Thomas Gerber
email: thomas.gerber@math.rwth-aachen.de

Nicolas Jacon
email: nicolas.jacon@univ-reims.fr

HARISH-CHANDRA SERIES IN FINITE UNITARY GROUPS AND CRYSTAL GRAPHS

Keywords: Mathematics Subject Classification. 20C33, 20C08, 20G42, 17B37, 81R50 Harish-Chandra series, unitary group, branching graph, Fock space, crystal basis, crystal graph

The distribution of the unipotent modules (in nondefining prime characteristic) of the finite unitary groups into Harish-Chandra series is investigated. We formulate a series of conjectures relating this distribution with the crystal graph of an integrable module for a certain quantum group. Evidence for our conjectures is presented, as well as proofs for some of their consequences for the crystal graphs involved. In the course of our work we also generalize Harish-Chandra theory for some of the finite classical groups, and we introduce their Harish-Chandra branching graphs.

Introduction

Harish-Chandra theory provides a means of labelling the simple modules of a finite group G of Lie type in non-defining characteristics, including 0. The set of simple modules of G (up to isomorphism) is partitioned into disjoint subsets, the Harish-Chandra series, each arising from a cuspidal simple module of a Levi subgroup of G. Inside each series, the modules are classified by the simple modules of an extended Iwahori-Hecke algebra arising from the cuspidal module representing the series. This yields, however, a rather indirect labelling of the simple modules, as it requires the classification of the cuspidal simple modules. Moreover, for each of these, the corresponding Iwahori-Hecke algebra has to be computed and its simple modules have to be classified. This program has been completed successfully by Lusztig for modules over fields of characteristic 0 (see [START_REF] Lusztig | Characters of reductive groups over a finite field[END_REF]). For modules over fields of positive characteristic, only partial results are known.

In some cases a different labelling of the simple modules of G is known. This arises from Lusztig's classification of the simple modules in characteristic 0, together with sufficient knowledge of Brauer's theory of decomposition numbers. This applies in particular to the general linear groups G = GL n (q) and the general unitary groups G = GU n (q), where the unipotent modules (in any non-defining characteristic) are labelled by partitions of n. For characteristic 0 this result is due to Lusztig and Srinivasan [START_REF] Lusztig | The characters of the finite unitary groups[END_REF], for prime characteristic it follows from work of Dipper [START_REF] Dipper | On the decomposition numbers of the finite general linear groups. II[END_REF] and Geck [START_REF] Geck | On the decomposition numbers of the finite unitary groups in nondefining characteristic[END_REF]. In these cases it is natural to ask how to determine the partition of the unipotent modules into Harish-Chandra series from these labels of the unipotent modules, i.e. from the partitions of n.

By work of Dipper and Du (see [START_REF] Dipper | Harish-Chandra vertices and Steinberg's tensor product theorem for general linear groups[END_REF]Section 4]), this can be done for the general linear groups. First attempts to find a similar description for the unitary groups are described in [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF]. It turned out, however, that this is possible only in a favourable case, the case of linear characteristic (see [START_REF] Gruber | Decomposition numbers of finite classical groups for linear primes[END_REF]Corollary 8.11] in conjunction with the above mentioned results by Dipper and Du). The general description of the Harish-Chandra series of the unitary groups and other classical groups is still open.

In this paper we present a series of conjectures which, when true and proved, will solve generalized versions of this problem, at least for large characteristics.

Let us now describe our main results and conjectures. As above, G denotes a finite group of Lie type, viewed as group with a split BNpair of characteristic p. We also let ℓ be a prime different from p. In this introduction, by a simple module for G we will always mean an absolutely simple module over a field of characteristic 0 or ℓ. In Section 2 we introduce a generalization of Harish-Chandra theory if G is a unitary, symplectic or orthogonal group of odd degree. Thus the Weyl group of G, as group with a BN-pair, is of type B. Instead of using all Levi subgroups for Harish-Chandra induction, we restrict to what we call pure Levi subgroups: those that arise from a connected subset of the Dynkin diagram of G which is either empty or else contains the first node ajacent to the double edge. This way we obtain more cuspidal modules, which we call weakly cuspidal. All main results of Harish-Chandra theory remain valid in this more general context. In particular, we obtain a distribution of the simple modules into weak Harish-Chandra series (Proposition 2.3). The usual Harish-Chandra series are unions of weak Harish-Chandra series. In characteristic 0, the two notions coincide for unipotent modules, as a Levi subgroup having a unipotent cuspidal module is pure by Lusztig's classification.

In Section 3 we prove some results on the endomorphism ring of a Harish-Chandra induced weakly cuspidal module. Theorem 3.2 states that, under some mild restrictions, this endomorphism ring is in fact an Iwahori-Hecke algebra of type B. Some information about the parameters of this algebra is also given. For example, if a simple weakly cuspidal module in characteristic ℓ lies in a block containing an ordinary cuspidal module, then the parameters of the two Iwahori-Hecke algebras are related through reduction modulo ℓ.

In Section 4 we define the Harish-Chandra branching graph for the unipotent modules of the classical groups considered. This graph records the socle composition factors of Harish-Chandra induced unipotent modules, very much in the spirit of Kleshchev's branching rules for modules of symmetric groups (see [START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. I[END_REF][START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. II[END_REF][START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux[END_REF][START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. IV[END_REF], in particular [40, Theorem 0.5]).

Section 5 contains our conjectures. These are restricted to the case of the unitary groups, since for classical groups apart from type A, no labelling of their unipotent modules in positive non-linear characteristic is known to date. We thus let G = GU n (q) from now on and we write e for the multiplicative order of -q in a field of characteristic ℓ. Following [27, Definition 5.3], we call ℓ linear for G, if e is even. For our conjectures, however, we assume that e is odd and larger than 1, so that in particular ℓ is non-linear for G. (The case e = 1, i.e. ℓ | q + 1 has been settled in [START_REF] Geck | Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type[END_REF].) Conjecture 5.4 concerns the relation between Harish-Chandra series of ordinary modules and those in characteristic ℓ. It predicts that if two unipotent modules of G, labelled by the partitions λ and µ, respectively, lie in the same weak Harish-Chandra series, then λ and µ have the same 2-core, i.e. the ordinary unipotent modules labelled by these two partitions also lie in the same Harish-Chandra series. In this sense the ℓ-modular weak Harish-Chandra series (of unipotent modules) form a refinement of the ordinary Harish-Chandra series. According to Conjecture 5.5, the e-core of λ should be a 2-core, if λ labels a weakly cuspidal unipotent module. This amounts to the assertion that if a unipotent ℓ-block contains a weakly cuspidal module and ℓ > n, then the block also contains an ordinary cuspidal module (not necessarily unipotent). Conjecture 5.7 relates the Harish-Chandra branching graphs with crystal graphs arising from canonical bases in submodules of Fock spaces of level 2, which are acted on by the quantum group U ′ v ( sl e ). This is in analogy to the case of Kleshchev's branching graph in characteristic p, which is isomorphic to the crystal graph of a Fock space of level 1 with an action of the quantum group U ′ v ( sl p ) (see [START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. I[END_REF][START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. II[END_REF][START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. III. Some corollaries and a problem of Mullineux[END_REF][START_REF] Kleshchev | Branching rules for modular representations of symmetric groups. IV[END_REF]). The conjecure is also put in perspective by the results of Shan [START_REF] Shan | Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras[END_REF] on the branching rules on the category O of the cyclotomic rational double affine Hecke algebras. Finally, Conjecture 5.8 is just a weaker form of Conjecture 5.7. Its statement gives an algorithm to compute the distribution of the unipotent modules in characteristic ℓ into weak Harish-Chandra series from the combinatorics of the crystal graph involved. In our conjectures we assume that ℓ is large enough compared to n, where we choose a bound to ensure that the ℓ-modular decomposition matrices of the Iwahori-Hecke algebras involved are generic, i.e. only depend on e and not on ℓ (see 5.7). In the computed examples, ℓ > n is good enough. We also compute the parameters of the Iwahori-Hecke algebra corresponding to a weakly cuspidal module under the assumption that Conjecture 5.5 holds true (Proposition 5.9). Finally, the truth of Conjectures 5.7 and 5.8 implies an isomorphism of certain connected components of crystal graphs with different parameters. This is discussed in 5. [START_REF] Dipper | Hecke algebras of type B n at roots of unity[END_REF].

In Section 6 we collect our evidence for the conjectures. In Theorem 6.2 we prove that Conjecture 5.8 holds for some subgraphs of the Harish-Chandra branching graph and the crystal graph, respectively. It is a generalization of the main result of Geck [START_REF] Geck | Modular principal series representations[END_REF] for principal series to other ordinary Harish-Chandra series. In Theorem 6.3 we prove that a version of Conjecture 5.5 holds for blocks of weight 1, i.e. blocks with cyclic defect groups.

In Section 7 we prove that the consequences implied by the conjectures for the crystal graphs are indeed true. This adds more evidence to our conjectures. Conjecture 5.8 implies that a weakly cuspidal module is labelled by a partition which gives rise to a highest weight vertex in the crystal graph. Such partitions can be characterized combinatorially (see [START_REF] Jacon | A combinatorial decomposition of higher level Fock spaces[END_REF]). We prove in Theorem 7.6 that the corresponding e-core is indeed a 2-core, as predicted by Conjecture 5.5. In [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF]Theorem 8.3] we had proved that the unipotent module of G labelled by the partition (1 n ) is cuspidal if and only if ℓ divides n or n-1. We prove that the analogous statement holds for corresponding vertices of the crystal graph (Proposition 7.5). Another consequence is stated in Corollary 7.7. Suppose that λ labels a weakly cuspidal module of G and that the 2-core of λ is different from λ and contains more than one node. Then there is a particular e-hook of λ such that the partition λ ′ obtained from λ by removing this e-hook also labels a weakly cuspidal module, and the two weakly cuspidal modules should give rise to isomorphic Harish-Chandra branching graphs. This is remarkable as n and ne have different parities and the modules of G = GU n (q) and GU n-e (q) are not directly related via Harish-Chandra induction. We prove in Theorem 7.8 that, as predicted in 5.10, the two connected components in question are isomorphic (as unlabelled) graphs. A further consequence of our conjectures is stated in Corollary 7.9: non-isomorphic composition factors of the socles of modules Harish-Chandra induced from G = GU n (q) to GU n+2 (q), lie in different ℓ-blocks.

Let us finally comment on the history of this paper. First notes of the second author date back to 1993, following the completion of [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF]. There, a general conjecture for the distribution of the simple modules of a unitary group into Harish-Chandra series for the linear prime case was presented. This conjecture was later verified in [START_REF] Gruber | Decomposition numbers of finite classical groups for linear primes[END_REF]. A further conjecture of [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF] for the case that ℓ divides q + 1 was proved in [START_REF] Geck | Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type[END_REF]. The conjectures in [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF] were based on explicit decomposition matrices of unipotent modules of GU n (q), computed by Gunter Malle. These decomposition matrices were completely known in the linear prime case for n ≤ 10 and published in [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF]. At that time, the information in the non-linear prime case was less comprehensive. Much more complete versions of these decomposition matrices and the distribution of the unipotent modules into Harish-Chandra series are now available by the recent work [START_REF] Dudas | Decomposition matrices for low rank unitary groups[END_REF] of Dudas and Malle.

Since the publication of [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF], many attempts have been made to find the combinatorial pattern behind the Harish-Chandra series of the unitary groups. The breakthrough occurred in 2009, when the second and last author shared an office during a special program at the Isaac Newton Institute in Cambridge. The paper [START_REF] Geck | Modular principal series representations[END_REF] by Geck and some other considerations of the second author suggested that the simple modules of certain Iwahori-Hecke algebras of type B should label some unipotent modules of the unitary groups. The paper [START_REF] Geck | Canonical basic sets in type B n[END_REF] by Geck and the third author on canonical basic sets then proposed the correct labelling by Uglov bipartitions. This set of bipartitions is defined through a certain crystal graph, called G c,e below. The two authors compared their results on these crystal graphs on the one hand, and on the known Harish-Chandra distribution on the other hand. Amazingly, the two results matched.

A generalization of Harish-Chandra theory

Here we introduce a generalization of Harish-Chandra theory for certain families of classical groups by restricting the set of admissible Levi subgroups.

2.1.

Let q be a power of the prime p. For a non-negative integer m let G := G m := G m (q) denote one of the following classical groups, where we label the cases according to the (twisted) Dynkin type of the groups:

( 2 A 2m-1 ): GU 2m (q), ( 2 A 2m ): GU 2m+1 (q), (B m ): SO 2m+1 (q), (C m ): Sp 2m (q). (We interpret GU 0 (q) and Sp 0 (q) as the trivial group.)

If m ≥ 1, the group G is a finite group with a split BN-pair of characteristic p, satisfying the commutator relations. In these cases, the Weyl group W of G is a Coxeter group of type B m , and we number the set S = {s 1 , . . . , s m } of fundamental reflections of W according to the following scheme.

