements.

(ii) Assume that G^{F} = GU_{n}(q). Then C_{λ} is cuspidal if and only if all parts of λ are distinct.

Proof. Let u ∈ C_{λ}^{F}. We claim that C_{λ} is cuspidal if and only if every split torus of G centralizing u lies in the centre of G. This can be seen as follows. Suppose C_{λ} is cuspidal and let S be any split torus of G centralizing u. Then the centralizer L := C_{G}(S) is a split Levi subgroup of G (see [2, Proposition 20.4]). Since C_{λ} is cuspidal, we must have L = G and hence S is contained in the centre of G. Conversely, assume that C_{λ} is not cuspidal, and let L be a proper split Levi subgroup of G such that u ∈ L^{F}. Then the centre of L contains a split torus S such that C_{G}(S) = L (see [2, Proposition 20.6]), and hence S is not contained in the centre of G.

By [1, E IV], the centralizer of u in G is a semidirect product of its unipotent radical with a connected reductive group M. In each case, the group M is given explicitly in [loc. cit.]. Let λ_{i} (i = 1, …, r) be the parts of λ and, for j ≥ 1, denote by r_{j} the number of λ_{i} equal to j.

(i) Assume that G^{F} = GL_{n}(q). Then M^{F} is a direct product of GL_{r_{i}}(q) for i ≥ 1. So M does not contain any non-central split torus of G if and only if r_{n} = 1, and r_{i} = 0, for i ≠ n. This condition means that λ = (n), i.e., C_{λ} is the class of regular unipotent elements of G.

(ii) Assume that G^{F} = GU_{n}(q). Then M^{F} is a direct product of GU_{r_{i}}(q) for i ≥ 1. So M does not contain any non-central split torus of G if and only if r_{i} ∈ {0, 1}, for all i. This condition means that all parts of λ are distinct.

(6.3) Denote by E(G^{F} , 1) the set of unipotent characters of G^{F}. Let

E(G^{F} , 1) = {χ_{1}, …, χ_{m}}.

For each i ∈ {1, …, m}, let C_{i} := ξ_{G}(χ_{i}) be the F -stable unipotent conjugacy class of G determined by Lusztig’s map ξ_{G} (see (5.1)). We choose notation so that dim \overline{C}_{i} < dim \overline{C}_{j} implies i < j. The special feature of our group G = GL_{n}(\overline{𝔽}_{q}) is that the restriction of ξ_{G} to the geometric conjugacy class E(G^{F} , 1) is bijective. Thus, the C_{i} are all distinct. Choose u_{i} ∈ C_{i}^{F}, and denote by γ_{u_{i}}^{G} the character of the corresponding GGGR. With these notations and by (5.1), we have that the matrix of scalar products

((χ_{i}, γ_{u_{j}}^{G}))_{1≤i,j≤m}

is non-singular and lower triangular (see also [18, Corollary B]).

(6.4) Fix i ∈ {1, …, m}, and let P_{i} be an F -stable parabolic subgroup of G with rational Levi decomposition P_{i} = U_{i}L_{i} such that C_{i} ∩ L_{i}^{F} ≠ ∅ and L_{i} has minimal dimension with this property. Note that the minimality of L_{i} implies that L_{i} is uniquely determined up to conjugation in G^{F}. We will therefore call it the Gelfand–Graev vertex (GG-vertex) of C_{i} and of the character χ_{i}. Choose u_{i} ∈ C_{i} ∩ L_{i}^{F} as in Lemma 5.4. We then consider the character

R_{L_{i}}^{G}(γ_{u_{i}}^{L_{i}}).

The choices of u_{i} and L_{i} imply that (u_{i})_{L_{i}} is a cuspidal class of L_{i}. As an immediate consequence of Proposition 5.5 and by the same arguments as in (6.3) we now have:

Proposition 6.5 The matrix of scalar products ((χ_{i}, R_{L_{j}}^{G}(γ_{u_{j}}^{L_{j}})))_{1≤i,j≤m} is non-singular and lower triangular.

(6.6) Let ℓ be a prime not dividing q. We are going to study properties of ℓ-blocks and ℓ-modular Brauer characters of G^{F}. The set

E_{ℓ}(G^{F} , 1) = ⋃_{t} E(G^{F} , t),

where t runs over all semisimple ℓ-elements in G^{∗}^{F}, is a union of ℓ-blocks of G^{F} (see [15]). Given a character χ of G^{F}, denote by χ̂ its restriction to the ℓ-regular elements. Then the characters {χ̂ | χ ∈ E(G^{F} , 1)} form a basic set of ℓ-modular Brauer characters of E_{ℓ}(G^{F} , 1) (see [(8A), loc. cit.]). Thus, in order to find the decomposition matrix of E_{ℓ}(G^{F} , 1) it is sufficient to determine the decomposition numbers for the characters χ ∈ E(G^{F} , 1).

Since every γ_{u_{i}}^{G} is induced from a unipotent subgroup of G^{F}, it is the character of a projective module for G^{F} (over a suitable valuation ring of characteristic 0 with residue class field of characteristic ℓ). From the definition of Harish-Chandra induction we see that R_{L_{i}}^{G}(γ_{u_{i}}^{L_{i}}) is a character of a projective module, too. As in [18], the statement of Proposition 6.5 therefore implies that the decomposition matrix of E_{ℓ}(G^{F} , s) has a lower uni-triangular shape, with the above ordering of the characters.

With the γ_{u_{i}}^{G} replaced by R_{L_{i}}^{G}(γ_{u_{i}}^{L_{i}}) we can now do much better.

(6.7) Using (6.6) we see that E_{ℓ}(G^{F} , 1) contains precisely m irreducible Brauer characters, φ_{1}, …, φ_{m} say. By the results on the triangular shape of the decomposition matrix we can arrange the labelling in such a way that, for fixed j, the Brauer character φ_{j} appears with multiplicity 1 as a constituent in χ̂_{j}, and does not appear in χ̂_{i}, for i < j. Let Φ_{1}, …, Φ_{m} be the characters of the projective indecomposable modules corresponding to the irreducible Brauer characters φ_{1}, …, φ_{m}. With the above labelling we see that, for fixed j, the character Φ_{j} is a summand in R_{L_{j}}^{G}(γ_{u_{j}}^{L_{j}}), and does not occur in R_{L_{i}}^{G}(γ_{u_{i}}^{L_{i}}) for i > j.

An ℓ-modular irreducible Brauer character φ is cuspidal if and only if the scalar product (φ, R_{L}^{G}(Φ)) is zero for every proper split Levi subgroup L of G and every projective character Φ of L^{F} (see [25, Lemma 4.1(c)]). With our above notations we therefore have the following result which also explains the notation of cuspidal classes.

