Decomposition numbers for the principal Φ_{2n}-block of \mathrm{Sp}_{4n}(q) and \mathrm{SO}_{4n+1}(q)
aa r X i v : . [ m a t h . R T ] F e b DECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) OLIVIER DUDAS AND EMILY NORTON
Abstract.
We compute the decomposition numbers of the unipotent characters lying inthe principal ℓ -block of a finite group of Lie type B n ( q ) or C n ( q ) when q is an odd primepower and ℓ is an odd prime number such that the order of q mod ℓ is 2 n . Along the way, weextend to these finite groups the results of [12] on the branching graph for Harish-Chandrainduction and restriction. Introduction
The representation theory of a finite group of Lie type G := G ( F q ) over a field of positivecharacteristic ℓ coprime to q has a close relationship to the representation theory of the Heckealgebra of its Weyl group. The decomposition matrix of the Hecke algebra always embeds asa submatrix of the decomposition matrix of G . When G = GL n ( q ) is the finite general lineargroup, the square unitriangular submatrix of the decomposition matrix of the unipotent blocksis the same as the decomposition matrix of the q -Schur algebra, a quasihereditary cover of theHecke algebra of the symmetric group S n . This is related to the fact that in characteristic0, GL n ( q ) has exactly one cuspidal irreducible unipotent representation as n ranges over N ,namely, the trivial representation of GL ( q ). When G is not of type A , less is understood aboutthe decomposition matrix of the unipotent blocks of G . There are more cuspidal unipotentrepresentations in characteristic 0 which give rise to multiple Hecke and quasihereditary alge-bras, all of which play a role in the unipotent blocks of G . However, in the general case, theknowledge of the decomposition numbers for these algebras is not enough to determine thoseof G . In [8] the first author initiated the use of Deligne–Lusztig characters to find the missingnumbers. This proved successful in determining decomposition matrices for finite groups of Lietype in small rank, see [27], [9], [10], [11].In this paper, we are concerned with groups of type B m and C m such as G = SO m +1 ( q )and G = Sp m ( q ) for odd q . If n is the order of q in F × ℓ , the complexity of the decompositionmatrix grows with m/n . For that reason we will consider the case where m = 2 n , which issomehow the simplest case outside of the cyclic defect case. Two situations arise: • (linear prime case) q has order n in F × ℓ , in which case n is necessarily odd, and therepresentation theory of G behaves as a type A phenomenon and can be deduced fromthe representation theory of q -Schur algebras of symmetric groups [26]; • (unitary prime case) q has order 2 n in F × ℓ . Explicit decomposition matrices in thatcase were obtained by Okuyama–Waki for m = 2 [32] and by Malle and the first authorfor m = 4 [10] and m = 6 [11].Our main result provides a generalisation of the unitary prime case to any even m . To ourknowledge it is the first general result for defect 2 blocks of finite groups of Lie type outside oftype A phenomena.Let Φ d ( q ) be the d -th cyclotomic polynomial evaluated at q . Main Theorem. (Theorems 2.2, 2.3, 2.4, 2.7, 2.8, and 2.10)
Let G be a finite group of Lie typeover F q of type B n or C n for q an odd prime power. Let ℓ be an odd prime number such thatthe order of q in F × ℓ is n . Then all but two decomposition numbers of the unipotent charactersin the principal ℓ -block of G are known. If Φ n ( q ) ℓ > n then both these numbers are and thedecomposition numbers are completely known. Under our assumptions on ℓ and q , studying the unipotent characters in the principal ℓ -block is a reasonable restriction. First, any other unipotent ℓ -block has defect 0 or 1, andits decomposition matrix is known by [20]. Second, the decomposition numbers of the non-unipotent characters in the principal ℓ -block of G may be recovered from those of the unipotentcharacters and from partial knowledge of the character table of G by [22, 21].The methods used to obtain the decomposition matrices are two-fold:(1) First, we use Harish-Chandra induction and restriction to produce projective indecom-posable modules (PIMs);(2) Second, we compute the missing PIMs (corresponding to cuspidal simple modules) usingsome partial information on the decomposition of Deligne–Lusztig characters on PIMs.For both of these steps, we use a truncated version of the Harish-Chandra induction and restric-tion coming from the categorical b sl n -action on unipotent representations defined in [13]. Therecent unitriangularity result in [4] allows us to compute the branching graph for this truncatedinduction, which provides the missing information for step (2) to be successful. Acknowledgments.
We thank Gunter Malle and Raph¨ael Rouquier for helpful conversations,and Gunter Malle for perspicacious comments on a draft of this paper. O. Dudas gratefully ac-knowledges financial support by the ANR, Project No ANR-16-CE40-0010-01E and by the grantSFB-TRR 195. E. Norton was supported by the grant SFB-TRR 195. E. Norton also thanksthe workshop “Categorification in quantum topology and beyond” at the Erwin Schr¨odingerInstitut, Vienna, January 2019.1.
Representation theory for types B and C Combinatorics.
Partitions and symbols.