(1) Put δ := 2, if G m (q) = GU 2m (q) or GU 2m+1 (q), and δ := 1, otherwise. Then the standard Levi subgroup L r,m-r of G has structure

❢ ❢ ❢ ❢ ❢ ❢ . . . s 1 s 2 s m-1 s m 2.2. A subset I ⊆ S is called left connected, if it is of the form I = {s 1 , s 2 , . . . , s r } for some 0 ≤ r ≤ m. The corresponding standard Levi subgroup L I of G is denoted by L r,m-r . A Levi subgroup L of G is called pure, if it is conjugate under N to
L r,m-r ∼ = G r (q) × GL 1 (q δ ) × • • • × GL 1 (q δ )
with mr factors GL 1 (q δ ), and with a natural embedding of the direct factors of L r,m-r into G.

Lemma. Let I and J be two left connected subsets of S, and let x ∈ D IJ , where D IJ ⊆ W denotes the set of distinguished double coset representatives with respect to the parabolic subgroups W I and W J of W . Then x I ∩ J is left connected.

Proof. We identify W with the set of permutations π of Ω := {±i | 1 ≤ i ≤ m} satisfying π(-i) = -π(i) for all 1 ≤ i ≤ m, where we choose notation such that s 1 acts as the transposition (1, -1), and s i as the double transposition (i -1, i)(-i, -i + 1) for 2 ≤ i ≤ m. If J = ∅, there is nothing to prove. Thus assume that J = {s 1 , . . . , s r } for some 1 ≤ r ≤ m. Then the set of W J -orbits on Ω equals {{±i | 1 ≤ i ≤ r}, {r + 1}, {-r -1}, . . . , {m}, {-m}}.

The set of W I -orbits on Ω has an analogous form and putting

Z := {1 ≤ i ≤ r | i is not fixed by x W I } we find that the set of ( x W I ∩ W J )-orbits on Ω equals (2) {{±i | i ∈ Z}, {j}, {-j} | 1 ≤ j ≤ m, j / ∈ Z}.
Indeed, if i, j ∈ Z, the elements (i, -i) and (i, j)(-j, -i) of W lie in x W I ∩ W J , and thus {±i | i ∈ Z} is an x W I ∩ W J -orbit.

On the other hand, if [START_REF] Carter | Finite groups of Lie type: Conjugacy classes and complex characters[END_REF]Theorem 2.7.4]). Suppose that s 1 / ∈ J ′ . Then W J ′ fixes {i ∈ Ω | i > 0} and thus Z = ∅ by [START_REF] Broué | Les l-blocs des groups GL(n, q) et U (n, q 2 ) et leurs structures locales[END_REF]. In turn, J ′ = ∅ and thus J ′ is left connected. If s 1 ∈ J ′ , then J ′ is left connected, as otherwise W J ′ would have at least two orbits on Ω of lengths larger than 1.

J ′ := x I ∩ J, then x W I ∩ W J = W J ′ , as x ∈ D IJ (see
Götz Pfeiffer has informed us of a different proof of the above result, using the descent algebra of W . Pfeiffer's proof also applies to Weyl groups of types A and D.

Proposition. Let L, M be pure Levi subgroups of G, and let x ∈ N. Then x L ∩ M is a pure Levi subgroup of G.

Proof. We may assume that L = L I and M = L J for I, J ⊆ S left connected. As x L I ∩ L J is conjugate in N to y L I ∩ L J , where y ∈ D IJ , we may also assume that x ∈ D IJ . Then x I ∩ J is left connected by the lemma. This completes the proof.

2.3.

Let k be a field of characteristic ℓ ≥ 0 with ℓ = p, such that k is a splitting field for all subgroups of G. We write kG-mod for the category of finite-dimensional left kG-modules. It is known that Harish-Chandra philosophy for kG carries over to the situation where L is replaced by L * . The first ideas in this direction go back to Grabmeier's thesis [START_REF] Grabmeier | Unzerlegbare Moduln mit trivialer Youngquelle und Darstellungstheorie der Schuralgebra[END_REF], who replaced Green correspondence in symmetric groups by a generalized Green correspondence with respect to Young subgroups. Further developments are due to Dipper and Fleischmann [START_REF] Dipper | Modular Harish-Chandra theory I[END_REF]. A comprehensive treatment including several new aspects can be found in [4, Chapter 1]. The crucial ingredient in this generalization is Proposition 2.2.

Let L ∈ L. We write R G L and * R G L for Harish-Chandra induction from kL-mod to kG-mod and Harish-Chandra restriction from kG-mod to kL-mod, respectively. For X ∈ kL-mod we put

H G,k (L, X) := End kG (R G L (X))
for the endomorphism algebra of R G L (X). Let X ∈ kG-mod. We say that X is weakly cuspidal, if * R G L (X) = 0 for all G = L ∈ L * . A pair (L, X) with L ∈ L * and X a weakly cuspidal simple kL-module is called a weakly cuspidal pair. Let (L, X) be a weakly cuspidal pair. Then the weak Harish-Chandra series defined by (L, X) consists of the simple kG-modules which are isomorphic to submodules of R G L (X). If Y ∈ kG-mod lies in the weak Harish-Chandra series defined by (L, X), then L ∈ L * is minimal with * R G L (Y ) = 0, and X is a composition factor of * R G L (Y ). We collect a few important facts about weak Harish-Chandra series.

Proposition. Let (L, X) be a weakly cuspidal pair.

(a

) Write R G L (X) = Y 1 ⊕ • • • ⊕ Y r with indecomposable modules Y i , 1 ≤ i ≤ r.
Then each Y i has a simple head Z i , which is also isomorphic to the socle of Y i . Moreover, Y i ∼ = Y j , if and only if Z i ∼ = Z j . The Harish-Chandra series defined by (L, X) consists of the kG-modules isomorphic to the Z i .

(b) The weak Harish-Chandra series partition the set of isomorphism types of the simple kG-modules.

(c) The weak Harish-Chandra series defined by (L, X) is contained in a usual Harish-Chandra series, and thus every usual Harish-Chandra series is partitioned into weak Harish-Chandra series.

Proof. It follows from [4, Theorems 1.20(iv), 2.27] that H G,k (L, X) is a symmetric k-algebra (notice that the cited results are also valid in our situation where L is replaced by L * ). This implies the statements of (a) (see, e.g. [START_REF] Cabanes | Representation theory of finite reductive groups[END_REF]Theorem 1.28]).

The proof of (b) is analogous to the proof in the usual Harish-Chandra theory.

To prove (c), let M ∈ L with M ≤ L, and let Z ∈ kM-mod be cuspidal (in the usual sense) such that X occurs in the socle of

R L M (Z). Then R G L (X) is a submodule of R G L (R L M (Z)) ∼ = R G M (Z)
, and thus every simple module in the socle of R G L (X) also occurs in the socle of R G M (Z) and hence in the usual Harish-Chandra series defined by (M, Z).

2.4. Let (L, X) be a weakly cuspidal pair. The following proposition gives information about those composition factors of R G L (X) that do not lie in the weak Harish-Chandra series defined by (L, X). The corresponding result for usual Harish-Chandra series is implicitly contained in [START_REF] Hiss | Harish-Chandra series of Brauer characters in a finite group with a split BN -pair[END_REF]Lemma 5.7] (see the remarks in [20, (2.2)]). Since this result is particularly relevant in the definition of the Harish-Chandra branching graph, and since it is not explicitly formulated in [START_REF] Hiss | Harish-Chandra series of Brauer characters in a finite group with a split BN -pair[END_REF]Lemma 5.7], and wrongly stated in [START_REF] Geck | Modular representations of finite groups of Lie type in non-defining characteristic[END_REF]Proposition 2.11(b)], we give a proof here.

Proposition. Let (L, X) be a weakly cuspidal pair, and let Y be a composition factor of R G L (X). Suppose that Y lies in the weak Harish-Chandra series defined by (M, Z), a weakly cuspidal pair.

Then there is x ∈ N such that x L ≤ M.

If x L = M, then Z ∼ = x X.
In particular, if Y does not lie in the weak Harish-Chandra series defined by (L, X), then |L| < |M|.

Proof. Let P (Z) denote the projective cover of Z. We have

0 = [P (Z), * R G M (Y )] = [R G M (P (Z)), Y ], the inequality arising from the fact that Z is a composition factor of * R G M (Y ), the equation arising from adjointness. As Y is a composition factor of R G L (X), we obtain 0 = [R G M (P (Z)), R G L (X)] = x∈D M,L [P (Z), R M M ∩ x L ( * R x L M ∩ x L ( x X))].
(Here, D M,L ⊆ N denotes a suitable set of representatives for double cosets with respect to parabolic subgroups of G with Levi complements M and L, respectively.) Thus there is

x ∈ D M,L such that [P (Z), R M M ∩ x L ( * R x L M ∩ x L ( x X
))] = 0. As (L, X) is a weakly cuspidal pair, so is ( x L, x X). It follows from Proposition 2.2 that M ∩ x L = x L, and thus x L ≤ M. If x L = M, we obtain [P (Z), x X] = 0, hence our claim.

2.5.

If char(k) = 0, a kG-module is unipotent, if it is simple and its character is unipotent. If ℓ > 0, a kG-module is unipotent, if it is simple and its Brauer character (with respect to a suitable ℓ-modular system) is a constituent of some unipotent character (restricted to ℓ ′ -elements).

As L * ⊆ L, every cuspidal kG-module X is weakly cuspidal. The converse is not true, as the following example shows. Let G = GU 6 (q) and suppose that ℓ > 6 and divides q 2q + 1. The Levi subgroup L = GL 3 (q 2 ) (a Levi complement of the stabilizer of a maximal isotropic subspace of the natural vector space of G), contains a cuspidal unipotent kL-module X by [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF]Theorem 7.6]. By applying [START_REF] Geck | Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type[END_REF]Lemma 3.16] and [START_REF] Gruber | Green vertex theory, Green correspondence, and Harish-Chandra induction[END_REF]Proposition 2.3.5] we find that R G L (X) is indecomposable. Let Y denote the unique head composition factor of R G L (X) (see [START_REF] Geck | Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type[END_REF]Theorem 2.4]). By construction, Y is not cuspidal, but weakly cuspidal. (The kG-module Y has label 2 3 in the notation of [START_REF] Dudas | Decomposition matrices for low rank unitary groups[END_REF]Table 8]).

Now suppose that ℓ = 0. Then a weakly cuspidal unipotent kGmodule is cuspidal. Indeed, GL n (q δ ) has a cuspidal unipotent module over k only if n = 1. In particular, if L ∈ L has a cuspidal unipotent module over k, then L ∈ L * . If X is a weakly cuspidal unipotent kGmodule and L ∈ L is minimal with * R G L (X) = 0, every constituent of * R G L (X) is cuspidal. Thus L ∈ L * and hence, as X is weakly cuspidal, L = G.

The endomorphism algebra of Harish-Chandra induced weakly cuspidal modules

In important special cases the endomorphism algebras H G,k (L, X) of weakly cuspidal pairs (L, X) are Iwahori-Hecke algebras. The result applies in particular when X is unipotent.

We keep the notation of Section 2, except that we assume that m ≥ 1 here.

Thus if G = G m (q) is one of the groups introduced in 2.1, then G has a split BN-pair of rank m. For a Levi subgroup L of G we put N (L) := (N G (L) ∩ N)L and W G (L) := N (L)/L.
Let ℓ be a prime not dividing q. We choose an ℓ-modular system (K, O, k) such that K is large enough for G. That is, O is a complete discrete valuation ring with field of fractions K of characteristic 0, and residue class field k of characteristic ℓ. Moreover, K is a splitting field for all subgroups of G.

3.1. Put r := m -1 and L := L r,1 ∈ L * . Thus L = M × T with M ∼ = G r (q)
and T ∼ = GL 1 (q δ ). (In case m = 1, either M is the trivial group, or cyclic of order q + 1 if G = GU 3 (q).) Let P denote the standard parabolic subgroup of G with Levi complement L and let U denote its unipotent radical. We have |W G (L)| = 2 and we let s ∈ N G (L) denote an inverse image of the involution in W G (L). We choose s of order 2 if G is unitary or orthogonal, and of order 4 with s 2 ∈ T if G is symplectic, and such that s centralizes M. (Such an s always exists.)

Let R be one of the rings K, O, or k. As M is an epimorphic image of P , we get a surjective homomorphism π : RP → RM. Consider the element

(3) y := u, u ′ ∈ U su ′ sus ∈ P su ′ sus ∈ RP.
Then z := π(y) ∈ Z(RM) as s centralizes M.

Lemma. With the above notation, z = (q -1)z ′ for some z ′ ∈ Z(RM).

In case G is a unitary group, we have z ′ = 1 + (q + 1)z ′′ for some z ′′ ∈ Z(RM).

Proof. We first claim that T ∼ = GL 1 (q δ ) acts on

U := {(u ′ , u) ∈ U × U | su ′ sus ∈ P } by x.(u ′ , u) := (sxs -1 u ′ sx -1 s -1 , xux -1 ), x ∈ T, (u ′ , u) ∈ U.
Indeed, ( 4)

s(sxs -1 u ′ sx -1 s -1 )s(xux -1 )s = (s 2 xs -2 )su ′ sus(s -1 x -1 s) for x ∈ T, (u ′ , u) ∈ U. As s normalizes T , the claim follows. Now π(x) = 1 for x ∈ T and thus (4) implies π(su ′ sus) = π(sv ′ svs) if (u ′ , u), (v ′ , v) ∈ U are in the same T -orbit.
The claims in the arguments below can be verified by a direct computation in G. Suppose that G is a unitary or symplectic group. For each

1 = u ∈ Z(U) there is a unique u ′ ∈ Z(U) such that (u ′ , u) ∈ U.
For every such pair we have π(su ′ sus) = 1. The elements (u ′ , u) ∈ U with u ∈ Z(U) lie in regular T -orbits, as T acts fixed point freely on U \ Z(U) by conjugation. This implies our result, as |Z(U)| = q and |T | = q δ -1. Now suppose that G is an orthogonal group. Then T acts with regular orbits on U \{1}, hence on U, again implying our result.