Proposition 6.8 If φ_{j} is cuspidal, then the class C_{j} must be cuspidal. In particular, the number of cuspidal irreducible Brauer characters of G^{F} is at most the number of cuspidal unipotent classes of G.

Proof. Assume that φ_{j} is cuspidal. Since the scalar product (φ_{j}, R_{L_{j}}^{G}(γ_{u_{j}}^{L_{j}})) is non-zero, L_{j} cannot be a proper split Levi subgroup. So we have L_{j} = G, and by the choice of L_{j} in (6.4) we conclude that C_{j} is cuspidal.

(6.9) (i) The class of regular unipotent elements is always cuspidal by Proposition 6.2. In the above ordering of classes, this class has label m. The corresponding character R_{L_{m}}^{G}(γ_{u_{m}}^{L_{m}}) is the usual Gelfand–Graev character Γ_{G} of G^{F} (see [6, Section 8.1]). As remarked in (4.1) it corresponds to a unique unipotent ℓ-modular Brauer character. We have called it the modular Steinberg character, and we have seen that it may or may not be cuspidal. This will be studied in detail in Sections 7,8,9.

(ii) Let us consider the case G^{F} = GL_{n}(q). By Proposition 6.2(i) and the choice of u_{i} and L_{i}, the class (u_{i})_{L_{i}} contains the regular unipotent elements of L_{i}, hence γ_{u_{i}}^{L_{i}} is the ordinary Gelfand–Graev character of L_{i}^{F}. This construction was already used in [24, Corollary 3.4], to give an alternative proof of the fact that the decomposition matrix of E_{ℓ}(GL_{n}(q), 1) has a lower unitriangular shape. So, the use of R_{L_{i}}^{G}(γ_{u_{i}}^{L_{i}}) in (6.4) may be viewed as a generalization of this construction to the case of the finite unitary groups (and to other finite groups of Lie type as well, of course).

Now assume that G^{F} = GU_{n}(q). By [6, Section 13.7], the group G^{F} has at most one ordinary cuspidal unipotent character, and there exists one if and only if n = s(s + 1)/2 for some s ≥ 1. In this case, the cuspidal unipotent character is labelled by the partition with parts (s, s−1, …, 2, 1).

Theorem 6.10 Assume that n = s(s + 1)/2 for some s ≥ 1. Then the reduction modulo ℓ of the unique cuspidal unipotent character of GU_{n}(q) is an irreducible, and hence cuspidal, Brauer character.

Proof. Let i_{0} ∈ {1, …, m} such that λ_{i_{0}} is the partition of n with parts (s, s − 1, …, 2, 1). Then the image of the cuspidal unipotent character χ under Lusztig’s map ξ_{G} is the class C_{i_{0}} (since D_{G}(χ) = ±χ, and by the remarks at the beginning of this section). We must show that the Brauer character φ_{i} is not a constituent of χ̂ if i ≠ i_{0}. This is clear for the case i > i_{0} since the decomposition matrix of E_{ℓ}(G^{F} , 1) has a lower triangular shape. Now let i < i_{0} and assume that φ_{i} is a constituent of χ̂. This implies that φ_{i} must be cuspidal, too (see [25, Lemma 4.3]). By Proposition 6.8, we conclude that C_{i} is a cuspidal class. So, by Proposition 6.2(ii), all parts of the partition λ_{i} associated with C_{i} are distinct. This implies that λ_{i} is greater than or equal to λ_{i_{0}} in the dominance order on partitions. Thus, we conclude that C_{i_{0}} ≤ C_{i}, hence dim \overline{C}_{i_{0}} ≤ dim \overline{C}_{i}, and so i_{0} ≤ i, a contradiction. Therefore, φ_{i} cannot be a constituent of χ̂ if i < i_{0}. The proof is complete.

(6.11) One consequence of Theorem 6.10 is that it completely determines one row in the ℓ-modular decomposition matrix of E_{ℓ}(G^{F} , 1), namely the one corresponding to the ordinary cuspidal unipotent character. Another consequence is that it also gives parts of certain columns of this decomposition matrix, namely by using Harish-Chandra induction from proper split Levi subgroups: Let G^{F} = GU_{n}(q) and assume that L is a split Levi subgroup of G and that L^{F} has a cuspidal unipotent character χ. Let O be a suitable valuation ring of characteristic 0 with residue class field of characteristic ℓ as in (2.1), and M be an OL^{F} -lattice with ordinary character χ. The previous theorem and the remarks in [17, Abschnitt 2.6] imply that the endomorphism ring of R_{L}^{G}(M) is an Iwahori–Hecke algebra over O associated with a certain finite Weyl group (which is isomorphic to the stabilizer of χ in N_{G}(L)/L). From a theorem of Dipper (see [10, Consequences 2.6]) we then conclude that the ℓ-modular decomposition matrix of this Iwahori–Hecke algebra is contained as a submatrix in the ℓ-modular decomposition matrix of E_{ℓ}(G^{F} , 1).

7 Cuspidal unipotent Brauer characters in the general linear groups

(7.1) Here we consider the general linear group G^{F} = GL_{n}(q), n ≥ 2, where q is a power of a prime p. We fix a prime ℓ different from p and write d for the order of q modulo ℓ. Let Φ_{m}(q) denote the m-th cyclotomic polynomial evaluated at q. Furthermore, 𝔽_{q} is the field with q elements and 𝔽̄_{q} its algebraic closure. The term Levi subgroup will always denote a split Levi subgroup. We begin with some elementary results on the general linear groups.

Lemma 7.2 The following statements are equivalent:

(a) ℓ divides the index of every proper Levi subgroup of GL_{n}(q).

(b) n = dℓ^{i} for some non-negative integer i.

(c) ℓ divides Φ_{n}(q).

Proof. We first prove that (a) implies (b). Write n = m_{−1} + dm with 0 ≤ m_{−1} < d and consider the ℓ-adic expansion of m,

m = m_{0} + m_{1}ℓ + … + m_{r}ℓ^{r},  0 ≤ m_{i} < ℓ, m_{r} ≠ 0.  (6)

Then

n = m_{−1} + m_{0}d + m_{1}dℓ + … + m_{r}dℓ^{r}  (7)

is the d-ℓ-adic expansion of n. To (7) we associate the Levi subgroup L^{F} isomorphic to

GL_{1}(q)^{m_{−1}} × GL_{d}(q)^{m_{0}} × GL_{dℓ}(q)^{m_{1}} × … × GL_{dℓ^{r}}(q)^{m_{r}},  (8)

where we have written GL_{s}(q)^{t} for the direct product of t copies of GL_{s}(q). Now L^{F} contains a Sylow ℓ-subgroup of GL_{n}(q) (see [36] for ℓ odd and [7, Theorem 1] for ℓ = 2). If (a) holds then L^{F} = GL_{n}(q). Furthermore, ℓ divides the order of GL_{n}(q), and so n = dℓ^{i} for some non-negative integer i.