Let m be a non-negative integer. A partition λ of m is a non-increasing sequence of non-negative integers λ ≥ λ ≥ · · · ≥ m . We call m the size of λ and we denote it by | λ | . Let s ∈ Z and λ be a partition of m . The charged β -setof λ is the set β s ( λ ) := { λ + s, λ + s − , . . . , λ i + s − i + 1 , . . . } . It is a subset of Z which contains all z ∈ Z such that z ≤ s − { non-zero parts of λ } .A bipartition λ = λ .λ of m consists of a pair ( λ , λ ) of partitions such that | λ | := | λ | + | λ | equals m . If λ (resp. λ ) is the empty partition we will write λ = λ . (resp. λ = .λ ). Given s = ( s , s ) in Z , a charged symbol with charge s is a pair Λ = ( X, Y ) where X = β s ( λ )and Y = β s ( λ ) for some bipartition λ = λ .λ . The defect of the charged symbol Λ is D = s − s . We set Λ † = ( Y, X ). It is a symbol with charge s ∗ := ( s , s ) and it is associatedto the bipartition λ .λ . Note that the defect of a symbol should not be confused with the otheruse of the word “defect” arising in representation theory of finite groups, namely the defect ofa block.Throughout this paper we shall only be working with charged symbols with odd defect andwith a specific charge. Given t ∈ Z , let us define(1.1) σ t := (cid:26) ( t, − − t ) if t is even,( − − t, t ) if t is odd . ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 3 Note that σ − − t = σ t . A symbol Λ = (
X, Y ) is a charged symbol with charge σ t for some t ∈ Z . If X = { x , x , x , . . . } and Y = { y , y , y , . . . } , we will represent Λ byΛ = (cid:18) x x x . . .y y y . . . (cid:19) . This convention differs from the usual convention, for example the one in
Chevie [31] since weallow both positive and negative defect, and since a symbol in
Chevie is necessarily truncated onthe right whereas our symbols are infinite to the right. Nevertheless this will be needed to havea consistent action of the i -induction operators on all symbols from the various Harish-Chandraseries, and to allow n to grow arbitrarily large, see § Remark 1.2.
The -row convention for representing the charged symbol Λ of a bipartition λ with charge s is the ◦ -rotation of the -abacus of λ with charge s as in [24] . We will sometimes find it useful to drop the notation of symbols and work with Youngdiagrams. The Young diagram of the bipartition λ is the set of triples Y ( λ ) := { ( x, y, j ) ∈ N × N × { , } | ≤ x ≤ { nonzero parts of λ j } , ≤ y ≤ λ jx } . An element b = ( x, y, j ) ∈ Y ( λ ) is called a box of the Young diagram. We will draw the Youngdiagram of a bipartition λ by putting the diagrams of λ and λ side by side. For a box ( x, y, j )in Y ( λ ), x represents the row and y the column, with the convention that rows are decreasingin length from top to bottom, as illustrated below for the example λ = 543 . Y (543 .
21) = · Cores and co-cores.
Let d be a positive integer and let Λ = ( X, Y ) be a symbol. A d -hookin the top row (resp. in the bottom row) of Λ is a pair ( x, x + d ) such that x + d ∈ X and x / ∈ X (resp. x + d ∈ Y and x / ∈ Y ). Removing a d -hook in the top row amounts to changing Λ to(( X r { x + d } ) ∪ { x } , Y ), and similarly for the bottom row. The d -core is the symbol obtainedby recursively removing all possible d -hooks. Removing or adding d -hooks does not change thedefect of the symbol.A d -co-hook of Λ is a pair ( x, x + d ) such that x + d ∈ X and x / ∈ Y or x + d ∈ Y and x / ∈ X .The co-hook is removed from Λ by removing x + d from X and adding x to Y , or removing x + d from Y and adding x to X , and then exchanging X and Y . Recursively removing all d -co-hooks yields the d -co-core of Λ.1.1.3. Families.
Let Λ = (
X, Y ) be a symbol. The composition ̟ Λ attached to the symbolis the non-increasing sequence ̟ Λ := ( ̟ ≥ ̟ ≥ ̟ ≥ · · · ) obtained by considering theunion of X and Y as a multiset. Since X and Y are β -sets of some partitions, any term in thecomposition ̟ Λ occurs at most twice (and all but finitely many terms appearing do).The dominance order on compositions defines a relation on symbols. We say that two symbolsΛ and Λ ′ lie in the same family and we write Λ ≡ Λ ′ if ̟ Λ = ̟ Λ ′ . In other words, two symbolsare in the same family if their multisets of entries are the same. We write Λ ⊳ Λ ′ and we saythat Λ ′ dominates Λ if ̟ Λ ⊳ ̟ Λ ′ , by which we mean ̟ Λ = ̟ Λ ′ and P ji =1 ̟ i ≤ P ji =1 ̟ ′ i for all j ≥
1. This defines a strict partial order on the set of symbols. We will write Λ E Λ ′ if Λ = Λ ′ or Λ ⊳ Λ ′ . Example 1.3.
The following four symbols (cid:18) − · · ·− − · · · (cid:19) , (cid:18) − − · · · − · · · (cid:19) , (cid:18) − − · · · − · · · (cid:19) , (cid:18) − · · · − − · · · (cid:19) OLIVIER DUDAS AND EMILY NORTON form a family attached to the composition (1 , , − , − , − , . . . ) . The first three symbols havecharge (0 , − and correspond to the bipartitions . , . and . , whereas the fourth symbolhas charge ( − , and corresponds to the empty bipartition. Note that we have only includedsymbols that have charge σ t for some t ∈ Z . Unipotent representations of finite reductive groups of type B and C . Representations of finite groups.