3.2.

Let R be one of K or k. If X is an indecomposable RG-module, we let ω X denote the central character of RG determined by the block containing X.

Let r be an integer with 0 ≤ r ≤ m and put n := mr. Let L := L r,n ∈ L * denote the standard Levi subgroup of G = G m (q) isomorphic to G r (q) × GL 1 (q δ ) n . Write M and T for the direct factors of L isomorphic to G r (q) and GL 1 (q δ ) n , respectively. Let X be a weakly cuspidal simple RM-module, extended trivially to an RL-module.

For R = K and X cuspidal, the following result is due to Lusztig (see [START_REF] Lusztig | Irreducible representations of finite classical groups[END_REF]Section 5]).

Theorem. With the above notation, H G,R (L, X) is an Iwahori-Hecke algebra corresponding to the Coxeter group of type B n , with parameters as in the following diagram.

(5)

❢ ❢ ❢ ❢ ❢ . . . Q q δ q δ q δ q δ
The parameter Q is determined as follows. Let U and z be as in 3.1, applied to G r+1 . Put γ := ω X (z) ∈ R and let ξ ∈ R be a solution of the quadratic equation

x 2 -γ x -|U| = 0. Then Q = ξγ |U| + 1.
Moreover, the following statements hold.

(a) Suppose that R = k and that X lies in a block containing a cusp-

idal KM-module Y . If Q is the parameter of H G,K (L, Y ) associated to the leftmost node of the diagram (5), then Q is the reduction modulo ℓ of Q. (b) If R = k and ℓ | q -1, then Q = 1. (c) If G is unitary, R = k and ℓ | q + 1, then Q = -1.
Proof. First notice that we have W G (L, X) = W G (L), and that W G (L) is isomorphic to a subgroup of W and a Coxeter group of type B m (see [START_REF] Howlett | Normalizers of parabolic subgroups of reflection groups[END_REF]). By Mackey's theorem and Proposition 2.2 (cf. the proof of [5, Proposition 9.2.4]) we also have

dim R (H G,R (L, X)) = |W G (L)|. Recall that N (L) = (N G (L) ∩ N)L. In our situation we have N (L) = M × C with T ≤ C and C/T ∼ = W G (L).
In particular, we may view X as an RN (L)-module on which C acts trivially. This implies that H G,R (L, X) is an Iwahori-Hecke algebra of type B m as in [START_REF] Geck | Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type[END_REF]Corollary 3.13].

The parameters not corresponding to the leftmost node of ( 5) can now be computed exactly as in the case where X is cuspidal and unipotent (see [START_REF] Geck | Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type[END_REF]Proposition 4.4]).

To determine Q we may assume that n = 1 (see the considerations in [21, (3.14)]). Thus G = G r+1 (q) and L ∼ = M × GL 1 (q δ ). We are thus in the situation of 3.1 and make use of the notation introduced there. Then H := H G,R (L, X) is 2-dimensional over R with basis elements B 1 and B s , where B 1 is the unit element of H and B s is defined as follows. We may realize

R G L (X) as R G L (X) = {f : G → X | f (hg) = h.f (g), for all h ∈ P, g ∈ G}. Then B s is defined by B s (f )(g) := 1 |U| u∈U f (sug), f ∈ R G L (X), g ∈ G,
as s ∈ C acts trivially on X. We have B 2 s = ζB 1 + ηB s with ζ = 1/|U|, and η such that the element y of (3) acts as the scalar |U|η on X. This is proved exactly as in [START_REF] Howlett | Induced cuspidal representations and generalised Hecke rings[END_REF]Proposition 3.14]. Now y acts in the same way on X as z = π(y). Since X is absolutely irreducible, z ∈ Z(RM) acts by the scalar ω X (z). Thus

|U|η = ω X (z) = γ. Put T s := ξB s , T 1 := B 1 . Then T 2 s = QT 1 + (Q -1)T s with Q = ξη + 1.
This gives our first claim.

To prove (a), put γ := ω Y (z), and let ξ be a solution of x 2 -γx-|U| = 0. Observe that γ, ξ ∈ O. Then the reduction modulo ℓ of γ equals ω X (z), and the reduction modulo

ℓ of ξ is a solution of x 2 -γx-|U| = 0. Thus the reduction modulo ℓ of Q := ξ η + 1 equals ξη + 1 = Q and (a) is proved. Suppose now that R = k. If ℓ | q -1,
we have γ = 0 by Lemma 3.1 and thus Q = 1. If G is unitary and ℓ | q + 1, we have γ = -2, again by Lemma 3.1. Also, |U| is an odd power of q, i.e. |U| = -1 in k, hence ξ = -1 and Q = -1. This completes our proof.

The Harish-Chandra branching graph

In this section we fix a power q of the prime p and a prime ℓ = p. We also let k denote an algebraically closed field of characteristic ℓ.

4.1. For m ∈ N, we let G := G m := G m (q) denote one of the groups of 2.1. Recall that G m is naturally embedded into G m+1 , by embedding G m into the pure Levi subgroup L m,1 ∼ = G m × GL 1 (q δ ) of G m+1
. By iterating, we obtain an embedding of G m into G m+j for every j ∈ N.

By kG-mod u we denote the full subcategory of kG-mod consisting of the modules that have a filtration by unipotent kG-modules. By the result of Broué and Michel [START_REF] Broué | Blocs et séries de Lusztig dans un groupe réductif fini[END_REF], and by [START_REF] Hiss | Regular and semisimple blocks of finite reductive groups[END_REF], kG-mod u is a direct sum of blocks of kG. The above embedding of G m into G m+j yields a functor

R m+j m : kG m -mod u → kG m+j -mod u , defined by R m+j m (X) := R G m+j L m,j (Infl L m,j Gm (X)), X ∈ kG m -mod u , where Infl L m,j Gm (X) denotes the trivial extension of X to L m,j ∼ = G m × GL 1 (q δ ) j . The adjoint functor * R m+j m : kG m+j -mod u → kG m -mod u , is given by * R m+j m (X) := Res L m,j Gm ( * R G m+j L m,j (X)), X ∈ kG m+j -mod u .
Let R m := R m (q) denote the Grothendieck group of kG m -mod u , and put

R := R(q) := m∈N R m .
For an object X ∈ kG m -mod u , we let [X] denote its image in R m .

4.2. The (twisted) Dynkin type of G is one of the symbols 2 A ι with ι ∈ {0, 1}, B or C, where GU r (q) has twisted Dynkin type 2 A ι with ι ≡ r (mod 2).

The Harish-Chandra branching graph G D,q,ℓ corresponding to q, ℓ and the (twisted) Dynkin type D is the directed graph whose vertices are the elements [X], where X is a simple object in kG m -mod u for some m ∈ N. Thus the vertices of G D,q,ℓ are the standard basis elements of R. We say that the vertex

[X] has rank m, if [X] ∈ R m . Let [X] and [Y ] be vertices in G D,q,ℓ . Then there is a directed edge from [X] to [Y ] if and only if there is m ∈ N such that [X] has rank m and [Y ] has rank m + 1, and such that Y is a head composition factor of R m+1 m (X). A vertex in G D,q,ℓ is called a source vertex, if it has only outgoing edges.
Every unipotent kG-module is self dual unless G = Sp 2n (q) and ℓ = 2. This follows from the fact that every ordinary unipotent character of G is rational valued (see [17, 5.1]) in conjunction with the fact that the unipotent characters form a basic set for the unipotent Brauer characters (see [15, Theorem A]) in the cases stated. Thus if ℓ is odd, or G is unitary or orthogonal, Y is a head composition factor of R m+1 m (X) if and only if Y is in the socle of R m+1 m (X). By adjunction, Y is a head composition factor of R m+1 m (X) if and only if X is in the socle of * R m+1 m (Y ), and Y is in the socle of R m+1 m (X) if and only if X is a head composition factor of * R m+1 m (Y ). An example for part of a Harish-Chandra branching graph is displayed in Table 1, where the vertices are represented by their labels. This can be proved with the help of the decomposition matrices computed in [START_REF] Dudas | Decomposition matrices for low rank unitary groups[END_REF] plus some ad hoc arguments.

4.3.

We have the following relation with the weak Harish-Chandra series of G.

Proposition. Let [X] be a vertex of rank n of G D,q,ℓ . Then [X] is a source vertex if and only if X ∈ kG m -mod u is weakly cuspidal.
Suppose that X is weakly cuspidal and let j ∈ N. View X as a module of L m,j via inflation. Then a simple object Y ∈ kG m+j -mod u lies in the weak (L m,j , X) Harish-Chandra series, if and only if there is a directed path from

[X] to [Y ] in G D,q,ℓ . Proof. Clearly, X is weakly cuspidal if m = 0. Assume that m ≥ 1.
Then X is weakly cuspidal if and only if * R m m-1 (X) = 0, which is the case if and only if [X] is a source vertex.

Assume now that X is weakly cuspidal, let j ∈ N and let [Y ] be a vertex of rank m + j. Suppose there is a path from

[X] to [Y ].
We proceed by induction on j to show that Y occurs in the head of R m+j m (X). If j = 0, there is nothing to prove. So assume that j > 0 and that the claim has been proved for j -1.

Let [Z] be a vertex of rank m + j -1 that occurs in a path from [X] to [Y ]. By induction, Z is a head composition factor of R m+j-1 m (X). By exactness, R m+j m+j-1 (Z) is a quotient of R m+j m+j-1 (R m+j-1 m (X)) ∼ = R m+j m (X).
As Y is a quotient of R m+j m+j-1 (Z), we are done. Suppose now that Y occurs in the head of R m+j m (X). We proceed by induction on j to show that there is a path from

[X] to [Y ], the cases j ≤ 1 being trivial. As Y is isomorphic to a quotient of R m+j m (X) ∼ = R m+j m+j-1 (R m+j-1 m (X)), there is a composition factor Z of R m+j-1 m (X) such that Y is a quotient of R m+j m+j-1 (Z). In particular, there is an edge from [Z] to [Y ]. If Z occurs in the head of R m+j-1 m (X), there is a path from [X] to [Z]
by induction, and we are done. Aiming at a contradiction, assume that Z does not occur in the head of R m+j-1 m (X). Then Z does not lie in the weak Harish-Chandra series of G m+j-1 defined by (L m,j-1 , X). It follows from Proposition 2.4 that Z lies in the weak Harish-Chandra series defined by (L m ′ ,m-m ′ +j-1 , X ′ ) for some m < m ′ and some weakly cuspidal module X ′ . In particular, Y lies in this weak Harish-Chandra series. This contradiction completes our proof.

Conjectures

Here we formulate a series of conjectures about the ℓ-modular Harish-Chandra series and the Harish-Chandra branching graph for the unitary groups.

5.1.

As always, we let q denote a power of a prime p, and we fix a prime ℓ different from p. The multiplicative order of -q modulo ℓ is denoted by e := e(q, ℓ). Thus e is the smallest positive integer such that ℓ divides (-q) e -1. In particular, e = 1 if ℓ = 2.

The symbol G always stands for the unitary group GU n (q). For fixed n, we choose an ℓ-modular system (K, O, k) such that K is large enough for G and with k algebraically closed. (Note that K and O may vary if n or q varies.) 5.2. The set of partitions of a non-negative integer n is denoted by P n and we write λ ⊢ n if λ ∈ P n . We put P := ∪ n∈N P n . Let λ ∈ P. Then λ (2) and λ (2) denote the 2-core and the 2-quotient of λ, respectively. (As in [13, Section 1], the 2-quotient is determined via a β-set for λ with an odd number of elements, where we use the term β-set in its original sense of being a finite set of non-negative integers as introduced in [36, p. 77f].) For a non-negative integer t we write ∆ t := (t, t -1, . . . , 1) for the triangular partition of t(t + 1)/2. Then λ (2) = ∆ t for some t ∈ N. Suppose that λ (2) = (µ 1 , µ 2 ). We then put λ(2) := (µ 1 , µ 2 ) if t is even, and λ(2) := (µ 2 , µ 1 ), otherwise. If µ = (µ 1 , µ 2 ) is a bipartition, we let Φ t (µ) denote the unique partition λ with λ (2) = ∆ t and λ(2) = (µ 1 , µ 2 ) (see [START_REF] James | The representation theory of the symmetric group[END_REF]Theorem 2.7.30]).

The set of bipartitions of n is denoted by P

n , and we put P (2) :=

∪ n∈N P (2) n . Finally, we write µ ⊢ 2 n if µ ∈ P (2) n . 5.3. Let G = GU n (q)
. By a result of Lusztig and Srinivasan [START_REF] Lusztig | The characters of the finite unitary groups[END_REF], the unipotent KG-modules are labelled by partitions of n. We write Y λ for the unipotent KG-module labelled by λ ∈ P n . Let λ and µ be partitions of n. It follows from the main result of Fong and Srinivasan [12, Theorem (7A)], that Y λ and Y µ lie in the same ℓ-block of G, if and only if λ and µ have the same e-core (for ℓ = 2, use [2, Théorème (3.9)]). The e-weight and the e-core of the ℓ-block containing Y λ are, by definition, the e-weight and the e-core of λ, respectively.