By [27, Lemma IX.8.1], (b) implies (c), and the implication from (c) to (a) is clear since Φ_{n}(q) divides the index of every proper Levi subgroup of GL_{n}(q).

Lemma 7.3 Suppose ℓ divides Φ_{n}(q). Then the Coxeter torus of GL_{n}(q) contains regular ℓ-elements.

Proof. Let ζ generate the Sylow ℓ-subgroup of the multiplicative group of 𝔽_{q^{n}}. Then 𝔽_{q}(ζ) = 𝔽_{q^{n}} since ζ is not contained in any proper intermediate field of 𝔽_{∥^{⋉}}/𝔽_{∥}, and so ζ has n conjugates in the Galois group of 𝔽_{q^{n}}/𝔽_{q}. It follows that the element diag(ζ, ζ^{q}, ζ^{q^{2}}, …, ζ^{q^{n−1}}) is conjugate in GL_{n}(𝔽̄_{q}) to a regular ℓ-element in the Coxeter torus of GL_{n}(q).

Lemma 7.4 Let 1_{m} denote the (m × m)-identity matrix. Suppose x ∈ GL_{n}(q) has no eigenvalue equal to 1. Then the centralizer in GL_{m+n}(q) of the matrix

( \begin{matrix}1_{m} & 0 \\ 0 & x \end{matrix} )

consists of all matrices of the form

( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )

with a ∈ GL_{m}(q) and with b ∈ GL_{n}(q) centralizing x.

Proof. This can be verified by direct calculation.

(7.5) We now investigate the question of when the modular Steinberg character of GL_{n}(q) is cuspidal.

Theorem 7.6 The ℓ-modular Steinberg character φ_{St} of GL_{n}(q) is cuspidal if and only if n = dℓ^{i} for some non-negative integer i.

Proof. If n = dℓ^{i}, then ℓ divides the index of every proper Levi subgroup by Lemma 7.2, and so φ_{St} is cuspidal by Theorem 4.2.

Suppose now that φ_{St} is cuspidal. Consider the Levi subgroup L^{F} corresponding to the d-ℓ-adic expansion of n as in (8), but with the factor GL_{1}(q)^{m_{−1}} replaced by GL_{m_{−1}}(q). Let D_{i} denote a Sylow ℓ-subgroup of the direct factor L_{i}^{F} = GL_{dℓ^{i}}(q) of L^{F}, 0 ≤ i ≤ r. Let D be the direct product of the D_{i} with the appropriate multiplicities. Then D is a Sylow ℓ-subgroup of L^{F} and also one of GL_{n}(q) by [36, 7].

We now wish to show that the centralizer of D in GL_{n}(q) is contained in L^{F}. This is done by induction on the number of direct factors L_{i}^{F} of L^{F}. Let x ∈ L_{r}^{F} denote a non-identity ℓ-element in the Coxeter torus of L_{r}^{F}, which exists by Lemma 7.3. Since the eigenvalues of x lie in a single orbit of a certain Galois group, no eigenvalue of x equals 1. By Lemma 7.4, any element of GL_{n}(q) centralizing D is thus of the form

( \begin{matrix}a & 0 \\ 0 & b \end{matrix} )

with a ∈ GL_{n−dℓ^{r}}(q) and b ∈ GL_{dℓ^{r}}(q). Furthermore, a centralizes a suitable Sylow ℓ-subgroup of GL_{n−dℓ^{r}}(q) and we are done by induction.

Thus D satisfies (1) and by Theorem 4.2 the semisimple vertex of φ_{St} is contained in L^{F}. By our assumption we must have GL_{n}(q) = L^{F} and so n = dℓ^{i} for some i or n = m_{−1}. If n = m_{−1}, the order of GL_{n}(q) is not divisible by ℓ, in which case φ_{St} is certainly not cuspidal.

(7.7) The first part of the following theorem is due to Dipper [9, Theorem 3.5]. A different proof has been given by James [28, Theorem 3.6], using Gelfand’s original construction of the cuspidal representations of GL_{n}(q). The second part is implicit in [13]. We give another proof for this part based on Theorem 4.2.

Theorem 7.8 In GL_{n}(q) the reduction modulo ℓ of every cuspidal ordinary character contained in a unipotent block is irreducible. Every ℓ-modular cuspidal unipotent character is liftable to an ordinary irreducible character.

Proof. Let χ be an ordinary irreducible cuspidal character of GL_{n}(q) contained in a unipotent block. Every ℓ-modular constituent of χ̂ is then also cuspidal by [25, Lemma 4.3(a)]. As we have seen in (6.9(ii)), a unipotent block contains at most one cuspidal Brauer character φ, and if so, φ = φ_{St}. Hence χ̂ is a multiple of φ_{St}. In particular, χ is a constituent of the Gelfand-Graev character. Since the latter is multiplicity free, χ̂ = φ_{St} by Brauer reciprocity. This proves the first part of the theorem.

To prove the second, let φ be a cuspidal ℓ-modular character in a unipotent block. Then we have φ = φ_{St} by (6.9(ii)), and so φ_{St} is cuspidal. Thus ℓ divides Φ_{n}(q). By Lemma 7.3, the Coxeter torus of GL_{n}(q) contains a regular ℓ-element. The corresponding Deligne-Lusztig character is, up to sign, an ordinary cuspidal character χ in a unipotent block of GL_{n}(q). By the first part of the theorem, χ is a lift of φ_{St}.

The above theorem is true without the restriction to unipotent blocks. One has to adapt the arguments to the union of blocks E_{ℓ}(G^{F} , s), where s is a semisimple ℓ′-element, and to Harish-Chandra induction of suitable Levi subgroups.

(7.9) We can now determine those Levi subgroups of GL_{n}(q) containing a cuspidal unipotent ℓ-modular character. Let

λ = (λ_{1}, λ_{2}, …, λ_{s}), λ_{1} ≥ λ_{2} ≥ … λ_{s} ≥ 0

be a partition of n and

L^{F} = GL_{λ_{1}}(q) × GL_{λ_{2}}(q) × … × GL_{λ_{s}}(q)

the corresponding Levi subgroup of GL_{n}(q). It is clear that an irreducible Brauer character

φ = φ_{1} ⊗ φ_{2} ⊗ … ⊗ φ_{s},  (9)

where φ_{i} is an irreducible ℓ-modular character of GL_{λ_{i}}(q), is cuspidal if and only if φ_{i} is cuspidal for all i. In GL_{1}(q) every irreducible character is cuspidal (but only the trivial character is unipotent). Thus φ in (9) is cuspidal if and only if every λ_{i} either equals 1 or else is of the form dℓ^{j} for some j ≥ 0. Changing notation for partitions, the semisimple vertices of irreducible unipotent Brauer characters thus correspond to partitions of the form

(1^{m_{−1}}, d^{m_{0}}, (dℓ)^{m_{1}}, …, (dℓ^{r})^{m_{r}}) ,  m_{i} ≥ 0.  (10)

Since every semisimple vertex contains exactly one cuspidal unipotent Brauer character, the ℓ-modular Harish-Chandra series are determined by partitions of the form (10). In the special case of dℓ > n, the Harish-Chandra series is determined by a pair (m_{−1}, m_{0}) of non-negative integers such that n = m_{−1} + dm_{0}.