Let G be any finite group and Λ a commutative ringwith unit. We denote by Λ G -mod the abelian category of finitely generated left Λ G -modules.The set of isomorphism classes of irreducible (or simple) objects will be denoted by Irr Λ G . Wewill write K ( Λ G -mod ) for the Grothendieck group of the category Λ G -mod .Let ℓ be a prime number. We shall work with representations over fields of characteristiczero and ℓ . For that purpose we fix an ℓ -modular system ( K , O , k ) where K is an extension of Q ℓ ,the ring of integers O of K over Z ℓ is a complete d.v.r and its residue field k has characteristic ℓ .Throughout this paper we will assume that this modular system is sufficiently large, so thatthe algebras K G and k G split for any finite group G considered, that is, so that all irreduciblerepresentations of G over K (resp. k ) remain irreducible over any field extension of K (resp. k ).We will usually identify K ( K G -mod ) with the space of virtual characters of G , and its basisIrr K G by the set of (ordinary) irreducible characters. We will denote by h− ; −i G the usual innerproduct on K ( K G -mod ).1.2.2. Finite reductive groups and Deligne–Lusztig characters.
We fix a non-negative integer m .Let G be a connected reductive group, quasi-simple of type B m or C m , defined over the finitefield F q . Let F : G −→ G be the corresponding Frobenius endomorphism. The finite group G := G F is a finite reductive group . If H is any closed subgroup of G we will denote by H := H F the corresponding finite group. We fix an F -stable maximal torus T of G contained in an F -stable Borel subgroup B of G . We denote by W m := N G ( T ) / T the corresponding Weyl group,which is of type B m . The choice of B defines a subset of simple reflections S = { s , s , . . . , s m } on which F acts trivially. They are labeled according to the following Coxeter diagram: s s s s m − s m The G -conjugacy classes of F -stable maximal tori are parametrized by the conjugacy classesof W m . Given w ∈ W m we will denote by T w a maximal torus of type w . Given θ a K -linearcharacter of T w , Deligne–Lusztig defined in [6] a virtual character R GT w ( θ ) of G over K . We willwrite R w := R GT w (1 T w ) for the Deligne–Lusztig character associated to the trivial character of T w . The irreducible constituents of the various R w ’s are the unipotent characters of G .1.2.3. Harish-Chandra induction and restriction.
Given I ⊆ S , we write W I for the subgroupof W generated by I and P I = B W I B for the corresponding standard parabolic subgroup of G . It has a Levi decomposition P I = L I ⋉ U I where L I is the unique Levi complement of P I containing T . The Harish-Chandra induction and restriction functors are defined by R GL I := Λ G/U I ⊗ Λ L I − and ∗ R GL I := Hom G ( Λ G/U I , − )where Λ is any of the rings K , O , k . A Λ G -module V is said to be cuspidal if ∗ R GL I ( V ) = 0 forall I ( S . When ℓ ∤ q , the functors ( ∗ R GL I , R GL I ) form a biadjoint pair of exact functors between Λ L I -mod and Λ G -mod . ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 5 When Λ = K these functors yield linear maps on characters of G which we will still denoteby R GL I and ∗ R GL I . When I = ∅ we have that L I = T is a split torus. In that case the Harish-Chandra induction and the Deligne–Lusztig map R GT defined above coincide, which justifies ournotation.1.2.4. Unipotent characters.