It was shown by Geck in [START_REF] Geck | On the decomposition numbers of the finite unitary groups in nondefining characteristic[END_REF] that if the Y λ , λ ⊢ n, are ordered downwards lexicographically, the corresponding matrix of ℓ-decomposition numbers is square and lower unitriangular. This defines a unique labelling of the unipotent kG-modules by partitions of n, and we write X µ for the unipotent kG-module labelled by µ ∈ P n . Thus X µ is determined by the following two conditions. Firstly, X µ occurs exactly once as a composition factor in a reduction modulo ℓ of Y µ , and secondly, if X µ is a composition factor in a reduction modulo ℓ of Y ν for some ν ∈ P n , then ν ≤ µ.

5.4.

Our first conjecture asserts a compatibility between ordinary and weak modular Harish-Chandra series.

Conjecture. Let G = GU n (q) and let µ, ν ∈ P n . If X µ and X ν lie in the same weak Harish-Chandra series of kG-modules, then µ and ν have the same 2-core, i.e. Y µ and Y ν lie in the same Harish-Chandra series of KG-modules. (In other words, the partition of P n arising from the weak ℓ-modular Harish-Chandra series is a refinement of the partition of P n arising from the ordinary Harish-Chandra series.) 5.5. Again let G = GU n (q). We also conjecture that if e is odd and ℓ > n, a weakly cuspidal unipotent module can only occur in an ℓ-block of G which contains a cuspidal simple KG-module (not necessarily unipotent). The following lemma investigates when this happens. For the notions of e-core and e-weight of a unipotent ℓ-block of G see 5.3.

Lemma.

Assume that e and ℓ are odd and let B denote a unipotent ℓblock of G of e-weight w. If B contains a cuspidal simple KG-module, the e-core of B is a 2-core. The converse of this statement holds if ℓ > ew.

Proof. Suppose that B contains some cuspidal simple KG-module Y . Then Y belongs to the Lusztig series of some ℓ-element x ∈ G (see [START_REF] Fong | The blocks of finite general linear and unitary groups[END_REF]Theorem (7A)

i)]). Put C := C G (x). Then C = C 0 × C 1 , where C 0 is the unitary group on the 1-eigenspace of x. Write Y = Y 0 ⊗ K Y 1 with cuspidal unipotent KC i -modules Y i , i = 0, 1.
Let µ ∈ P be the partition labelling Y 0 . Then µ is a 2-core, and in turn, the e-core of µ is a 2-core as well. By [12, Theorem (7A) and Proposition (4F)], the e-core of µ equals the e-core of B and our claim follows.

Now assume that the e-core of B is the 2-core ∆ s , and put m ′ := s(s + 1)/2. As ℓ > ew, there is an ℓ-element x in G with C := C G (x) = (q e + 1) w × GU m ′ (q), where (q e + 1) w denotes a direct product of w factors of the cyclic group of order q e + 1 (and n = we + m ′ ). Let Z denote the cuspidal unipotent KC-module labelled by ∆ s (if s ≤ 1, then C is a maximal torus and Z is the trivial module), and let Y be the simple KG-module corresponding to Z under Lusztig's Jordan decomposition. Then Y is cuspidal by [43, 7.8.2], and Y lies in B, again by [START_REF] Fong | The blocks of finite general linear and unitary groups[END_REF]Theorem (7A) and Proposition (4F)].

Conjecture. Suppose that e is odd and ℓ > n. Let λ ∈ P n . If X λ is weakly cuspidal, then the e-core of λ is a 2-core.

It follows from [START_REF] Gruber | Decomposition numbers of finite classical groups for linear primes[END_REF]Corollary 8.8] that if e is even, then X λ is cuspidal if and only if λ is a 2-core. (In this case, λ also is an e-core.)

Assuming that Conjecture 5.5 holds, the parameter Q of a weakly cuspidal unipotent kG-module X λ of G can be computed from the ecore of λ by Corollary 5.9 below. 5.6. To present our next conjectures, we first have to introduce the Fock space of level 2 and its corresponding crystal graph. The results summarized below are due to Jimbo, Misra, Miwa and Okado [START_REF] Jimbo | Combinatorics of representations of U q ( sl(n)) at q = 0[END_REF] and Uglov [START_REF] Uglov | Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials[END_REF]. For a detailed exposition see also [START_REF] Geck | Representations of Hecke algebras at roots of unity[END_REF]Chapter 6].

A charged bipartition is a pair (µ, c), written as |µ, c with µ ∈ P (2) and c ∈ Z 2 . Fix c = (c 1 , c 2 ) ∈ Z 2 , and let v denote an indeterminate.

The Fock space (of level 2) and charge c is the Q(v)-vector space

F c := m∈N µ⊢ 2 m Q(v)|µ, c .
Assume that e ≥ 2. There is an action of the quantum group Let us now describe, following [START_REF] Geck | Representations of Hecke algebras at roots of unity[END_REF], how to compute the good inodes of |µ, c , and thus the graph G c,e , algorithmically. A node of µ = (µ 1 , µ 2 ) is a triple (a, b, j), where (a, b) is a node in the Young diagram of µ j , for j = 1, 2. A node γ of µ is called addable (respectively removable) if µ ∪ {γ} (respectively µ\{γ}) is still a bipartition. The content of γ = (a, b, j) is the integer cont(γ) = b-a+c j . The residue of γ is the element of {0, 1, . . . , e -1} defined by res(γ) = cont(γ) mod e. For 0 ≤ i ≤ e -1, γ is called an i-node if res(γ) = i.

U ′ v ( sl e ) on F c such that F c is an integrable U ′ v ( sl e )-module
Fix i ∈ {0, 1, . . . , e -1}, and define an order on the set of addable and removable i-nodes of µ by setting

γ ≺ c γ ′ if cont(γ) < cont(γ ′ ) or cont(γ) = cont(γ ′ ) and j > j ′ .
Sort these sets of nodes according to ≺ c , starting from the smallest one.

Encode each addable (respectively removable) i-node by the letter A (respectively R), and delete recursively all occurences of consecutive letters RA. This yields a word of the form A α i R β i , which is called the reduced i-word of µ. Note that by Kashiwara's crystal theory [38, Section 4.2], we have the following expression for the weight of the vector |µ, c :

(6) wt(µ, c) = e-1 i=0 (α i -β i )Λ i .
Let γ be the rightmost addable (respectively leftmost removable) inode in the reduced i-word of µ. Then γ is called the good addable (respectively good removable) i-node of µ.

Each connected component of G c,e is isomorphic to the crystal of a simple highest weight module of U ′ v ( sl e ), whose highest weight vector is the unique source vertex of the component. The rank of a vertex |µ, c of G c,e is m, if µ ⊢ 2 m. We write G ≤m c,e for the induced subgraph of G c,e containing the vertices of rank at most m.

As an example, the graph G ≤3 (0,0),3 is displayed in Table 3. 5.7. Let t be a non-negative integer, put r := t(t + 1)/2 and ι := r (mod 2) ∈ {0, 1}. Then KGU r (q) has a unipotent cuspidal module Y , and (GU r (q), Y ) determines a Harish-Chandra series of unipotent KGU r+2m (q)-modules for every m ∈ N. Recall from 4.2 that G2 Aι,q,ℓ denotes the Harish-Chandra branching graph corresponding to q, ℓ and the groups GU 2j+ι (q), j ≥ 0. As we are dealing exclusively with unitary groups in this section, we shall replace the index 2 A ι by ι in the symbol for the graph. The vertices of G ι,q,ℓ correspond to the isomorphism classes of the unipotent kGU 2j+ι (q)-modules, where j runs through the set of non-negative integers. We may thus label the vertices of G ι,q,ℓ by the set ∪ j∈N P 2j+ι .

To formulate our next conjecture, we assume that Conjecture 5.4 holds. Under this assumption, the induced subgraph G t ι,q,ℓ of G ι,q,ℓ whose vertices correspond to the set of partitions with 2-core ∆ t , is a union of connected components of G ι,q,ℓ . We write Gt ι,q,ℓ for the graph with vertices P (2) , and a directed edge µ → ν, if and only if there is a directed edge in G t ι,q,ℓ between the vertices Φ t (µ) and Φ t (ν). This change of labelling of G t ι,q,ℓ is motivated by the result of Fong and Srinivasan in [START_REF] Fong | Brauer trees in classical groups[END_REF]Appendix], where they match the two different labellings of the ordinary unipotent modules of the unitary groups. If µ ⊢ 2 m is a vertex of Gt ι,q,ℓ , the rank of this vertex is m. For a non-negative integer d we let Gt,≤d ι,q,ℓ denote the induced subgraph of Gt ι,q,ℓ containing the vertices of rank at most d.

We shall also need certain bounds to pin down the scope of the conjectures. Recall that q and ℓ, hence e are fixed from the outset. Let ξ ∈ C denote a primitive root of unity of order 2e. For non-negative integers m, t with m = 0, let us write H m,ξ,t for the Iwahori-Hecke algebra of type B m over Q[ξ] with parameters ξ 2t+1 (corresponding to s 1 in the diagram (1)) and ξ 2 , if m > 1 (corresponding to s 2 , . . . , s m ). Then Q[ξ] is a splitting field for H m,ξ,t . This follows from the construction of the simple H m,ξ,t -modules by Dipper, James and Murphy (see [START_REF] Dipper | Hecke algebras of type B n at roots of unity[END_REF]Section 6]), as the radical of a bilinear form is insensitive to field extensions. Let ℓ ′ be a prime number such that 2e divides ℓ ′ -1 and let ξ ∈ F * ℓ ′ denote an element of order 2e. Then there is a decomposition map d ℓ ′ : R 0 (H m,ξ,t ) → R 0 (H ℓ ′ m, ξ,t ), where H ℓ ′ m, ξ,t is the Iwahori-Hecke algebra of type B m over F ℓ ′ with parameters ξ2t+1 and ξ2 , and where R 0 (A) denotes the Grothendieck group of the finite dimensional algebra A. The map d ℓ ′ arises from the specialization Z[ξ] → F ℓ ′ , ξ → ξ. By the general argument of Geck given in [16, 2.7], there is a positive integer N, depending only on m, e and t, such that d ℓ ′ is trivial for all primes ℓ ′ > N with 2e | ℓ ′ -1. Let N(e, m, t) denote the smallest positive integer with this property, and put N(e, 0, t) := 1. Next, for a positive integer n, set (7) Ñ (e, n) := max{n, N(e, m, t) | t(t + 1)/2 ≤ n, 2m ≤ n}.

Let us motivate this definition. If G = GU n (q) and Y is a cuspidal unipotent KL-module for a (pure

) Levi subgroup L of G, then End KG (R G L (Y )
) is an Iwahori-Hecke algebra of type B m with parameters q 2t+1 and q 2 , where t(t + 1)/2 + 2m = n. Also, if Conjecture 5.5 is true and (L, X) is a weakly cuspidal pair in G, the algebra End kG (R G L (X)) is an Iwahori-Hecke algebra over k with parameters q 2s+1 and q 2 with s(s + 1)/2 ≤ n (see Proposition 5.9 below). If e is odd, the order of q in F ℓ ⊆ k equals 2e. Finally, put [START_REF] Dipper | Harish-Chandra vertices and Steinberg's tensor product theorem for general linear groups[END_REF] b(e, ℓ) := max{n | ℓ > Ñ(e, n ′ ) for all 1 ≤ n ′ ≤ n}.

As Ñ(e, n) ≥ n for all positive integers n, the bound b(e, ℓ) is well defined and tends to infinity with ℓ. We are now ready to formulate our next conjecture.

Conjecture. Let the notation be as above. Assume that e is odd, larger than 1, and put c := (t + (1e)/2, 0). Then with b := ⌊b(e, ℓ)/2⌋ the graph Gt,≤b ι,q,ℓ equals G ≤b c,e , if the colouring of the edges of the latter graph is neglected.

5.8.

Let G = GU n (q). As the Harish-Chandra series of unipotent kGmodules can be read off from the Harish-Chandra branching graph by Proposition 4.3, the truth of Conjecture 5.7 would give an algorithm to determine the partition of the kG-modules into weak Harish-Chandra series from the labels of the modules, at least if ℓ is large enough. In particular, the question of whether X λ is weakly cuspidal, can be read off from λ.

Conjecture. Let λ ∈ P n ′ for some n ′ ≤ n of the same parity as n. Let t ∈ N such that λ (2) = ∆ t and let µ = λ(2) (see 5.2). Assume that e is odd, larger than 1, and ℓ > Ñ(e, n) (see (7) for the definition of Ñ(e, n)), and put c := (t + (1e)/2, 0). Then X λ is weakly cuspidal, if and only if |µ, c is a source vertex in G c,e .

Suppose that X λ is weakly cuspidal and let ρ ∈ P n . Then X ρ lies in the weak Harish-Chandra series defined by X λ , if and only if ρ (2) = λ (2) = ∆ t , and |ρ (2) , c lies in the connected component of G c,e containing |µ, c , i.e. |ρ (2) , c is obtained from |µ, c by adding a sequence of good nodes. 5.9. Let (L, X) be a weakly cuspidal pair of G = GU n (q). Provided Conjecture 5.5 is true, we can compute the parameters of H G,k (L, X).