Which irreducible Brauer characters lie in the Harish-Chandra series determined by (10)? Dipper and Du have announced a complete solution to this question, so we do not want to pursue it here.

8 Cuspidal unipotent Brauer characters in the unitary groups

(8.1) Let G^{F} = GU_{n}(q), n ≥ 1, denote the unitary group over 𝔽_{q}. It will be convenient to let GU_{0}(q) denote the trivial group. Fix a prime ℓ not dividing q. In contrast to the previous section, let d denote the order of −q modulo ℓ. If x is a real number, we shall write |x| for the absolute value of x and [x] for the largest integer not exceeding x. The p′-part of the integer z will be denoted by z_{p′}. As above, a Levi subgroup will mean a split Levi subgroup. We first give a criterion similar to the one in Lemma 7.2 of when ℓ divides the index of every proper Levi subgroup. This is more complicated than in the case of the linear groups, since Ennola duality (the passage from q to −q) does not send split Levi subgroups to split Levi subgroups.

Lemma 8.2 For n ≥ 2 the following statements are equivalent:

(a) ℓ divides the index of every proper Levi subgroup of GU_{n}(q).

(b) d is odd and divides one of n or n − 1.

Proof. The result is clear for n = 2, so we assume that n ≥ 3 in the following. To prove the implication from (a) to (b) we consider the Levi subgroup isomorphic to GU_{n−2}(q) × GL_{1}(q^{2}). The p′-part of its index in GU_{n}(q) is

\frac{(q^{n} − (−1)^{n})(q^{n−1} − (−1)^{n−1})}{q^{2} − 1}.

This implies that d divides one of n or n − 1.

Let

L^{F} = \{ \begin{matrix}GL_{(n−1)/2}(q^{2}) × GU_{1}(q), & if n is odd, \\ GL_{n/2}(q^{2}), & if n is even. \end{matrix}

In both cases, the p′-part of the index of L^{F} in G^{F} is a product of terms q^{i} + 1, where i is odd. Since ℓ divides one of these, d has to be odd.

To prove that (b) implies (a) we have to show that Φ_{d}(−q) divides the index of every proper parabolic subgroup of GU_{n}(q). For this purpose it suffices, of course, to consider the maximal parabolic subgroups of G^{F}. Their Levi subgroups are of the form

L_{m}^{F} = GL_{m}(q^{2}) × GU_{n−2m}(q),  1 ≤ m ≤ [\frac{n}{2}] .  (11)

We have

|GU_{n}(q)|_{p′} = |∏_{t=1}^{∞} Φ_{t}(−q)^{[n/t]}| ,

and

|L_{m}^{F}|_{p′} = |∏_{t=1}^{∞} Φ_{t}(−q)^{[(n−2m)/t]} ∏_{t=1}^{∞} Φ_{t}(q^{2})^{[m/t]}| ,

and so

|GU_{n}(q): L_{m}^{F}|_{p′} = |∏_{t=1}^{∞} Φ_{t}(−q)^{[n/t]−[(n−2m)/t]} ∏_{t=1}^{∞} Φ_{t}(q^{2})^{−[m/t]}| .  (12)

We have to show that Φ_{t}(−q) occurs with positive multiplicity in (12) if t is odd and divides one of n or n − 1.

Now Φ_{t}(q^{2}) = Φ_{t}(−q)Φ_{2t}(−q) for odd t and Φ_{t}(q^{2}) = Φ_{2t}(−q) for even t. Hence for odd t, the multiplicity of Φ_{t}(−q) in (12) equals

[\frac{n}{t}] − [\frac{n − 2m}{t}] − [\frac{m}{t}] .  (13)

Now

[\frac{n − 2m}{t}] + [\frac{m}{t}] ≤ \frac{n − m}{t} < \frac{n}{t},

where the latter inequality follows from m ≥ 1. Thus if t divides n, the multiplicity (13) is positive.

Now let t > 1 be an odd divisor of n − 1. Write n − 1 = at, a ∈ ℤ, and m = bt + s with b ∈ ℤ and 0 ≤ s < t. Then

[\frac{n − 2m}{t}] + [\frac{m}{t}] = [a − 2b + \frac{1 − 2s}{t}] + [b + \frac{s}{t}] = a − b + [\frac{1 − 2s}{t}] < a,

since m = s ≥ 1 if b = 0. Thus in this case, too, the multiplicity (13) is positive.

Theorem 8.3 The ℓ-modular Steinberg character φ_{St} of GU_{n}(q) is cuspidal if and only if d, the order of −q modulo ℓ, is odd and divides one of n or n − 1. In particular, φ_{St} is always cuspidal if ℓ divides q + 1.

Proof. If d is odd and divides one of n or n − 1, then ℓ divides the index of every proper Levi subgroup of GU_{n}(q) by Lemma 8.2, and φ_{St} is cuspidal by Theorem 4.2.

Now suppose that φ_{St} is cuspidal. For n = 2 and n = 3 see Section 9. So assume n ≥ 4 in the following. We first wish to show that d divides one of n or n − 1. For this purpose we may assume that d ≥ 3. Consider the Levi subgroup

L^{F} ≅ GU_{n−2}(q) × GL_{1}(q^{2}).

By [15, Proposition (1C)] we have

R_{L}^{G}(χ_{(1^{n−2})} ⊗ 1) = χ_{(3,1^{n−3})} + χ_{(2^{2},1^{n−4})} + χ_{(1^{n})}.

Here, χ_{λ} denotes the unipotent character of the unitary group corresponding to the partition λ as explained in the introduction to Section 6. If χ_{(1^{n})}, the Steinberg character of GU_{n}(q), does neither lie in the ℓ-block containing χ_{(3,1^{n−3})} nor in the ℓ-block containing χ_{(2^{2},1^{n−4})}, then the Harish-Chandra induction of the Gelfand-Graev character of L^{F} to G has an indecomposable direct summand containing the Steinberg character but no other unipotent character. This direct summand must be the component of the Gelfand-Graev character of G which lies in the unipotent blocks. This implies that φ_{St} is not cuspidal, contrary to our assumption.