We recall here Lusztig’s parametrization of unipotent charactersof G (see for example [30, § G is a quasi-simple group of type B m or C m . Thefinite group G admits a cuspidal unipotent character if and only if m = t + t for some t ≥ σ t ,see (1.1), which is given byΛ = (cid:18) t t − . . . . . . . . . − − t − − t . . . (cid:19) if t is even , (cid:18) − − t − − t . . .t t − . . . . . . . . . (cid:19) if t is odd . More generally, if t + t ≤ m , one can consider the standard Levi subgroup L of G of type B t + t or C t + t . Then the unipotent characters of G lying in the Harish–Chandra series of thecuspidal unipotent character of L correspond to bipartitions λ = λ .λ of size r = m − t − t .In that case will we write (cid:2) λ .λ (cid:3) B t t for the corresponding unipotent character, or (cid:2) Λ (cid:3) if Λis the symbol of charge σ t attached to λ . With our convention, the character with smallestdegree in the series is [ r. ] B t t when t is even but [ .r ] B t t when t is odd. It agrees with theconvention in Chevie [31] when t is even but when t is odd, the components of the bipartitionmust be swapped.The decomposition of the Deligne–Lusztig characters R w in terms of symbols was determinedby Lusztig. Given an irreducible character χ of W m over K we can form the almost character R χ := 1 | W m | X w ∈ W m χ ( w ) R w . Then using [30, Thm. 4.23] one can compute the multiplicity (cid:10) R χ ; (cid:2) Λ (cid:3)(cid:11) G of the unipotentcharacter [Λ] for any symbol Λ. Two examples of computations of Deligne–Lusztig charactersare given in the appendix.1.2.5. Unipotent ℓ -blocks. By an ℓ -block B of G we mean a minimal 2-sided ideal of the groupalgebra O G . We have B = O Gb for a unique primitive central idempotent b of O G . Wewill write Irr K B for the ordinary irreducible characters lying in B , that is, those irreduciblecharacters χ ∈ Irr K G such that χ ( b ) = 0. An ℓ -block B is unipotent if it contains at leastone unipotent character. In particular, the principal ℓ -block, which is the block containing thetrivial character, is unipotent. We will denote by O G -umod the category of representations overthe sum of the unipotent ℓ -blocks of G .Assume now that ℓ and q are odd. The ℓ -blocks of G were classified by Fong–Srinivasan in[19]. There are two situations, depending on whether ℓ is “linear” or “unitary”. Let d be themultiplicative order of q in F × ℓ . • If d is odd, ℓ is said to be a linear prime for G . In that case two unipotent characters (cid:2) Λ (cid:3) and (cid:2) Λ ′ (cid:3) lie in the same block if and only if the symbols Λ and Λ ′ have the same d -core. The number of d -hooks that must be removed to reach the d -core is called theweight of the block. OLIVIER DUDAS AND EMILY NORTON • If d is even, ℓ is said to be a unitary prime for G . Set e := d/
2, the order of q in F × ℓ . Then two unipotent characters (cid:2) Λ (cid:3) and (cid:2) Λ ′ (cid:3) lie in the same block if and only ifthe symbols Λ and Λ ′ have the same e -co-core. The number of e -co-hooks that mustbe removed to reach the e -co-core is called the weight of the block.We will often refer to an ℓ -block B as a Φ d -block, where Φ d stands for the d -th cyclotomicpolynomial. This is justified by the fact that many of the properties of B depend only on d rather than on ℓ , see [2, Thm. 5.24]. For example, if ℓ > d then any defect group of B isisomorphic to ( Z / Φ d ( q ) ℓ ) r where r is the weight of the block.1.2.6. Decomposition matrix.
Recall that ( K , O , k ) is an ℓ -modular system which is sufficientlylarge for G . Then every K G -module admits an integral form over O , which can then be reducedmodulo ℓ to a k G -module. The image of that module in the Grothendieck group does notdepend on the choice of the integral form and we obtain a linear map dec : K ( K G -mod ) −→ K ( k G -mod )called the decomposition map. The decomposition matrix is the matrix of this map in thebases Irr K G and Irr k G . It respects the block decomposition so that we can talk about thedecomposition matrix of an ℓ -block. Dually, every projective k G -module P lifts to a uniqueprojective O G -module e P , up to isomorphism. By the character of P we mean the characterof the K G -module K e P . Brauer reciprocity states that the decomposition matrix of G is alsothe matrix whose columns are the characters of the PIMs (the projective indecomposable k G -modules) in the basis Irr K G .In this paper we shall only be interested in the decomposition matrix of unipotent ℓ -blocks. Itis a reasonable restriction since any block is conjecturally Morita equivalent to some unipotentblock [1]. This was proved for a large class of non-unipotent blocks in [3]. We say that a PIM isunipotent if it belongs to a unipotent ℓ -block. When ℓ is odd, the unipotent characters form abasic set of the unipotent ℓ -blocks [22, 21]. If in addition q is odd, this basic set is unitriangularwith respect to the order on families [4] (see [33, §
4] for the description of the order in terms ofsymbols). This means that there is a labeling of the unipotent PIMs by symbols such that (cid:10) K e P Λ ; [Λ ′ ] (cid:11) G = (cid:26) ′ , ′ Λ . To avoid cumbersome notation we will denote by Ψ [Λ] the unipotent part of the character ofthe PIM corresponding to the unipotent character (cid:2) Λ (cid:3) by unitriangularity.Similarly, we say that a simple k G -module is unipotent if it belongs to a unipotent ℓ -block.The unitriangularity of the decomposition matrix gives a natural labeling of the unipotentsimple k G -modules by unipotent characters. If (cid:2) Λ (cid:3) is a unipotent character, we will denote by S [Λ] the corresponding simple k G -module.1.3. Branching rules.