Proposition. Assume that e is odd and that ℓ > n. Suppose that Conjecture 5.5 holds. Then, in the situation and notation of Theorem 3.2, it follows that X lies in a kM-block B whose e-core equals the 2-core ∆ s for some s ≥ 0. Moreover, Q = q 2s+1 . Proof. By Conjecture 5.5, the e-core of the partition labelling X (viewed as unipotent kM-module) is a 2-core, ∆ s , say. Now ℓ > n by assumption, and thus ℓ > ew, if w denotes the e-weight of the block B of kM containing X. Thus Lemma 5.5 implies that B contains a cuspidal simple KM-module Y . By Theorem 3.2(a), the parameter Q is equal to the corresponding parameter of the Iwahori-Hecke algebra H G,K (L, Y ). By the results of Lusztig [43, Section 5], we have Q = q 2s+1 . 5.10. If the Conjectures 5.5, 5.7 and 5.8 are true, Proposition 5.9 implies a compatibility between certain connected components of the crystal graph.

Let (L, X λ ) be a weakly cuspidal pair of G = GU n (q). Assume that e is odd and that ℓ > Ñ (e, n). Let λ (2) = ∆ t , put r := t(t + 1)/2 and write n = r + 2m. Then we may assume that L = GU r (q) × GL 1 (q 2 ) m . Suppose now that the e-core of λ equals ∆ s . (The e-core of λ should be a 2-core by Conjecture 5.5.) By Theorem 3.2 and Proposition 5.9, we have that H G,k (L, X λ ) is an Iwahori-Hecke algebra of type B m with parameters q 2s+1 and q 2 . According to [ On the other hand, by Conjecture 5.8, this Harish-Chandra series should also be labelled by the set of bipartitions arising from λ(2) by adding a sequence of m good nodes with respect to the charge (t + (1e)/2, 0).

The compatibility of the two labellings is guaranteed by Theorem 7.8 below.

5.11. We give an example for the phenomenon discussed above. Suppose that e = 3 and L := GU 4 (q) × GL 1 (q 2 ) m , a pure Levi subgroup of G := GU 4+2m (q). The Steinberg kL-module X (1 4 ) is cuspidal. As the 2core of (1 4) is trivial we have t = 0. Suppsoe that Conjecture 5.7 is valid for G and ℓ. Then the kG-modules in the (L, X (1 4 ) )-Harish-Chandra series correspond to the vertices at distance m of the bipartition (-, 1 2 ) in the crystal graph for e = 3 and charge (-1, 0).

The Iwahori-Hecke H G,k (L, X (1 4 ) ) is of type B m with parameters q 3 and q 2 , as s = 1. Its simple modules are labelled by the Uglovbipartitions of m corresponding to e = 3 and charge (0, 0).

Some evidence

Here we present the evidence for our conjectures. Keep the notation of Section 5. We also assume that e is odd and larger than 1 in this section.

6.1. Conjecture 5.7 holds for e = 3, 5 and the groups GU n (q) for n ≤ 10, if ℓ > n. In these cases, most of the decomposition numbers and the Harish-Chandra series have been computed by Dudas and Malle [START_REF] Dudas | Decomposition matrices for low rank unitary groups[END_REF]. The Harish-Chandra branching graphs can be determined from this information using some additional arguments. The corresponding crystal graphs can be computed with the GAP3 programs written by one of the authors (see [33]).

6.2.

In addition to the cases mentioned above, we have checked that Conjecture 5.8 holds for n = 12 and e = 3 if ℓ ≥ 13. There are more cases where Conjecture 5.8 is known to be true.

Theorem. Let 0 ≤ t < (e -1)/2 be an integer, put r := t(t + 1)/2 and let λ := ∆ t .

Let m ∈ N, put n := r + 2m and G = GU n (q). Then

L := L r,m ∼ = GU r (q) × GL 1 (q 2 ) m
is a pure Levi subgroup of G and X λ is a cuspidal unipotent kL-module.

If ℓ > Ñ(e, n) (cf. ( 7)), the unipotent kG-module X ρ lies in the weak Harish-Chandra series defined by (L, X λ ) if and only if Proof. The cuspidal unipotent KG-module Y λ of GU r (q) reduces irreducibly to the unipotent kG-module X λ (see [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF]Theorem 6.10]). In particular, X λ is cuspidal.

ρ(2) ∈ Φ (t+(1-e)/2,0)
Let Xλ denote an OL-lattice in Y λ . The endomorphism algebra H G,O (L, Xλ ) is an Iwahori-Hecke algebra over O of type B m with parameters q 2t+1 and q 2 . By a result of Dipper [7, Theorem 4.9], the ℓ-modular decomposition matrix of H G,O (L, Xλ ) is embedded into the decomposition matrix of the unipotent KG-modules as a submatrix.

Notice that λ is an e-core by our assumption on t. Hence Y λ is of ℓ-defect 0 and thus X λ and Xλ are projective. It follows that R G L ( Xλ ) is projective as well. The corresponding columns of the decomposition matrix of OG are exactly the columns of the decomposition matrix of H G,O (L, Xλ ), extended by zeroes. Let Ẑ be an indecomposable summand of R G L ( Xλ ) and let Y ρ be a composition factor of K ⊗ O Ẑ with ρ maximal in the lexicographic order on partitions of n. If Y κ is another composition factor of K ⊗ O Ẑ, then κ ρ by [START_REF] Geck | On the decomposition numbers of the finite unitary groups in nondefining characteristic[END_REF] (see also [18, (2.23)]). In particular, the head of k ⊗ O Ẑ equals X ρ by the labelling of the unipotent kG-modules described in 5.3. Thus X ρ lies in the Harish-Chandra series defined by (L, X λ ). Every element of this series arises in this way.

To proceed, we will make use of the notion of a canonical basic set as defined in [START_REF] Geck | Representations of Hecke algebras at roots of unity[END_REF]Definition 3.2.1]. Applying the results of [18, Section 3], we obtain the following facts. Firstly, the Iwahori-Hecke algebra H G,k (L, X λ ) has a canonical basic set with respect to Lusztig's afunction on H G,k (L, X λ ) if ℓ > Ñ(e, n) (see (7) and [START_REF] Geck | Modular principal series representations[END_REF]Corollary 3.3]). Secondly, this canonical basic set agrees with the canonical basic set of a suitable specialization of a generic Iwahori-Hecke algebra to an Iwahori-Hecke algebra H (2e) K of type B m , whose parameters are powers of a 2eth root of unity (see [START_REF] Geck | Modular principal series representations[END_REF]Lemma 3.1]). The canonical basic set of H G,k (L, X λ ) (or rather of the algebra H The simple H G,K (L, Y λ )-modules correspond to the simple constituents of R G L (Y λ ) via Fitting correspondence. The first are labelled by bipartitions of m, the latter ones by the partitions of n with 2-core ∆ t .

The matching between the two labellings is given by µ ↔ Φ t (µ) for µ ∈ P

(2) m (see the Appendix of [START_REF] Fong | Brauer trees in classical groups[END_REF]). Suppose that µ, µ ′ ∈ P

(2) m with Φ t (µ) Φ t (µ ′ ). Then a µ ′ ≤ a µ , where a denotes Lusztig's a-function of the Iwahori-Hecke algebra H G,K (L, Y λ ). This inequaltity follows from the results summarized in [18, 2.5] together with Lusztig's formula for the a-function described in [23, Example 1.3.9, Case 3]. Through the embedding of the decomposition matrix of H G,O (L, Xλ ), the members of the canonical basic set thus correspond to the composition factors of R G L (Y λ ) which are at the top of their respective columns in the decomposition matrix of OG. As these top composition factors label the kG-modules in the Harish-Chandra series of kG defined by (L, X λ ), our claim follows. Theorem 6.2 is true without the assumption that t < (e -1)/2 if Conjecture 5.4 holds. Indeed, in this case every unipotent kG-module in the (L, X λ )-series is labelled by a partition with 2-core ∆ t . Let P (X λ ) ∈ OL-mod denote the projective cover of X λ . Again by [7, Theorem 4.9], the decomposition matrix of R G L ( P (X λ )) contains the decomposition matrix of H G,O (L, Y λ ) as a submatrix (with a row of the latter labelled by µ ⊢ 2 m corresponding to a row of the former labelled by Φ t (µ)). Let Ẑ be an indecomposable summand of R G L ( P (X λ )) such that the head of k ⊗ O Ẑ lies in the Harish-Chandra series defined by (L, X λ ). Put Z := K ⊗ O Ẑ, and let Y ρ be a unipotent composition factor of Z with ρ maximal. Then X ρ is the head of k ⊗ O Ẑ, and hence ρ (2) = ∆ t . It follows as in the proof above that ρ(2) ∈ Φ (t+(1-e)/2,0) e,m . 6.3. Let G = GU n (q). Then Conjecture 5.5 holds in the following form for unipotent blocks of e-weight 1 of G (for the notions of e-core and e-weight of a unipotent ℓ-block of G see 5.3).

Theorem. Assume that e is odd and let B be a unipotent ℓ-block of G of e-weight 1. Then B contains a weakly cuspidal kG-module, if and only if the e-core of B is a 2-core.

Proof. Suppose first that the e-core of B is a 2-core. Then B contains a cuspidal simple KG-module by Lemma 5.5 as e < ℓ. In particular, B contains a cuspidal unipotent kG-module.

Now suppose that the e-core of B is not a 2-core. Let s(B) denote the Scopes number of B (see [30, 7.2] for the definition of s(B)). Our assumption implies that s(B) ≥ 1. Indeed, consider an e-abacus diagram (in the sense of [36, p. 78f] or [13, Section 1]) for the e-core of B. Since the latter is not a 2-core, there is 0 ≤ i ≤ e -1 such that the number of beads on string i is at least one larger than the number of beads on string i -2, if 2 ≤ i ≤ e -1, and at least two larger than the number of beads on string e-2 or e-1, if i = 0 or 1, respectively. This exactly means s(B) ≥ 1. The Reduction Theorem and its consequence [START_REF] Hiss | Scopes reduction and Morita equivalence classes of blocks in finite classical groups[END_REF]Theorems 7.10,8.1] now imply that every projective kG-module of B is obtained from Harish-Chandra induction of a projective kGmodule of GU n-2 (q) × GL 1 (q 2 ). In particular, B contains no weakly cuspidal kG-module.

6.4.

Let e be odd. We now determine all partitions µ ∈ P of e-weight 1 such that X µ is weakly cuspidal. For 0 ≤ t ≤ (e -1)/2 let µ t,e := (t, t -1, . . . 3, 2, 1 e+1 ), and for 0 ≤ t < (e -1)/2 let ν t,e := (t + 2, t + 1, . . . , 3, 2, 1 e-2t-2 ).

(we understand µ 0,e = 1 e and µ 1,e = 1 e+1 ). For t = (e -1)/2, we also put ν t,e := µ t,e .

Proposition. Let G = GU n (q). Assume that e is odd and let µ ∈ P n have e-weight 1. Then X µ is weakly cuspidal if and only if n = t(t + 1)/2 + e for some 0 ≤ t ≤ (e -1)/2 and µ ∈ {µ t,e , ν t,e }.

Proof. The result is trivially true for e = 1. Thus assume that e ≥ 3 in the following. Let B denote the unipotent ℓ-block of G containing X µ .

Assume first that X µ is weakly cuspidal. Then, by Theorem 6.3, the e-core of µ is a 2-core, ∆ t , say. In particular, n = t(t + 1)/2 + e. As ∆ t is an e-core, we have 0 ≤ t ≤ (e -1)/2. By [13, (6A)], the partitions µ t,e and ν t,e label the unipotent KG-modules in B connected to the exceptional vertex of the Brauer tree of B (there is only one such if t = (e -1)/2).

Suppose that µ ∈ {µ t,e , ν t,e }. Let µ ′ ∈ {µ t,e , ν t,e } such that Y µ and Y µ ′ lie on the same side of the exceptional vertex in the Brauer tree of B. Then µ and µ ′ have the same 2-core ∆ s , say, again by [13, (6A)]. If µ ′ = µ t,e , we clearly have s < t, and thus ∆ s is an e-core. If µ ′ = ν t,e , then s = t+2, and ∆ s is an e-core if e ≥ 2t+5, and of e-weight 1 if e = 2t+3. In the latter case, n = t(t + 1)/2 + (2t + 3) = (t + 2)(t + 3)/2 = |∆ s |, and thus µ = ∆ s = ν t,e , a contradiction. Thus in any case ∆ s is an e-core, and so X ∆s is projective. Using [13, (6A)] once more, we find that X µ lies in the Harish-Chandra series defined by (L, X ∆s ), where L is the pure standard Levi subgroup of G corresponding to GU |∆s| (q). In particular, X µ is not weakly cuspidal, contradicting our assumption. Now assume that µ is one of µ t,e or ν t,e . Then the e-core of µ equals ∆ t , and X µ corresponds to the edge of the Brauer tree linking Y µ with the exceptional vertex. Now B contains a cuspidal nonunipotent KG-module Y by Lemma 5.5 (or rather its proof). Hence Y is attached to the exceptional vertex, which thus labels cuspidal simple KG-modules. It follows that X µ is cuspidal, completing our proof.

More evidence for our conjectures is given in the next section where we prove some consequences of our conjectures for the crystal graph.