Thus χ_{(1^{n})} and χ_{(3,1^{n−3})} or χ_{(1^{n})} and χ_{(2^{2},1^{n−4})} lie in a common ℓ-block. By the Nakayama conjecture for GU_{n}(q) [15], the first case implies that d divides n, and the second that d divides n − 1. It remains to show that d is odd.

Assume if possible that d = 2e is even. Then e is the order of q^{2} modulo ℓ. Write n = 2em + ϵ, where ϵ = 0 if d divides n, and ϵ = 1 otherwise. The ℓ-adic expansion (6) of m determines the Levi subgroup

L^{F} = GL_{e}(q^{2})^{m_{0}} × GL_{eℓ}(q^{2})^{m_{1}} × … × GL_{eℓ^{r}}(q^{2})^{m_{r}} × GU_{ϵ}(q)

of G^{F}. We put L_{i}^{F} = GL_{eℓ^{i}}(q^{2}) for the linear direct factors of L^{F}. Let D_{i} denote a Sylow ℓ-subgroup of L_{i}^{F}, and define D := D_{0}^{m_{0}} ×D_{1}^{m_{1}} ×…×D_{r}^{m_{r}} ≤ L^{F}. Then D is a Sylow subgroup of GL_{em}(q^{2}) ≤ G^{F} by the results of [36, 7], and thus also one of G^{F}. We shall show below that C_{G^{F}} (D) ≤ L^{F}. But this implies that φ_{St} is not cuspidal by Theorem 4.2, a contradiction.

Let x denote a non-trivial ℓ-element in the Coxeter torus of GL_{eℓ^{r}}(q^{2}) which exists by Lemma 7.3. Then D_{r} contains an element conjugate to

( \begin{matrix}x & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & F_{1}(x) \end{matrix} ) ,  (14)

where 1 denotes the (n−2eℓ^{r})×(n−2eℓ^{r})-identity matrix and F_{1} a suitable Frobenius morphism of GL_{eℓ^{r}}(𝔽̄_{q}). The eigenvalues of x are all of the form ζ^{q^{2i}} for a suitable ℓ-element ζ in the multiplicative group of a finite extension field of 𝔽_{∥^{⊭}}. The eigenvalues of F_{1}(x) are the (−q)-th powers of the eigenvalues of x. Since d is even, this implies that the sets of eigenvalues of x and of F_{1}(x) are disjoint. Since, furthermore, none of these eigenvalues equals 1, it is easily checked that an element of GU_{n}(q) centralizing (14) must lie in GL_{eℓ^{r}}(q^{2}) × GU_{n−2eℓ^{r}}(q). By way of induction this implies C_{G^{F}} (D) ≤ L^{F}.

If n = 5 and d = 3, then there is only one proper Levi subgroup containing a Sylow ℓ subgroup D of GU_{5}(q), namely L^{F} = GU_{3}(q) × GL_{1}(q^{2}). In this situation, however, condition (1) is not satisfied. Let X be an OL^{F} - lattice in the ordinary Steinberg module of L^{F}. The splitting of R_{L}^{G}(X) into indecomposable summands is achieved by the block distribution. This example shows that condition (1) is too strong in some cases, and that we do need the results of Fong and Srinivasan on the distribution of the characters into ℓ-blocks in the above proof.

(8.4) We have shown in Section 6 that if n = s(s+1)/2 for some s ≥ 1, then the ordinary cuspidal unipotent character remains irreducible modulo ℓ for every prime ℓ not dividing q. This then gives a modular cuspidal unipotent character. We shall now show that if d is even and ℓ > n, then there is at most one such modular cuspidal unipotent character. So for even d the situation is similar to that for the general linear groups. It is not clear whether the assumption ℓ > n is really necessary for this result.

Theorem 8.5 Suppose that d is even and that ℓ > n. Then G^{F} = GU_{n}(q) has a cuspidal unipotent ℓ-modular character if and only if n = s(s + 1)/2 for some s ≥ 1. In this case, there is exactly one, namely the reduction modulo ℓ of the cuspidal unipotent ordinary character, which lies in a block of defect zero.

Proof. Let M be a simple unipotent kG^{F} -module. Suppose first that M lies in a block with a non-trivial defect group D. Since ℓ > n, it follows from [4, 5.4(1)] and the description of the defect groups in [loc. cit., 5.24(2)] that the centralizer of D is contained in a proper Levi subgroup L^{F} of G^{F}. In particular, D satisfies condition (1). Furthermore, D is abelian, since ℓ > n. By Corollary 2.7, the L-vertex of M is contained in L^{F}. In particular, M is not cuspidal.

Suppose that G^{F} has a cuspidal unipotent ℓ-modular character. Then it must lie in a block of defect zero, and its lift must be a cuspidal ordinary character. Since every unipotent irreducible Brauer character is a constituent of some ordinary unipotent character, the lift must be a unipotent character. Hence n = s(s + 1)/2.

If n = s(s + 1)/2, and d is even, then the cuspidal unipotent ordinary character lies in a block of defect 0, since the Young diagram corresponding to its partition has only hooks of odd lengths.

We remark that the condition ℓ > n can be weakened to the conditions that the Sylow ℓ-subgroup of G^{F} is abelian, and that the non-trivial defect groups of the unipotent ℓ-blocks are contained in proper Levi subgroups and satisfy (1).

(8.6) As an example for the results and methods developed so far, we take the group G^{F} = GU_{8}(q) and assume that ℓ ≥ 5 and divides q^{2} + 1. Thus d = 4. Consider the Levi subgroup L^{F} = GL_{2}(q^{2}) × GL_{2}(q^{2}). By Theorem 7.6, the ℓ-modular Steinberg character of L^{F} is cuspidal. Let M be the corresponding cuspidal unipotent kL^{F} -module. We are interested in the structure of R_{L}^{G}(M).

Observe that W (L^{F}) is a group of permutation matrices isomorphic to a dihedral group of order 8. Choose a Coxeter torus T^{F} of L^{F} which is invariant under W (L^{F}) and let D be a Sylow ℓ-subgroup of L^{F} contained in T^{F}. Then D also is a Sylow subgroup of G^{F}. As in the proof of Theorem 8.3, the centralizer of D in G^{F} is contained in L^{F} and thus equals T^{F}. Therefore N_{G^{F}} (D) = N_{G^{F}} (T^{F}) ≤ L^{F} W (L^{F}) = N_{G^{F}} (L^{F}).

Since M is invariant in N_{G^{F}} (L^{F}), we have W (L^{F} , M) = W (L^{F}), and so all assumptions of Proposition 2.9 are satisfied. Since every unipotent Brauer character of G^{F} is real valued, we have that R_{L}^{G}(M) is semisimple

and splits into irreducibles in the same way as Z := M ↑_{L^{F}}^{N_{G^{F}} (L^{F})}. Now we have to determine the structure of the latter module. We shall show below that M is extendable to its inertia group N_{G^{F}} (L^{F}). If M′ is such an extension, then, by Clifford theory, Z is isomorphic to the tensor product of M′ with the regular module of W (L^{F}), and thus has 5 pairwise nonisomorphic constituents.