We recall and complete in this section the main result in [13, §
6] onthe branching rules for Harish-Chandra induction and restriction for unipotent representationsof groups of type B and C . Throughout this section we will assume that ℓ and q are odd, andthat d , the multiplicative order of q in F × ℓ , is even. In particular we have d > ℓ is odd, the main result of this section, Theorem 1.12, remains valid for any group of type B or C . ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 7 Level 2 Fock spaces.
Let λ = λ .λ be a bipartition, and s ∈ Z . The charged content ofa box b = ( x, y, j ) in the Young diagram Y ( λ ) is co s ( b ) = y − x + s j . Given another bipartition µ and c ∈ Z , we write µ r λ = c if there exists a box b of µ withcharged content co s ( b ) = c such that the Young diagram of λ is obtained from the Youngdiagram of µ by removing the box b .Let { e i , f i } i ∈ ,...,d − be the Chevalley generators of the affine Lie algebra b sl d . The Fock space F ( s ) with charge s is the b sl d -module equipped with a C -basis | λ , s i labeled by bipartitions onwhich the Chevalley generators act by e i | λ , s i = X j ≡ i mod d λ r µ = j | µ , s i and f i | λ , s i = X j ≡ i mod d µ r λ = j | µ , s i . Note that the action of b sl d depends only on the class of s and s in Z /d .1.3.2. Order on bipartitions.
We consider here an order on bipartitions defined by Dunkl–Griffeth in [14, § λ be a bipartition and let Y ( λ ) be its Young diagram. If b = ( x, y, j )is a box in Y ( λ ) we write j ( b ) := j . If µ is another bipartition and s ∈ Z we write λ (cid:22) s µ iffor all α ∈ R and j = 1 , { b ∈ Y ( λ ) | co s ( b ) − j ( b ) d > α or co s ( b ) − j ( b ) d α and j ( b ) ≤ j }≤ { b ∈ Y ( µ ) | co s ( b ) − j ( b ) d > α or co s ( b ) − j ( b ) d α and j ( b ) ≤ j } . This is exactly the order ≤ c defined in [14] with r = 2, c = d − and d = − d = ( s − s ) /d +1 /
2. Note that changing s to s + ( s, s ) for any s ∈ Z does not change the order. We will needthe following lemma which relates the Dunkl–Griffeth order on bipartitions to the dominanceorder on charged symbols defined in § Lemma 1.4.
Let σ ∈ Z and set s := σ + (0 , d ) . Let λ , µ be two bipartitions, and let Λ , Λ ′ be the corresponding symbols of charge σ . Then λ (cid:22) s µ = ⇒ Λ ⊳ Λ ′ or Λ ≡ Λ ′ . Proof.
First observe that given a box b in the Young diagram of λ we have co s ( b ) − j ( b ) d co σ ( b ) − d . We deduce that λ (cid:22) s µ if and only if for all α ∈ R and j = 1 , { b ∈ Y ( λ ) | co σ ( b ) > α or co σ ( b ) = α and j ( b ) ≤ j }≤ { b ∈ Y ( µ ) | co σ ( b ) > α or co σ ( b ) = α and j ( b ) ≤ j } . Let us consider the extended Young diagram e Y ( λ ), defined as the set of boxes b = ( x, y, j )with j = 1 , x ≥ y ≤ λ jx . Unlike the usual Young diagram we do not assume y ≥ e Y ( λ ) r Y ( λ )does not depend on the bipartition λ , and the number of boxes with a given content is finite,therefore one can replace Y ( λ ) and Y ( µ ) by e Y ( λ ) and e Y ( µ ) in (1.5).Working with extended Young diagrams makes the computations easier in (1.5). Indeed, if ̟ Λ := ( ̟ ≥ ̟ ≥ ̟ ≥ · · · ) is the composition attached to Λ, that is, the multiset given by OLIVIER DUDAS AND EMILY NORTON the union of β σ ( λ ) and β σ ( λ ) (see § { b ∈ e Y ( λ ) | co σ ( b ) ≥ α } = X ̟ k ≥ α ( ̟ k − ⌈ α ⌉ ) . To show the claim we can assume without loss of generality that α ∈ Z since the contents areintegers. Each row in e Y ( λ ) corresponds to an element ̟ k , and the highest content in that rowequals ̟ k −
1. Consequently, this row contains a box of content β (and only one) if and onlyif ̟ k − ≥ β . Therefore { b ∈ e Y ( λ ) | co σ ( b ) ≥ α } = X β ≥ α { b ∈ e Y ( λ ) | co σ ( b ) = β } = X β ≥ α { k ≥ | ̟ k − ≥ β } = X k,β̟ k − ≥ β ≥ α X k X β̟ k − ≥ β ≥ α X k̟ k − ≥ α ( ̟ k − α ) . Let ̟ Λ ′ = ( ̟ ′ ≥ ̟ ′ ≥ ̟ ′ ≥ · · · ) be the composition attached to the symbol Λ ′ . We deducefrom (1.5) with j = 2 that for all α ∈ R (1.6) X ̟ k ≥ α ( ̟ k − ⌈ α ⌉ ) ≤ X ̟ ′ k ≥ α ( ̟ ′ k − ⌈ α ⌉ ) . Now, if i ≥ i X k =1 ( ̟ k − ̟ ′ i ) ≤ X ̟ k ≥ ̟ ′ i ( ̟ k − ̟ ′ i ) ≤ X ̟ ′ k ≥ ̟ ′ i ( ̟ ′ k − ̟ ′ i ) = i X k =1 ( ̟ ′ k − ̟ ′ i )where we used (1.6) with α = ̟ ′ i for the second inequality. This shows that P ik =1 ̟ k ≤ P ik =1 ̟ ′ k for all i ≥ (cid:3) Categorification of unipotent representations.