Some properties of the crystal graph

The conjectures formulated in Section 5 imply some combinatorial properties of the crystal graphs involved. In this final section we prove some of these properties. Throughout this section we let e and t be nonnegative integers with e odd and larger than 1. (Contrary to previous usage, the letter k no longer denotes a field, but just an integer.) 7.1. Following [23, 6.5.17], we define a 1-runner abacus to be a subset A of Z such that -j ∈ A and j ∈ A for all j ≥ n and some 0 = n ∈ N. Let A be a 1-runner abacus. We enumerate the elements of A by a 1 , a 2 , . . . with a 1 > a 2 > • • • . The elements of Z \ A are called the holes of A. If we define λ j to be the number of holes of A less than a j , j = 1, 2, . . ., then λ := (λ 1 , λ 2 , . . .) is the partition associated to A. The charge of A is the integer a 1λ 1 . Let n be a positive integer such that {-j | j ≥ n} ⊆ A. Then the number of elements of A larger than -n equals n plus the charge of A. Moreover, a β-set for λ, in the sense of [35, p. 2], is obtained by adding n -1 to the elements of

A \ {-j | j ≥ n} = {x ∈ A | x ≥ -(n -1)
} to make them all nonnegative. Let A and A ′ be 1-runner abaci with associated partitions λ and λ ′ and charges c and c ′ . Then

A = A ′ if and only if λ = λ ′ and c = c ′ . Also, if A ⊆ A ′ and |A ′ \ A| = 1, then c ′ = c + 1.
By a symbol we mean a pair B := (B 1 , B 2 ) of 1-runner abaci. The components B 1 and B 2 are also called the first and second row of B, respectively. If µ i and c i are the partition associated to B i and the charge of B i , respectively, i = 1, 2, we also write B = B(µ, c) with µ = (µ 1 , µ 2 ) and c = (c 1 , c 2 ). Let c = (c 1 , c 2 ) ∈ Z 2 and let µ ∈ P (2) be a bipartition. Then B(µ, c) can be computed as follows (see [34, 2.2]). Let µ = (µ 1 , µ 2 ) with µ i = (µ i j ) j≥1 and µ

i j ≥ µ i j+1 ≥ 0 for j ≥ 1 and i = 1, 2. Then B(µ, c) = (B(µ, c) 1 , B(µ, c) 2 ) with B(µ, c) i := (B(µ, c) i j ) j≥1
, where B(µ, c) i j := µ i jj + c i + 1 for i = 1, 2 and j ≥ 1.

7.2. Put c = (t + (1e)/2, 0) and let µ = (µ 1 , µ 2 ) be a bipartition. To B(µ, c) we associate the 1-runner abacus

A e (µ, c) := {2j + e | j ∈ B(µ, c) 1 } ∪ {2j | j ∈ B(µ, c) 2 }.
In order to determine the partition associated to A e (µ, c), choose an even positive integer n = 2m such that {-j | j ≥ n -1} ⊆ A e (µ, c) and put

Ā := {x + n | x ∈ A e (µ, c), x ≥ -n}.
Then Ā is a β-set for the partition associated to A e (µ, c) with 0, 1 ∈ Ā.

Let Ā1 := {(x -1)/2 | x ∈ Ā, x odd} and Ā2 := {x/2 | x ∈ Ā, x even}. Then Ā1 = {j + (e -1)/2 + m | j ∈ B(µ, c) 1 , j ≥ -m -(e -1)/2} and Ā2 = {j + m | j ∈ B(µ, c) 2 , j ≥ -m}.
In particular, Āi is a β-set for µ

i , i = 1, 2 and | Ā1 | = | Ā2 | + t.
The latter equality follows from the remarks in the first paragraph of 7.1.

Lemma. The partition associated to A e (µ, c) equals Φ t (µ).

Proof. Use the notation introduced above. Then

| Ā| = (| Ā1 | + | Ā2 |) ≡ t (mod 2).
Thus Ā is a β-set for the partition with 2-core ∆ t , and 2-quotient (computed with respect to a β-set with an odd number of elements) (µ 2 , µ 1 ) if t is odd, and (µ 1 , µ 2 ) if t is even. This implies our claim. 7.3. Let c = (t+(1-e)/2, 0) and let µ ∈ P (2) . We are interested in the operation of deleting e-hooks from Φ t (µ). On A e (µ, c), this amounts to replacing an element y ∈ A e (µ, c) with ye ∈ A e (µ, c) by ye. If y is odd, this replacement corresponds to the operation of deleting j = (ye)/2 from B(µ, c) 1 and inserting j into B(µ, c) 2 . If y is even, this replacement corresponds to the operation of deleting j = y/2 from B(µ, c) 2 and inserting je into B(µ, c) 1 . This leads to the following operations on symbols, to which we refer as elementary operations.

(a) Delete an element j in the first row, which is not in the second row, and insert j in the second row. (b) Delete an element j in the second row, such that je is not in the first row, and insert je in the first row.

Iterating the two operations we end up with a symbol for which no such operation is possible. Even though the resulting symbol does not depend on the order in which we perform these operations, we decide to do the former operation first if possible, and always take the largest possible j so that each step in the algorithm is well defined. This gives the following elementary operations in a more restrictive sense. (a ′ ) Delete the largest element j in the first row, which is not in the second row, and insert j in the second row. (b ′ ) If every element in the first row is contained in the second row, delete the largest element j in the second row, such that je is not in the first row, and insert je in the first row.

Proposition. Put λ := Φ t (µ). Let µ ′ = ((µ ′ ) 1 , (µ ′ ) 2 ) ∈ P (2) and c ′ ∈ Z 2 such that B(µ ′ , c ′ ) is obtained from B(µ, c
) by an elementary operation of type (a) or (b).

Applying this elementary operation corresponds to removing an ehook from λ. Denote by λ ′ the resulting partition, and let t ′ be such that λ ′

(2) = ∆ t ′ . Suppose that μ = (μ 1 , μ2 ) is the bipartition such that Φ t ′ (μ) = λ ′ .
Then t ′ = t + 2, if the elementary operation applied is of type (b). If the elementary operation applied is of type (a), then

t ′ =    t -2, if t ≥ 2, 0, if t = 1, 1, if t = 0.
Moreover, μ = µ ′ , if t and t ′ have the same parity, ((µ ′ ) 2 , (µ ′ ) 1 ), otherwise, Proof. Consider a β-set Ā for Φ t (µ) as constructed in 7.2. An elementary operation results in replacing an element x of Ā by xe yielding the β-set Ā′ for λ ′ . (Notice that Ā′ is constructed from B(µ ′ , c ′ ) in the same way as Ā from B(µ, c).) Moreover, x is even or odd, if the elementary operation is of type (b) or (a), respectively. In the former case, the number of odd elements of Ā increases by 1, and thus t ′ = t+2. In the latter case, the number of odd elements of Ā decreases by 1.

Hence t ′ = t -2 if t ≥ 2, t ′ = 0 if t = 1, and t ′ = 1 if t = 0.
If the parity of t is the same as that of t ′ , then the constructions of Φ t (µ) and of Φ t ′ (μ) are the same, namely we have λ (2) = (µ 1 , µ 2 ) and (λ ′ ) (2) = (μ 1 , μ2 ) (respectively λ (2) = (µ 2 , µ 1 ) and (λ ′ ) (2) = (μ 2 , μ1 )) if t is even (respectively odd). Therefore, one can read off μ directly on the symbol B(µ ′ , c ′ ) (or on the β-sets Āi , i = 1, 2). It follows that μ = µ ′ .

On the other hand, if t and t ′ have different parities (say, without loss of generality, t even and t ′ odd), then the construction of Φ t ′ (μ) requires a permutation, unlike that of Φ t (µ). Therefore, one needs to permute the components of the bipartition one reads off B(µ ′ , c ′ ), i.e. μ = ((µ ′ ) 2 , (µ ′ ) 1 )

As an example, consider the bipartition µ = ((5 3 , 4 2 ), ( 6)), let e = 3 and t = 5. Then c = (4, 0) and

B(µ, c) = • • • -2 -1 6 • • • -2 -1 4 5 7 8 9 .
The associated 1-runner abacus A 3 (µ, c) can be represented as follows:

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 t t q t q q q q q q q q q t t t q q q t q t q t

With the notation of 7.2, taking n = 2 we obtain the β-set Ā = {0, 1, 3, 13, 14, 15, 19, 21, 23} for the partition λ := Φ 5 (µ) associated to A 3 (µ, c). We also have Ā1 = {0, 1, 6, 7, 9, 10, 11} and Ā2 = {0, 7}, which are β-sets for µ 1 = (5 3 , 4 2 ) and µ 2 = (6) respectively. Notice that λ = (15, 14, 13, 10 3 , 1). An elementary operation of type (a ′ ) on the symbol yields

B(µ ′ , c ′ ) = • • • -2 -1 6 9 • • • -2 -1 4 5 7 8 ,
with µ ′ = ((5 2 , 4 2 ), [START_REF] Dipper | Harish-Chandra vertices and Steinberg's tensor product theorem for general linear groups[END_REF][START_REF] Dipper | On the decomposition numbers of the finite general linear groups. II[END_REF]) and c ′ = (3, 1)). The 1-runner abacus A 3 (µ ′ , c ′ ) can be pictured as follows:

-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 t t q t q q q q q q q q q t t t q q q t t t q q

We obtain Ā′ = {0, 1, 3, 13, 14, 15, 19, 20, 21}, again using n = 2. Next, ( Ā′ ) 1 = {0, 1, 6, 7, 9, 10} and ( Ā) 2 = {0, 7, 10}, which are β-sets for (5 2 , 4 2 ) and (8, 6) respectively. The partition associated to A 3 (µ ′ , c ′ ) is λ ′ = (13 3 , 10 3 , 1) which is obtained from λ by removing a 3-hook. We have λ ′

(2) = ∆ 3 , i.e. t ′ = 3, and Φ 3 (µ ′ ) = λ ′ . 7.4. In the following we will make use of the notion of an e-period of a symbol (see [START_REF] Jacon | A combinatorial decomposition of higher level Fock spaces[END_REF]Definition 2.2]) and the concept of totally periodic symbols (see [START_REF] Jacon | A combinatorial decomposition of higher level Fock spaces[END_REF]Definition 5.4]). Let |µ, c be a charged bipartition. In our special situation, an e-period of B(µ, c) is a sequence (i 1 , k 1 ), (i 2 , k 2 ), . . . , (i e , k e ) of pairs of integers with

2 ≥ k 1 ≥ k 2 ≥ • • • ≥ k e ≥ 1 such that B(µ, c) k l i l = m-l +1
for some integer m. Moreover, m is the largest element in B(µ, c) 1 ∪ B(µ, c) 2 , and if ml + 1 ∈ B(µ, c) 1 for some 1 ≤ l ≤ e, then k l = 1. Suppose that B(µ, c) has an e-period (i 1 , k 1 ), (i 2 , k 2 ), . . . , (i e , k e ). Then this e-period is unique and the entries B(µ, c) k l i l of B(µ, c) are called the elements of the period. Removing these elements from B(µ, c), we obtain the symbol B(µ ′ , c ′ ) corresponding to a charged bipartition |µ ′ , c ′ which may or may not have an e-period. If iterating this procedure ends up in a symbol B(ν, d) such that ν is the empty bipartition, then B(µ, c) is called totally periodic.

By [34, Theorem 5.9], the symbol B(µ, c) is totally periodic, if and only if |µ, c is a highest weight vertex of G c,e . If B(µ, c) is totally periodic, then for each entry j in B(µ, c), there is a symbol B ′ , obtained from B(µ, c) by removing a sequence of e-periods, and an e-period (i 1 , k 1 ), . . . , (i e , k e ) of B ′ , such that j = (B ′ ) k l i l for some 1 ≤ l ≤ e. By a slight abuse of terminology, we say that j is contained in the period (i 1 , k 1 ), . . . , (i e , k e ) of B.

Let B ′ denote the symbol obtained from B(µ, c) by applying an elementary operation.

Lemma. If B(µ, c) is totally periodic, so is B ′ .

Proof. Suppose first that B ′ is obtained from B(µ, c) by an elementary operation (a). Moving j from row 1 to row 2 transforms the period (i 1 , k 1 ), . . . , (i e , k e ) containing j into a period (i ′ 1 , k ′ 1 ), . . . , (i ′ e , k ′ e ) such that (B ′ )

k ′ l i ′ l = B(µ, c) k l
i l for all l. In particular, B ′ is also totally periodic.

Suppose now that B ′ is obtained from B(µ, c) by an elementary operation (b). Deleting j from row 2 and inserting je in row 1 transforms the period (i 1 , k 1 ), . . . , (i e , k e ) containing j into a period

(i ′ 1 , k ′ 1 ), . . . , (i ′ e , k ′ e ) such that (B ′ ) k ′ l i ′ l = B(µ, c) k l i l -1 for all l < e -1 and (B ′ ) k ′ e i ′ e = B(µ, c) ke
iee. In particular, B ′ is also totally periodic.

7.5.

Let G = GU n (q), and let ℓ and e be as in 5.1. In [START_REF] Geck | Cuspidal unipotent Brauer characters[END_REF]Theorem 8.3] it is proved that X (1 n ) is cuspidal if and only if e is odd and divides n or n -1. This is consistent with Conjecture 5.8, as will be shown below. Let λ = (1 n ). Then the 2-core of λ equals ∆ t with t = 0 if n is even, and t = 1 if n is odd. Also λ(2) = (-, 1 m ) with m = ⌊n/2⌋; notice that n = 2m + t.