It thus remains to show that M is extendable to N_{G^{F}} (L^{F}). Suppose this is not the case. Then every constituent of Z occurs at least twice and the same is true for the semisimple module R_{L}^{G}(M). Let P_{M} denote the projective cover of M. Then R_{L}^{G}(P_{M}) is projective and maps onto R_{L}^{G}(M). Hence the projective cover X of R_{L}^{G}(M) is a direct summand of R_{L}^{G}(P_{M}) and every indecomposable component of X occurs at least twice. Since M is liftable to an ordinary character of the form ±R_{T,ϑ}^{G}, where ϑ is a suitable character of T^{F} of ℓ-power order, the Brauer character of R_{L}^{G}(M) is ±R̂_{T,1}^{G}. The scalar product of ±R̂_{T,1}^{G} with the Gelfand-Graev character of G^{F} is non-zero. Since R_{L}^{G}(M) is semisimple, this implies that the Steinberg component Γ_{St} of the Gelfand-Graev representation is a direct summand of X. However, the ordinary character of R_{L}^{G}(P_{M}) contains the Steinberg character of G^{F} exactly once as a constituent, contradicting the above considerations.

We finally remark that any lift of M has an inertia group strictly smaller than N_{G^{F}} (L^{F}). We therefore cannot apply the method described in (6.11) using the Iwahori-Hecke algebra of a suitable lift of M.

9 Modular Harish-Chandra series for the unitary groups

In this section we consider the distribution into series of the ℓ-modular unipotent Brauer characters of the general unitary group GU_{n}(q). The results are far from being complete. The problem seems to be considerably simpler if d, the order of −q modulo ℓ is even. In this case we make a precise conjecture on the distribution into Harish-Chandra series, and we can prove this conjecture in some special cases. There is also a conjecture for d = 1, which is probably the most difficult case as far as decomposition numbers are concerned.

We keep the notation of the preceding sections. In addition we write λ′ for the partition conjugate to λ. The irreducible unipotent ℓ-modular and ordinary characters are parametrized by partitions μ ⊢ n, and we write φ_{μ} and χ_{μ} for the corresponding modular respectively ordinary character (see (6.7)).

Recall that the split Levi subgroups of GU_{n}(q) have the shape

GL_{n_{1}}(q^{2}) × ⋯ × GL_{n_{r}}(q^{2}) × GU_{n_{0}}(q),

with n_{0} + 2 ∑_{i≥1} n_{i} = n.

(9.1) In order to state our conjectures we have to introduce some more notation. Let λ be a partition. We write |λ| := m, if λ ⊢ m. The 2-core and the 2-quotient of λ will be denoted by λ_{0} and (λ_{1}, λ_{2}) respectively. It is well known that there are unique partitions λ^{(−1)}, λ^{(0)}, λ^{(1)}, …, λ^{(r)}, such that λ^{(−1)} is d-regular and λ^{(i)} is ℓ-regular for 0 ≤ i ≤ r and

λ′ = λ^{(−1)}′ + dλ^{(0)}′ + dℓλ^{(1)}′ + … + dℓ^{r}λ^{(r)}′.  (15)

This is the d-ℓ-adic expansion of λ. If d = 1 we have that λ^{(−1)} is the empty partition and (15) is called the ℓ-adic expansion of λ.

Conjecture 9.2 Let d = 2e be even. Then the unipotent Brauer character φ_{μ} of GU_{n}(q) lies in the ℓ-modular Harish-Chandra series (L^{F} , ϑ) with

L^{F} = GL_{1}(q^{2})^{m_{−1}} ×GL_{e}(q^{2})^{m_{0}} ×GL_{eℓ}(q^{2})^{m_{1}} ×⋯×GL_{eℓ^{r}}(q^{2})^{m_{r}} ×GU_{n_{0}}(q),

ϑ = φ_{St} ⊗ φ_{St} ⊗ φ_{St} ⊗ ⋯ ⊗ φ_{St} ⊗ φ_{μ_{0}},

where n_{0} = |μ_{0}| and the m_{i} are obtained from the e-ℓ-adic expansion (15) of μ_{1} and μ_{2} as m_{i} = |μ_{1}^{(i)}| + |μ_{2}^{(i)}|, i ≥ −1.

Note that in particular this would imply that for even d the only cuspidal ℓ-modular unipotent characters are the ordinary cuspidal ones. This has been shown to be true for large enough ℓ in Theorem 8.5. At present we can prove this conjecture only in two special cases.

Proposition 9.3 Let d = 2e be even and assume that φ_{μ} lies in a block with cyclic defect group. Then Conjecture 9.2 holds.

Proof. The case of cyclic defect was studied by Fong and Srinivasan [16]. According to their results, φ_{μ} lies in a block with cyclic defect group if μ has precisely one d-hook. Hence we have μ̃ ⊢ n − d for the d-core μ̃ of μ. Since d is even, removing a d-hook can also be achieved by removing e 2-hooks, and hence the 2-core of μ̃ coincides with the one of μ. Consider the Levi subgroup

M^{F} = GL_{1}(q^{2}) × GL_{e−1}(q^{2}) × GU_{n−d}(q),

with unipotent characters

χ_{i} = χ_{(1)} ⊗ χ_{ν_{i}} ⊗ χμ̃, 1 ≤ i ≤ e − 1,

where ν_{i} is the hook partition of e − 1 of leg length i − 1. By definition of d, the order of GL_{1}(q^{2}) × GL_{e−1}(q^{2}) is not divisible by ℓ, hence χ_{i} is projective. It was shown in [loc. cit., 5A] that R_{M}^{G}(χ_{i}) contains all projective indecomposables in the block of χ_{μ} except for those involving the exceptional node. In particular this implies that all characters not directly connected to the exceptional node lie in the same series as some χ̂_{i}. Since all factors of χ_{i} have defect zero, by Theorem 6.10 this series is the same as its ordinary Harish-Chandra series, which obviously is given by (L^{F} , λ), where

L^{F} = GL_{1}(q^{2}) × GL_{1}(q^{2})^{e−1} × GL_{1}(q^{2})^{(n−d−n_{0})/2} × GU_{n_{0}}(q)

and

λ = 1 ⊗ 1 ⊗ 1 ⊗ χ̂_{μ_{0}}.

On the other hand, those characters not connected to the exceptional node are precisely the ones having e-regular 2-quotient by [loc. cit.], which proves the assertion of the proposition in this case.