We recall here the categorification resultof [13, § m ≥ G m := Sp m ( q ). Using the (unique) standard Levi subgroup G m − × GL ( q ) of G m we can form the chain of subgroups { } = G ⊂ G ⊂ · · · ⊂ G m − ⊂ G m ⊂ · · · . Since d , the multiplicative order of q in F × ℓ is even, the group GL ( q ) is an ℓ ′ -groupand the Harish-Chandra induction and restriction induce exact functors between k G m − -umod and k G m -umod for all m . We can form the abelian category k G • -umod := M m ≥ k G m -umod of the modules over all the unipotent ℓ -blocks of the various groups G m . We will denote by F and E the endofunctors of this category induced by Harish-Chandra induction and restrictionrespectively.Since ℓ is odd, the unipotent characters form a basic set for the unipotent blocks [22, 21]. Inparticular K ( k G • -umod ) has a Z -basis given by the image of the unipotent characters underthe decomposition map, see § § B t + t -series are labeled by bipartitions, or equivalently by symbols of charge σ t . Therefore they are ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 9 in bijection with the standard basis of any Fock space F ( s ). For our purpose we will considerthe charges s t defined by(1.7) s t := σ t + 12 (0 , d ) = ( ( t, − − t + d ) if t is even,( − − t, t + d ) if t is odd , for all t ≥
0. The previous discussion shows that there is an isomorphism of C -vector spaces(1.8) M t ≥ F ( s t ) ∼ −→ C ⊗ Z K ( k G • -umod ) | λ , s t i 7−→ dec (cid:2) λ (cid:3) B t t which sends an element of the standard basis to the image under the decomposition map of thecorresponding unipotent character. One of the main results in [13, §
6] is a categorification of(1.8). It gives, for every i = 0 , . . . , d − F i , E i ) of k G • -umod such that F = d − M i =0 F i and E = d − M i =0 E i called i -induction and i -restriction functors , which induce an action of b sl d on K ( k G • -umod )making (1.8) an isomorphism of b sl d -modules. Remark 1.9. In [13, (6.3)] the authors used the charge s ∗ t instead of s t when t is odd. This doesnot affect the categorification result since the Fock spaces F ( s ∗ t ) and F ( s t ) are clearly isomorphic,but it explains the discrepancy in our notation for unipotent characters with the one in Chevie .With our convention, under the isomorphism (1.8) , the action of f i on a symbol Λ = (cid:18) x x x . . .y y y . . . (cid:19) is given by increasing by any x j equal to i modulo d or any y j equal to i + d/ modulo d , whenpossible. Crystal graph and branching rules.
Fix a charge s = ( s , s ) ∈ Z , let d ∈ Z ≥ (herewe do not require d to be even), and consider the Fock space F ( s ). Let | λ , s i be a chargedbipartition in F ( s ). A box b of λ is removable if µ := λ \ b is a bipartition. Then b is called an addable box of the bipartition µ . For each i ∈ Z /d Z , define the i -word of | λ , s i as follows: listall the addable and removable boxes b of | λ , s i such that co s ( b ) ≡ i mod d in increasing orderfrom left to right according to their value in Z , with the convention that if co s ( b ) = co s ( b ′ )and b ∈ λ , b ′ ∈ λ , then b ′ is smaller than b . Now replace each addable box in the list bythe symbol + and each removable box in the list by the symbol − . The resulting string ofpluses and minuses is called the i -word of | λ , s i . The reduced i -word of | λ , s i is then found fromthe i -word by recursively canceling all adjacent pairs ( − +). The reduced i -word is of the form(+) a ( − ) b for some a, b ∈ Z ≥ . The Kashiwara operator ˜ f i adds the addable i -box correspondingto the rightmost + in the reduced i -word of | λ , s i , or if there is no + in the reduced i -word thenit acts by 0. Likewise, the Kashiwara operator ˜ e i removes the removable i -box correspondingto the leftmost − in the reduced i -word of | λ , s i , or if there is no − in the reduced i -word thenit acts by 0. The directed graph with vertices all bipartitions and ( Z /d Z -colored) edges λ i → µ if and only if µ = ˜ f i ( λ ), i ∈ Z /d Z , is called the b sl d -crystal on F ( s ) [28, Section 3], [17, Theorem2.8]. Example 1.10.