Proposition. Let e ≥ 3 be an odd integer, let m ∈ N and t ∈ {0, 1}. Proof. The proof proceeds by induction on m, the case m = 0 being clear. Assume that m > 0 and let s, s ′ ∈ {(e -1)/2, (e -3)/2} with s = s ′ . The symbol B of |(-, 1 m ), (-s, 0) equals

B = • • • -m 2 -m 3 -m • • • -s + 1 -s + 2 • • • 1 • • • -m 1 -m 2 -m • • • -s .
Let B ′ be the symbol obtained by removing the e-period from B. If m < e -1, we find

B ′ = • • • -(e -1) • • • -m 2 -m • • • -s • • • -(e -1)
,

and if m = e -1, we have

B ′ = • • • -(e -1) -(e -3) • • • -s • • • -(e - 1
) .

In the latter two cases, B ′ does not have an e-period and thus B is not totally periodic. On the other hand, e does not divide one of 2m -1, 2m, or 2m + 1, as 1 ≤ m ≤ e -1.

If m ≥ e, then Lemma. Suppose that B is totally e-periodic, that B 1 ⊆ B 2 and that je ∈ B 1 for all j ∈ B 2 with j ≥ m for some m ∈ Z.

B ′ = • • • -m 2 -m 3 -m • • • -e + 2 -e + 3 • • • -s • • • -m 1 -m 2 -m • • • -e + 1 .
Then for k = 1, 2 we have j-1 ∈ B k for all j ∈ B k with j ≥ m-e+1.

Proof. Let j ∈ B k with j -1 ∈ B k . Then j -1 ∈ B 1 and the period of B containing j ends in j. The first element in this period is j + e -1, and j + e -1 ∈ B 2 . As j -1 = j + e -1e, it follows that j + e -1 < m, hence our claim.

Put c = (t+(1-e)/2, 0). If Conjecture 5.8 is true, the highest weight vectors of the crystal graph G c,e label the weakly cuspidal unipotent GU n (q)-modules for large enough primes ℓ with e = e(q, ℓ). More explicitly, a weakly cuspidal GU n (q)-module X λ with λ (2) = ∆ t should be labelled by the highest weight vector | λ(2) , c . Moreover, if X λ is weakly cuspidal, the e-core of λ should be a 2-core by Conjecture 5.5.

Recall that λ with λ (2) = ∆ t and λ(2) are related by λ = Φ t ( λ(2) ).

Theorem. Let the notation be as above. Let µ ∈ P (2) be such that |µ, c is a highest weight vertex in G c,e . Then the e-core of Φ t (µ) is a 2-core.

Proof. Starting with B(µ, c), we apply a sequence of elementary operations, until we reach a symbol B ′ , which does not allow any such operation. Starting with A e (µ, c), the corresponding sequence of operations results in a 1-runner abacus A ′ , such that ye ∈ A ′ for all y ∈ A ′ . By Lemma 7.2, the partition associated to A ′ is the e-core of Φ t (µ).

The symbol B ′ is totally e-periodic by Lemma 7.4, and satisfies the assumptions of the above lemma for all m ∈ Z. Hence for k = 1, 2, we have j -1 ∈ (B ′ ) k for every j ∈ (B ′ ) k . This implies that x -2 ∈ A ′ for all x ∈ A ′ . In particular, the partition associated to A ′ is a 2-core.

Alternatively, one can use Uglov's approach to the Fock space to give a less combinatorial, more conceptual proof of the above theorem, which we are now going to sketch. Consider, for s ∈ Z, the space of semi-infinite wedge products Λ s+∞/2 , as it is defined in [47, §4]. We do not need the precise definition of this space here but we need to know that there are three ways to index the elements of its standard basis ("the semi-infinite ordered wedges"):

-by the set of elements denoted by |λ, s where λ ∈ P; -by the set of elements denoted by |µ, c , where µ ∈ P (2) and c = (c 1 , c 2 ) ∈ Z 2 is such that c 1 + c 2 = s. The way to pass from |λ, s to |µ, c is purely combinatorial; -by the set of elements denoted by |λ (e) , c (e) where λ (e) is the e-quotient of λ and c (e) = (c 1 , . . . , c e ) ∈ Z e satisfies e i=1 = s and parametrizes the e-core of λ. (Although this property is not used below, it is needed to understand the reference to Uglov's work at the end of this paragraph.)

Setting u := -v -1 , we have three actions of the algebras U ′ v ( sl e ), U ′ u ( sl 2 ) and another algebra H (the Heisenberg algebra) on the space Λ s+∞/2 . Moreover these three actions commute and we have the following decomposition (see [START_REF] Uglov | Canonical bases of higher-level q-deformed Fock spaces and Kazhdan-Lusztig polynomials[END_REF]Theorem 4.8]): [47, (38)], leads to an alternative proof of the above theorem. From now on, let |µ, c be a highest weight vertex in G c,e . Let |µ ′ , c ′ be a charged bipartition such that B(µ ′ , c ′ ) is the symbol obtained from B(µ, c) by applying one of the elementary operations described in 7.3 (a ′ ), (b ′ ). By Lemma 7.2, this implies in particular that Φ t (µ) is not an e-core.

Λ s+∞/2 = c∈A 2 e (s) U ′ v (
Lemma. Under the above hypothesis, |µ ′ , c ′ is a highest weight vertex and there is a crystal isomorphism B(µ, c) ≃ B(µ ′ , c ′ ).

Proof. By Lemma 7.4, we know that B(µ ′ , c ′ ) is totally periodic, and thus |µ ′ , c ′ is a highest weight vertex by [START_REF] Jacon | A combinatorial decomposition of higher level Fock spaces[END_REF]Theorem 5.9]. By the discussion at the beginning of this paragraph, it remains to show that the reduced i-words of |µ, c and |µ ′ , c ′ coincide for all 0 ≤ i ≤ e -1. Denote these words by w i (µ, c) and w i (µ ′ , c ′ ). In this proof, we use for more clarity the notation A k (j) (respectively R k (j)) instead of simply j -1 are congruent modulo e, and R 1 (j -1)A 2 (m) and R 2 (j)A 1 (j) vanish. Assume now that j is not the largest element of B(µ, c) 1 . First we consider the case that j + 1 / ∈ B(µ, c) 1 . The fact that B(µ, c) is totally periodic then implies that j 2 . We obtain the following five subcases.

+ 1 ∈ B(µ, c) 2 if j -1 / ∈ B(µ, c) 1 and j -1 ∈ B(µ, c) 2 if j -1 ∈ B(µ, c) 1 and j + 1 ∈ B(µ, c)
•

If j -1 / ∈ B(µ, c) 1 , j + 1 ∈ B(µ, c) 2 and j -1 / ∈ B(µ, c) 2 , then R 2 (j)A 1 (j) vanishes and R 1 (j -1) becomes R 2 (j -1). • If j -1 / ∈ B(µ, c) 1 , j + 1 ∈ B(µ, c) 2 and j -1 ∈ B(µ, c) 2 , then R 2 (j)A 1 (j) vanishes and R 2 (j -1)A 1 (j -1) appears. • If j -1 ∈ B(µ, c) 1 , j + 1 ∈ B(µ, c) 2 and j -1 / ∈ B(µ, c) 2 , then A 1 (j) becomes A 2 (j) and R 2 (j -1)A 1 (j -1) appears. • If j -1 ∈ B(µ, c) 1 , j + 1 ∈ B(µ, c) 2 and j -1 / ∈ B(µ, c) 2 , then R 2 (j)A 1 (j) vanishes and R 2 (j -1)A 1 (j -1) appears. • If j -1 ∈ B(µ, c) k , k = 1, 2 and j + 1 ∈ B(µ, c) 2 , then R 2 (j)A 1 (j) vanishes and A 2 (j -1) becomes A 1 (j -1). If j + 1 ∈ B(µ, c) 1 , then j + 1 ∈ B(µ, c) 2 otherwise j would not be moved. • If j -1 / ∈ B(µ, c) 2 and j -1 / ∈ B(µ, c) 1 , then R 1 (j -1) becomes R 2 (j -1) and R 2 (j) becomes R 1 (j). • The case j -1 ∈ B(µ, c) 2 and j -1 / ∈ B(µ, c) 1 can not occur as B(µ, c) is totally periodic. • If j -1 / ∈ B(µ, c) 2 and j -1 ∈ B(µ, c) 1 , then R 2 (j) becomes R 1 (j) and R 2 (j -1)A 1 (j -1) appears. • If j -1 ∈ B(µ, c) 2 and j -1 ∈ B(µ, c) 1 , then R 2 (j) becomes R 1 (j)
and A 2 (j -1) becomes A 1 (j -1). Suppose now that we apply operation (b ′ ), that is to say, that we delete j from B(µ, c) 2 and insert je in B(µ, c) 1 . This implies in particular that all elements of B(µ, c) 1 are in B(µ, c) 2 . Again, assume first that j is the largest element of B(µ, c) 2 . As B(µ, c) is totally e-periodic, j -1 appears in B(µ, c), hence j -1 ∈ B(µ, c) 2 . Denote by l the largest element of B(µ, c) 1 . Suppose first that je > l.

• If je > l + 1, then A 2 (j) becomes A 1 (je), and R 1 (j -1e)A 2 (j -1) appears. • If je = l + 1, then A 2 (j) becomes A 1 (je) and A 1 (je -1)

becomes A 2 (j -1). Now assume that je < l. Note that in this case je + 1 ∈ B(µ, c) 1 . Indeed, j is the first element in the period of B(µ, c), and je + 1 the last. As l ≥ je + 1 and l lies in the first row, so does je + 1.

• If je -1 / ∈ B(µ, c) 1 , then R 1 (je)A 2 (j) vanishes and R 1 (je -1)A 2 (j -1) appears.

• If je -1 ∈ B(µ, c) 1 , then A 1 (je -1) becomes A 2 (j -1)

and R 1 (je)A 2 (j) vanishes. Finally, assume that j is not the largest element of B(µ, c) 2 and let j ′ denote the largest element of B(µ, c) 2 . Then j ′e ∈ B(µ, c) 1 , as our operation of type (b ′ ) always moves the largest possible element. Hence l ≥ j ′e > je. Now j is the largest element of B(µ, c) 2 such that je is not in B(µ, c) 1 . By Lemma 7.6, this implies that for k = 1, 2 and every r > je + 1 we have r -1 ∈ B(µ, c) k if r ∈ B(µ, c) k . Hence all integers in the interval [je + 1, j ′ ] and [je + 1, l] are contained in B(µ, c) 2 and B(µ, c) 1 , respectively. This implies in particular that je ∈ B(µ, c) 2 as otherwise the element je + 1 of the second row must be the last element in its period. But then the element je + 1 of the first row must lie in an earlier period, which is impossible. This leaves to check the following possibilities.

• If je -1 ∈ B(µ, c) 1 it is also contained in B(µ, c) 2 , and R 1 (je) becomes R 2 (j) and A 1 (je -1) becomes A 2 (j -1). • If je -1 ∈ B(µ, c) 2 and je -1 ∈ B(µ, c) 1 , then R 1 (je) becomes R 2 (j) and R 1 (je -1)A 1 (j -1) appear. • If je -1 ∈ B(µ, c) 2 and je -1 ∈ B(µ, c) 1 , then R 1 (je) becomes R 2 (j) and R 1 (je -1)A 1 (j -1) appear. In each case, we see that w i (µ) = w i (µ ′ ), for all i = 1, . . . , e -1.

We record a first consequence of the above lemma. Let t ′ ∈ N and μ ∈ P (2) be such that λ ′ := Φ t ′ (μ) equals the partition obtained from λ := Φ t (µ) be removing the e-hook which corresponds to the elementary operation transforming B(µ, c) into B(µ ′ , c ′ ). (See Proposition 7.3 how to compute t ′ and μ.) Suppose that t and t ′ have the same parity and put c := (t ′ + (1e)/2, 0). Then μ = µ ′ and c is obtained from c ′ by adding or subtracting 1 to each of its components. By definition of the crystal graph, it is clear that translating each component of the charge by some fixed integer, results in the same graph with an overall translation of the labels of the arrows. In particular, |μ, c is a highest weight vertex.

Corollary. Let the notation be as in Section 5. Suppose that Conjecture 5.7 is true. Then the Harish-Chandra branching graphs corresponding to the weakly cuspidal modules X λ and X λ ′ are isomorphic (up to rank b(e, ℓ); cf. ( 8)).

Proof. It follows from the considerations preceeding the corollary, that B(µ, c) and B(μ, c) are isomorphic up to a global shift of the arrow labels.

This corollary shows that the validity of Conjecture 5.7 would yield a remarkable connection between the Harish-Chandra theory of unitary groups of odd and even degrees. Indeed, let ρ ∈ P be such that X ρ is weakly cuspidal. Suppose that ρ is not a 2-core, and that the 2-core of ρ is a partition of an integer at least 3. Then, by Proposition 7.3 and the proof of Theorem 7.6, there is a partition ρ ′ obtained from ρ by removing an e-hook such that X ρ ′ is weakly cuspidal. Moreover, the assumptions of the corollary are satisfied for the pair (ρ, ρ ′ ) and hence the Harish-Chandra branching graphs beginning in X ρ and X ρ ′ agree up to some rank. As e is odd, ρ and ρ ′ are partitions of integers of different parity. For the time being, this phenomenon can not be explained by Harish-Chandra theory.