It remains to consider the characters connected to the exceptional node. Again Fong and Srinivasan show that the corresponding projective indecomposable characters do not occur in R_{M}^{G}(χ) for any unipotent χ. If instead we start from the Levi subgroup

M′^{F} = GL_{e}(q^{2}) × GU_{n−d}(q),

then the modular Steinberg character φ_{St} of its first factor is cuspidal by Theorem 7.6. It is connected to the exceptional node in the Brauer tree for GL_{e}(q^{2}). By [16] the projective indecomposable modules corresponding to the exceptional node in the tree for χ_{μ} are direct summands of R_{M′}^{G}(Φ), where Φ is the projective cover of φ_{St} ⊗ χ̂μ̃. Furthermore, R_{M′}^{G}(Φ) has no other components lying in the unipotent blocks. Hence the two irreducible Brauer characters which are connected to the exceptional node lie in the same Harish-Chandra series as φ_{St} ⊗ χ̂μ̃, which again can easily be seen to coincide with the one of (L′^{F} , λ) for

L′^{F} = GL_{e}(q^{2}) × GL_{1}(q^{2})^{(n−d−n_{0})/2} × GU_{n_{0}}(q),
λ = φ_{St} ⊗ 1 ⊗ χ̂_{μ_{0}}.

The unipotent characters connected to the exceptional node have a 2-quotient consisting of one e-regular partition and one partition possessing exactly one vertical e-hook by [loc. cit.]. This completes the proof.

We can also prove Conjecture 9.2 in another important special case, namely when the 2-core of a partition consists of a pair of e-regular partitions and ℓ is large.

Proposition 9.4 Suppose that d = 2e is even and that ℓ > n. Then φ_{μ} lies in the Harish-Chandra series corresponding to

L^{F} = GL_{1}(q^{2})^{(n−n_{0})/2} × GU_{n_{0}}(q),
ϑ = 1 ⊗ φ_{μ_{0}}

if and only if μ_{1} and μ_{2} are both e-regular partitions.

Proof. First observe that the cuspidal unipotent Brauer characters in the statement of the proposition are liftable to defect 0 characters, hence are projective. The assertion now follows at once from the considerations in (6.11) and the main results of Dipper and James in [14, Section 5].

We hope to obtain more general results concerning Conjecture 9.2 in a subsequent paper.

(9.5) For odd d the situation is much more complicated. This comes from the fact that removing odd d-hooks does not preserve the 2-core, in contrast to the situation for even d-hooks. Thus one of the main arguments in the proof of Proposition 9.3 does not carry through. For the single case of d = 1, at least the converse is true: removing 2-hooks preserves the 1-core. In this case 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. This is the same as saying that all parts of μ′_{1} are distinct. By Proposition 6.8, the Brauer character φ_{μ} can only be cuspidal if μ′ is 2-regular. Our observations for small rank lead to the following conjecture.

Conjecture 9.6 Let d = 1. Then φ_{μ} is cuspidal if and only if μ′ is 2regular. The unipotent character φ_{μ} of GU_{n}(q) lies in the ℓ-modular Harish-Chandra series (L^{F} , λ) with

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

where μ_{1} ⊢ m and the n_{i} are obtained from the ℓ-adic expansion of μ_{2} as n_{i} = |μ_{2}^{(i)}|.

Hence in this case the number of cuspidal unipotent characters would be maximal possible, namely equal to the number of cuspidal classes (see Proposition 6.8). Note that here the unipotent characters form a single ℓ-block.

(9.7) For even d and n ≤ 10 we have computed the unipotent decomposition matrices of GU_{n}(q) and the distribution of unipotent characters into modular Harish-Chandra series. These results are given below in form of tables. In all these cases, Conjecture 9.2 can be verified by inspection. In the case of cyclic defect groups, the Brauer trees were determined by Fong and Srinivasan in [16]. The distribution into ℓ-modular Harish-Chandra series in these cases is contained in Proposition 9.3.

For d = 2, i.e., ℓ | q − 1, we only have to consider those cases where ℓ divides the order of the Weyl group W^{F}. If ℓ does not divide the order of W^{F} but ℓ | q −1, the unipotent decomposition matrix of G^{F} is an identity matrix by [23, Korollar 5.3.2]. This leaves the cases ℓ = 3 | q −1, 6 ≤ n ≤ 10 and ℓ = 5 | q −1, n = 10 to consider. If ℓ | q −1 the decomposition matrix of the Weyl group is contained as a submatrix in the unipotent decomposition matrix of G^{F} (see [19, Proposition 4.6]). This gives already the largest part of the tables in this case. The missing columns were obtained by inducing the Gelfand-Graev characters of suitable Levi subgroups and by applying the Propositions 2.5 and 2.11 and Theorem 8.3. The results are given in form of adjustment matrices as introduced by James in [29, Appendix 1]. The unipotent decomposition matrix for GU_{n}(q) is then obtained from these as described by James. Note that in the case that ℓ divides q −1, the generic unipotent decomposition matrix is the identity matrix.

The remaining cases are d = 4, n = 8, 9, 10 and ℓ ≥ 5. Here we made use of the following techniques. First we computed the decomposition matrices of certain Hecke algebras corresponding to ordinary cuspidal unipotent characters. This information was used as described in (6.11). The computations were done with the help of the computer algebra system GAP [35] and with the same methods as in [20]. Then, of course, we used Harish-Chandra induction of projective characters from proper Levi subgroups and the block distribution of the ordinary characters as determined by Fong and Srinivasan in [15]. Lower bounds for decomposition numbers were obtained from relations coming from non-unipotent ordinary characters in the blocks. Finally, the case d = 4 and n = 8 has already been treated in Example (8.6).

Adjustment matrix for l = 5 | q − 1

2A_{9}(q)

10 1
91 . 1
81^{2} 1 . 1
71^{3} . 1 . 1
61^{4} . . 1 . 1
51^{5} … 1 . 1
41^{6} …. 1 . 1
31^{7} . . . . . 1 . 1
21^{8} . . . . . . 1 . 1
1^{10} . . . . . . . 1 . 1

Adjustment matrices for l = 3 | q − 1

\begin{matrix}^{2}A_{5}(q) \\ 6 & 1 \\ 51 & . & 1 \\ 41^{2} & 1 & . & 1 \\ 31^{3} & . & 1 & . & 1 \\ 21^{4} & . & . & 1 & . & 1 \\ 1^{6} & . & . & . & 1 & . & 1 \end{matrix}  \begin{matrix}^{2}A_{6}(q) \\ 7 & 1 \\ 52 & 1 & 1 \\ 421 & . & . & 1 \\ 321^{2} & . & 1 & . & 1 \\ 2^{2}1^{3} & . & . & 1 & . & 1 \\ 1^{7} & . & . & . & . & 1 & 1 \end{matrix}

2A_{7}(q)