Let λ = 421 . , s = (1 , , and d = 4 . | λ , s i = 1 2 3 40 1 - - - - · - - - - - - - - - - Let us find the -word of | λ , s i . The addable -boxes ( x, y, j ) , where x is the row, y is thecolumn, and j is the component, are: (9 , , , (6 , , , (3 , , , (1 , , . The removable -boxesare: (6 , , , (3 , , , (1 , , . Ordering them we obtain the -word: (9 , ,
2) (6 , ,
2) (6 , ,
1) (3 , ,
2) (3 , ,
1) (1 , ,
2) (1 , , − − + + − Iterating cancellations of all adjacent ( − +) , we obtain the reduced -word: (9 , ,
2) (6 , ,
2) (1 , , − Thus ˜ f adds the box (6 , , to | λ , s i , and ˜ e removes the box (1 , , from | λ , s i . That is, ˜ f ( λ ) = 421 . and ˜ e ( λ ) = 321 . . We will often work with symbols instead of Young diagrams, so it is useful to describe howthe operators ˜ f i and ˜ e i act on symbols. Let [ λ ] B t t be a bipartition in the B t + t series, andlet Λ = ( X, Y ) be its symbol with charge σ t . Recall that this does not depend on d . Instead ofchanging the symbol to depend on d , we define the action so that it depends on d .Let d ∈ N with d ≥
2. An addable i -box of λ is x ∈ X such that x + 1 / ∈ X and x ≡ i mod d . A removable i -box of λ is x / ∈ X such that x + 1 ∈ X and x ≡ i mod d . The nextcondition records the dependence on d . An addable i -box of λ is y ∈ Y such that y + 1 / ∈ Y and y + d ≡ i mod d . A removable i -box of λ is y / ∈ Y such that y + 1 ∈ Y and y + d ≡ i mod d . We then order all addable and removable i -boxes of λ by the following rule: • x is less than x if and only if x < x in Z , • y is less than y if and only if y < y in Z , • x is less than y if and only if y + d > x in Z , • y is less than x if and only if x ≥ y + d in Z ,for all addable and removable boxes x, x , x of λ and y, y , y of λ . We then form the i -word of λ by listing its addable and removable i -boxes in increasing order as just defined. Thecancellation rule is the same as before, and we obtain the good addable i -box and the goodremovable i -box of λ (if they exist) as described before. Example 1.11.
Take λ = 31 . and consider it as a bipartition in the principal series, so that t = 0 . We have σ = (0 , − and thus the symbol associated to [ λ ] is Λ = (cid:18) XY (cid:19) = (cid:18) − − − . . . − − − . . . (cid:19) Now suppose that d = 6 . Let us calculate the action of ˜ f and ˜ e on Λ . The addable -boxes of λ are given by , − ∈ X , and the removable -boxes of λ are given by / ∈ Y . Ordering them ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 11 according to our rule, the -word of λ is: + − + − ∈ X / ∈ Y ∈ X We then cancel the occurrence of ( − +) , yielding + as the reduced -word and − ∈ X as thegood addable -box. There is no good removable -box, so ˜ e Λ = 0 . Adding the good addable -box to λ yields: ˜ f Λ = (cid:18) − − − . . . − − − . . . (cid:19) . Now we recall how the combinatorics of crystals relates to the representation theory ofunipotent blocks. We can define the colored branching graph whose vertices are labeled by theunipotent simple k G m -modules for all m ≥ T i −→ T ′ if T ′ appears inthe head of F i ( T ), or equivalently if T appears in the socle of E i ( T ′ ). Theorem 1.12.