7.8. We finally prove a property of the crystal graph which is implied by the considerations in 5.10. Let λ ∈ P with λ (2) = ∆ t and λ(2) = µ. Put c := (t+(1-e)/2, 0). Assume that |µ, c is a highest weight vector in G c,e . By Theorem 7.6, the e-core of λ is a 2-core, ∆ s , say, for some non-negative integer s. Put s := (s + (1e)/2, 0).

Theorem. With the notation introduced above, there is a graph isomorphism B(µ, c) ≃ B((-, -), s), up to a shift of the labels of the arrows.

Proof. We apply the algorithm used to compute the e-core of λ = Φ t (µ) described in the proof of Theorem 7.6. Applying a sequence of elementary operations of types (a ′ ) and (b ′ ) to B(µ, c), we end up with the symbol B((-, -), d) for some charge d = (d 1 , d 2 ).

We may as well apply the corresponding sequence of moves to the β-set Ā for λ = Φ t (µ) as constructed in 7.2. This results in a β-set Ā′ for ∆ s . The number of odd elements of Ā exceeds its number of even elements by t = c 1 -(1e)/2c 2 . If the number of odd elements of Ā′ is not smaller than the number of its even elements, the difference between the two numbers equals s. Otherwise, there are s + 1 more even numbers in Ā′ than odd ones. An operation of type (a ′ ) decreases the first component of the current charge by 1 and increases the second component by 1. The corresponding move on the β-set replaces an odd number by an even one. The analogous remarks apply for elementary operations of type (b ′ ). We thus find Note that there should be a way to relate these elementary crystal isomorphisms with the so-called canonical crystal isomorphism of [START_REF] Gerber | Crystal isomorphisms in Fock spaces and Schensted correspondence in affine type A, Algebr. Represent. Theory[END_REF]. 7.9. Put c := (t + (1e)/2, 0). Let µ = (µ 1 , µ 2 ) be a bipartition. For 0 ≤ j ≤ e -1, let f j denote the associated Kashiwara operator on G c,e (see 5.6).

Proposition. Let 0 ≤ j 1 = j 2 ≤ e -1. Suppose that f j i .|µ, c = 0 for i = 1, 2. Write f j i .|µ, c = |ν i , c , i = 1, 2. Then the e-cores of Φ t (ν 1 ) and of Φ t (ν 2 ) are distinct.

Proof. Let 0 ≤ j ≤ e -1. First note that if f j .|µ, c = 0 then (1) B(µ, c) 1 = B( f j .µ, c) 1 and B( f j .µ, c) 2 = B(µ, c) 2 ∪{k}\{k-1}

for k ∈ Z such that k ≡ j (mod e), or (2) B(µ, c) 2 = B( f j .µ, c) 2 and B( f j .µ, c) 1 = B(µ, c) 1 ∪{k}\{k-1}

for k ∈ Z such that k ≡ j (mod e). We have seen in 7.6 how to compute the e-cores of Φ t (ν i ), i = 1, 2. In this procedure, some of the elements x in B(ν i , c), i = 1, 2, must be replaced by xk.e for some k ∈ N. If the e-core of Φ t (ν 1 ) equals the e-core of Φ t (ν 2 ), this implies that at the end of these procedures, we obtain the same symbols. However, this is impossible as j 1 ≡ j 2 (mod e).

Table 1. The Harish-Chandra branching graph for GU n (q), 1 ≤ n ≤ 7 odd, ℓ | q 2q + 1, ℓ > 7

1 3 } } ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ ! ! ❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ 1 3 | | ③ ③ ③ ③ ③ ③ ③ ③ " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ 5 7⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ ⑦ 31 2 } } ③ ③ ③ ③ ③ ③ ③ ③ 32 } } ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ 2 2 1 " " ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ 1 5 " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ 7

  a standard Levi subgroup L I with I left connected. The set of all pure Levi subgroups of G is denoted by L * , whereas L denotes the set of all N-conjugates of all standard Levi subgroups of G. If L ∈ L * , a pure Levi subgroup of L is an element M ∈ L * with M ≤ L. Notice that the set of N-conjugacy classes in L * is linearly ordered in the following sense. Let L, M ∈ L * . Then |L| < |M| if and only if there is x ∈ N such that x L ≤ M. In particular, |L| = |M| if and only if L and M are conjugate in N.

  and |µ, c is a weight vector for every m ∈ N and µ ⊢ 2 m. Moreover, |(-, -), c is a highest weight vector and U ′ v ( sl e ).|(-, -), c is isomorphic to V (Λ(c)), the simple highest weight module with weight Λ(c) = Λ c 1 mod e + Λ c 2 mod e , where the Λ i , 0 ≤ i ≤ e -1 denote the fundamental weights of sl e . We write F c,e when we view F c as a U ′ v ( sl e )-module. There is a crystal graph G c,e describing the canonical basis of F c,e . The vertices of G c,e are all charged bipartitions |µ, c , µ ⊢ 2 m, m ∈ N. There is a directed, coloured edge |µ, c i → |ν, c if and only if ν is obtained from µ by adding a good i-node, where the colours i are in the range 0 ≤ i ≤ e -1. The associated Kashiwara operator fi acts on G c,e by mapping the vertex |µ, c to |ν, c if and only if there is an edge |µ, c i → |ν, c , and to 0, otherwise (see e.g. [23, 6.1]).

22 ,

 22 Theorem 5.4 and Example 5.6], the irreducible modules of this Iwahori-Hecke algebra are labelled by the set Φ of Uglov bipartitions of m. (See [22, Definition 4.4]; the Uglov bipartitions Φ c e,m are simply the bipartitions of m labelling the vertices of the connected component of the crystal graph G c,e containing |(-, -), c .) By the generalization of [21, Theorem 2.4] to weakly cuspidal modules, the elements of the (L, X λ )-Harish-Chandra series of kG are labelled by these bipartitions (see also Proposition 2.3(a)).

  denotes the set of Uglov bipartitions of m.)

4 ,

 4 Example 5.6]. The elements of this canonical basic set are labelled by the set of Uglov bipartitions of m.

  Put c := (t + (1e)/2, 0). Then the vertex |(-, 1 m ), c of G c,e is a highest weight vertex, if and only if e | 2m + t or e | 2m + t -1.

Thus B ′ 7 . 6 .

 76 is the symbol of |(-, 1 m-s-1 ), (-s ′ , 0) . Now B is totally eperiodic if and only if B ′ is totally e-periodic. By induction, B ′ is totally e-periodic if and only if e | 2m -2s -2 or e | 2m -2s -3 in case s ′ = (e -1)/2, respectively if and only if e | 2m -2s -2 or e | 2m -2s -1 in case s ′ = (e -3)/2. Suppose first that s ′ = (e -1)/2. Then s = (e -3)/2 and thus 2m -2s -2 = 2m + 1e. The claim follows. The other case works analogously. Let |µ, c be a charged bipartition, put B := B(µ, c) and B k := B(µ, c) k for k = 1, 2.

7. 7 .

 7 For a highest weight vertex |µ, c , write B(µ, c) for the connected component of G c,e containing µ. If |ν, d is another highest weight vertex such that wt(ν, d) = wt(µ, c), then [1, Lemma 6.9] implies that U ′ v ( sl e ).|µ, c ∼ = U ′ v ( sl e ).|ν, d ∼ = V (Λ), where V (Λ) is the irreducible highest weight U ′ v ( sl e )-module of highest weight Λ = wt(ν, d) = wt(µ, c). Therefore, we get B(µ, c) ≃ B(ν, d). Moreover, by the characterization (6), the weights of |µ, c and |ν, d coincide if these two charged bipartitions have the same reduced i-word for all 0 ≤ i ≤ e -1.

( 9 ) d 1 - 2 or ( 10 ) d 2 -

 912102 d 2 = s + (1e)/d 1 = s + (1 + e)/2. By Lemma 7.7, we have a crystal isomorphism B(µ, c) ≃ B((-, -), d). If we set d ′ = (d 2e, d 1 ), we also have a crystal isomorphism B(µ, c) ≃ B((-, -), d ′ ) (see [23, 6.2.9, 6.2.17]). By the remark preceding Corollary 7.7, we obtain B(µ, c) ≃ B((-, -), (d 1d 2 , 0)) and B(µ, c) ≃ B((-, -), (d 2ed 1 , 0)) up to an overall shift of the labels of the arrows. Applying Identities (9) respectively (10), we see that s + (1e)/2 equals d 1d 2 in the first case and d 2 -e-d 1 in the second. This concludes our proof.

  sl e ).H.U ′ u ( sl 2 ).|(-, -), c , where A 2 e (s) is the set of elements c ∈ Z 2 such that c 1c 2 ≤ e and c 1 + c 2 = s. In addition, if we fix c, the associated Fock space of level 2 is a U ′ v ( sl e )-submodule of Λ s+∞/2 (that is the actions are compatible). Let i ∈ {0, 1}. Denote by E i and F i the Chevalley operators of U ′ u ( sl 2 ). Regarding the action of E i on the set of charged bipartitions following Uglov's work, we see that |µ, d appears in the expansion of E i .|λ, c if and only if the symbol of |µ, d is obtained from the symbol of |λ, c by one of the two elementary operations (a) and (b) described in 7.3. This thus gives an algebraic interpretation of these transformations on symbols. Moreover, combining this interpretation with the discussion in [47, §4.3] and in particular

Table 2 .

 2 The Harish-Chandra branching graph for GU n (q), 1 ≤ n ≤ 7 odd, ℓ | 2q + 1, ℓ > 7, cont.

	51 2	32 2	52	3 2 1	421	31 4	2 2 1 3	1 7	321 2
				21 6⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥	! ! ❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈
			41 6⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥		21 3 ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ ! !
		61	43		41 3	21 5	2 3 1

Acknowledgements

Above all, we thank Meinolf Geck and Gunter Malle for inspiring this work in many ways and for their constant interest. We also thank Klaus Lux and Pham Huu Tiep for inviting the second author to the University of Arizona. During his visit parts of this paper were written. We thank Götz Pfeiffer for helpful discussions regarding Lemma 2.2, and Joseph Chuang and Matthew Fayers for independently suggesting the result of Corollary 7.9. We are also grateful to the two referees for carefully reading our manuscript, suggesting various improvements and preventing us from a few slips.

Finally, we very much thank the Isaac Newton Institute for Mathematical Sciences for inviting two of us to the special programme Algebraic Lie Theory (12 January to 26 June 2009), where the ground for the conjectures in this paper was laid.

A (respectively R) to encode the addable (respectively removable) node of content j lying in component k of B(µ, c). Note that the contents of the addable and removable nodes of a bipartition are the elements j -1 and j, respectively, for j in the corresponding symbol (provided j encodes a non-zero part). In fact, a removable node of content j -1 corresponds to an element j ∈ B(µ, c) k such that j -1 ∈ B(µ, c) k , and an addable node of content j corresponds to an element j ∈ B(µ, c) k such that j + 1 ∈ B(µ, c) k . Therefore, since an elementary operation affects either just one element j or just j and je, the only differences that can occur between w i (µ, c) and w i (µ ′ , c ′ ) are with letters A and R corresponding to nodes of content j -1, j, je -1 and je. We review the only possible changes by enumerating the cases.

Suppose first that we apply the elementary operation (a ′ ), that is to say we move j from row 1 of B(µ, c) to row 2. Moreover, j is the largest element in B(µ, c) 1 for which j ∈ B(µ, c) 2 . Denote by l the largest element of B(µ, c) 2 . To begin with, assume that j is the largest element of B(µ, c) 1 .

If j > l, then j is the first element of its period, and thus j -1 ∈ B(µ, c) 1 . Moreover, either

• j > l + 1, in which case the elementary operation takes A 1 (j) to A 2 (j) and creates an occurence of R 2 (j -1)A 1 (j -1), which cancels in the reduced i-word (for i = j -1 mod e), or • j = l + 1, in which case A 1 (j) in B(µ, c) becomes A 2 (j) in B(µ ′ , c ′ ), and A 2 (j -1) becomes A 1 (j -1). If j < l, the following possibilities arise.

• If j +1 / ∈ B(µ, c) 2 , then again j is the first element of its period, and thus j -1 ∈ B(µ, c) 1 . Moreover, either * j -1 / ∈ B(µ, c) 2 , and A 1 (j) becomes A 2 (j) and R 2 (j -1)A 1 (j -1) appears, or * j -1 ∈ B(µ, c) 2 , and A 2 (j -1) becomes A 1 (j -1) and A 1 (j) becomes A 2 (j).

in which case

A 2 (j -1) becomes A 1 (j -1) and R 2 (j)A 1 (j) vanishes, or * j -1 ∈ B(µ, c) 1 and j -1 / ∈ B(µ, c) 2 , in which case R 1 (j -1) becomes R 2 (j -1) and R 2 (j)A 1 (j) vanishes, or * j -1 ∈ B(µ, c) 1 and j -1 ∈ B(µ, c) 2 , in which case j is the last element in its period; if m ≥ j + 1 is the smallest element of B(µ, c) 2 with m + 1 ∈ B(µ, c) 2 , then m and Table 3. The crystal graph G ≤3 (0,0),3

-.-