8  1
71 . 1
62 . . 1
53 … 1
431 1 … 1
42^{2} . . 1 . . 1
3^{2}1^{2} … 1 . . 1
32^{2}1 . 1 . . . . . 1
2^{3}1^{2} . . . . . 1 . . 1
2^{2}1^{4} . . . . . . 1 . . 1
21^{6} …. 1 . . . . . 1
1^{8} . . . . . . . 1 … 1

2A_{8}(q)

9  1
81 . 1
71^{2} . . 1
63 . 1 . 1
621 …. 1
54 1 …. 1
52^{2} . . 1 … 1
4^{2}1 . . . . . . . 1
432 … 1 …. 1
3^{2}21 . . . . . . . . . 1
3^{2}1^{3} . . . . . . . 1 . . 1
32^{3} . . . . . . 1 …. 1
321^{4} . . . . . 1 . . . . . . 1
31^{6} . . . . . . . . . . 1 . . 1
2^{4}1 …. 1 . . . . . . . . . 1
2^{3}1^{3} . . . . . . . . . 1 . . . . . 1
21^{7} . . . . . . . . . . . . . . . 1 1
1^{9} . . . . . . . . . . . . . . 1 . . 1

2A_{9}(q)\begin{matrix}10 & 1 \\ 91 & . & 1 \\ 82 & . & . & 1 \\ 81^{2} & . & . & . & 1 \\ 73 & . & . & . & . & 1 \\ 71^{3} & . & . & . & . & . & 1 \\ 64 & . & . & . & . & . & . & 1 \\ 631 & . & . & . & 1 & . & . & . & 1 \\ 621^{2} & . & . & . & . & . & . & . & . & 1 \\ 5^{2} & . & . & . & . & . & . & . & . & . & 1 \\ 532 & . & . & . & . & . & . & . & . & . & . & 1 \\ 52^{2}1 & . & . & . & . & . & 1 & . & . & . & . & . & 1 \\ 4^{2}2 & . & . & . & . & . & . & 1 & . & . & . & . & . & 1 \\ 4^{2}1^{2} & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 \\ 43^{2} & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 431^{3} & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 42^{3} & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 \\ 421^{4} & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 \\ 41^{6} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & 1 \\ 3^{3}1 & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 3^{2}2^{2} & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & 1 \\ 3^{2}21^{2} & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & 1 \\ 32^{2}1^{3} & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 31^{7} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & 1 \\ 2^{5} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 \\ 2^{4}1^{2} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & 1 \\ 2^{3}1^{4} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 2^{2}1^{6} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 \\ 21^{8} & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 1^{10} & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \end{matrix}

37

Unipotent decomposition matrices for d = 4 (so ℓ | q^{2} + 1)

2A_{7}(q)

8  1
71 . 1
62 . . 1
61^{2} 1 . . 1
53 …. 1
4^{2} . . . . . 1
521 . . . . . . 1
431 … 1 … 1
51^{3} . 1 . . . . . . 1
42^{2} . . . . . . . . . 1
421^{2} . . . . . 1 …. 1
3^{2}2 . . . . . 1 . . . . . 1
3^{2}1^{2} . . . . . . . . . . . . 1
41^{4} 1 . . 1 … 1 . . . . . 1
32^{2}1 . . . . . . . . 1 . . . . . 1
321^{3} . . . . . . . . . . . . . . . 1
2^{4} . . . . . 1 …. 1 1 …. 1
2^{3}1^{2} . . 1 . . . . . . . . . . . . . . 1
31^{5} . 1 . . . . . . 1 . . . . . 1 … 1
2^{2}1^{4} …. 1 . . . . . . . . . . . . . . 1
21^{6} 1 . . . . . . . . . . . . 1 . . . . . . 1
1^{8} . 1 . . . . . . . . . . . . . . . . 1 . . 1

2A_{8}(q) \begin{matrix}9 & 1 \\ 81 & . & 1 \\ 72 & 1 & . & 1 \\ 71^{2} & . & . & . & 1 \\ 63 & . & . & . & . & 1 \\ 621 & . & . & . & . & . & 1 \\ 61^{3} & . & . & . & . & . & . & 1 \\ 54 & . & . & 1 & . & . & . & . & 1 \\ 531 & . & . & . & . & . & . & . & . & 1 \\ 52^{2} & . & . & . & . & . & . & . & . & . & 1 \\ 521^{2} & 1 & . & 1 & . & . & . & . & 1 & . & . & 1 \\ 51^{4} & . & . & . & . & . & . & . & . & 1 & . & . & 1 \\ 4^{2}1 & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 432 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 431^{2} & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 \\ 42^{2}1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 421^{3} & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & 1 \\ 41^{5} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 \\ 3^{3} & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & 1 \\ 3^{2}21 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 3^{2}1^{3} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 32^{3} & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 32^{2}1^{2} & . & . & . & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & 1 & . & . & . & 1 \\ 321^{4} & 1 & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 31^{6} & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 2^{4}1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & 1 \\ 2^{3}1^{3} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 2^{2}1^{5} & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & 1 & . & 1 \\ 21^{7} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & 1 \\ 1^{9} & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 \end{matrix}

39

2A_{9}(q) \begin{matrix}10 & 1 \\ 91 & . & 1 \\ 82 & . & . & 1 \\ 81^{2} & . & . & . & 1 \\ 73 & . & . & . & . & 1 \\ 721 & . & . & . & . & . & 1 \\ 71^{3} & . & . & . & . & . & . & 1 \\ 64 & . & . & . & . & . & . & . & 1 \\ 631 & . & . & . & . & . & . & . & . & 1 \\ 62^{2} & . & . & 1 & . & . & . & . & . & . & 1 \\ 621^{2} & . & . & . & . & . & . & . & 1 & . & . & 1 \\ 61^{4} & 1 & . & . & . & . & . & . & . & 1 & . & . & 1 \\ 5^{2} & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 541 & . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 \\ 532 & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 \\ 531^{2} & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & 1 \\ 52^{2}1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 521^{3} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 51^{5} & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 \\ 4^{2}2 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 4^{2}1^{2} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 43^{2} & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 4321 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 431^{3} & . & . & . & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 42^{3} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & 1 \\ 42^{2}1^{2} & . & . & 1 & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & 1 \\ 421^{4} & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 41^{6} & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 3^{3}1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 3^{2}2^{2} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & 1 \\ 3^{2}21^{2} & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 3^{2}1^{4} & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & 1 \\ 32^{3}1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 32^{2}1^{3} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 321^{5} & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 \\ 31^{7} & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 2^{5} & . & . & . & . & . & . & . & 1 & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & 1 \\ 2^{4}1^{2} & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & 1 & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & 1 \\ 2^{3}1^{4} & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 2^{2}1^{6} & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & 1 \\ 21^{8} & 1 & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \\ 1^{10} & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & 1 \end{matrix}

40

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