The map | λ , s t i 7→ S [ λ ] Bt t induces an isomorphism between the union of the crystal graphs of F ( s t ) for t ≥ and thecolored branching graph of Harish-Chandra induction and restriction.Proof. The proof follows the arguments given in the proof of [12, Thm. 4.37]. Let t ≥
0. Thereis a perfect basis of the Fock space F ( s t ) coming from an upper global basis of a quantumdeformation of F ( s t ) defined by Uglov in [35]. We denote this basis by b ∨ ( | λ , s t i ). It isunitriangular in the standard basis, with respect to the order (cid:22) s t defined in § b ∨ ( | λ , s t i ) ∈ | λ , s t i + X λ ≺ s t µ C | µ , s t i . On the other hand, since the decomposition matrix is unitriangular, we have, for every unipotentcharacter (cid:2) Λ (cid:3) attached to a symbol Λ(1.14) dec (cid:0)(cid:2) Λ (cid:3)(cid:1) ∈ S [Λ] + X Λ ⊳ Λ ′ Z S [Λ ′ ] in K ( k G • -umod ). Now let ψ be the isomorphism in (1.8) sending | λ , s t i to dec (cid:0)(cid:2) Λ (cid:3)(cid:1) where Λis the symbol of charge σ t attached to λ . Lemma 1.4 tells us that for all µ such that λ ≺ s t µ ,it holds that Λ ⊳ Λ ′ or Λ ≡ Λ ′ , where Λ ′ is the symbol of charge σ t attached to µ . Since ψ ( | µ , s t i ) ∈ S [Λ ′ ] + P Λ ′ ⊳ Λ ′′ C S [Λ ′′ ] by (1.14) we deduce that ψ ( | µ , s t i ) ∈ S [Λ ′ ] + X Λ ⊳ Λ ′′ C S [Λ ′′ ] . This together with (1.13) gives(1.15) ψ (cid:0) b ∨ ( | λ , s t i ) (cid:1) ∈ S [Λ] + X Λ ≺ Λ ′ C S [Λ ′ ] where Λ ≺ Λ ′ is the transitive closure of the relationΛ ⊳ Λ ′ , or Λ ′ ↔ µ with λ ≺ s t µ . If we define ϕ to be the bijection that sends S [Λ] to ψ (cid:0) b ∨ ( | λ , s t i ) (cid:1) , two perfect bases of L t ≥ F ( s t ), then (1.15) shows that ϕ satifies the assumptions of [12, Prop. 1.14] with respectto the order ≺ and therefore induces a crystal isomorphism. (cid:3) The decomposition matrix of the principal Φ n -block of Sp n ( q ) and SO n +1 ( q )We fix an integer n ≥
0. Throughout this section G = G n will denote one of the finite groupsSO n +1 ( q ) or Sp n ( q ) for q a power of an odd prime. We are interested in the decompositionmatrix of the principal ℓ -block of G when the order of q in F × ℓ equals 2 n . We will often referto this block as the principal Φ n -block of G (see § Unipotent characters in the principal Φ n -block. Let B be the principal Φ n -blockof G and b be the corresponding block idempotent. By § B are labeled by symbols of rank 2 n and n -co-core equal to thesymbol(2.1) (cid:18) − − . . . − − . . . (cid:19) . Consequently one obtains such symbols by adding two n -co-hooks to the symbol (2.1). Depend-ing on which rows the co-hooks are inserted in, one gets symbols of defect 1, − B -series, or the B -series of G respectively, see § Principal series . There are three families of symbols of defect 1 obtained by adding two n -co-hooks to the symbol (2.1). • For 0 ≤ j ≤ n and 0 < i < n , the symbol (cid:18) n − i − . . . b − j . . .n − j − . . . b − i . . . (cid:19) corresponding to the principal series character (cid:2) ( n − i )1 j . ( n − j +1)1 i − (cid:3) . There are n − • For 0 ≤ j < n , the symbol (cid:18) n − j − . . . b − j . . . − . . . . . . . . . (cid:19) corresponding to the principal series character (cid:2) (2 n − j )1 j . (cid:3) . There are 2 n suchsymbols. • For 0 < i ≤ n , the symbol (cid:18) − . . . . . . . . . n − i − . . . b − i . . . (cid:19) corresponding to the principal series character (cid:2) . (2 n − i +1)1 i − (cid:3) . Again, there are2 n such symbols.(2) B -series . The symbols of defect − (cid:18) − . . . b − i . . . b − j . . .n − i n − j − . . . . . . . . . . . . . . . (cid:19) for 0 ≤ i < j ≤ n . There are n ( n + 1) / (cid:2) i j − i − . ( n − i − n − j ) (cid:3) B , all of which lie in the B -series.In terms of bipartitions and their Young diagrams, the unipotent characters [ λ ] B in the principal Φ n -block are labeled by λ such that λ fits inside the rectangle 2 n − ,and λ is the reflection across a diagonal line of slope 1 of the skew shape 2 n − \ λ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 13 (in particular, λ fits in the rectangle ( n − ). This is best illustrated with a picture(here, for n = 7, we have shaded λ = 2 . · (3) B -series . The symbols of defect 5 are given by (cid:18) n − i n − j − . . . . . . . . . . . . . . . − . . . b − i . . . b − j . . . (cid:19) for 0 < i < j < n . There are ( n − n − / (cid:2) ( n − i − n − j − . i − j − i − (cid:3) B , all lying in the B -series.In terms of bipartitions and their Young diagrams, the unipotent characters [ λ ] B in the principal Φ n -block are labeled by λ such that λ fits inside the rectangle 2 n − ,and λ is the reflection across a diagonal line of slope 1 of the skew shape 2 n − \ λ (in particular, λ fits in the rectangle ( n − ). Again, this is best illustrated with apicture (here, for n = 7, we have shaded λ = 41 . ): · Consequently there are 2 n + 3 n unipotent characters in the principal Φ n -block. By [2], thenumber of unipotent characters in the principal Φ n -block may also be counted by the numberof complex irreducible representations of the complex reflection group G (2 n, , Induced columns.
In this section, we describe the columns of the decomposition matrixof the principal Φ n -block obtained by Harish-Chandra induction from proper Levi subgroups.These account for all but four columns of the decomposition matrix. In all cases, we find thateach non-cuspidal column in the principal Φ n -block is the character of an induced PIM, cutto the principal block.Since a group of type A r has no cuspidal unipotent module (over k ) unless r = 0 or r ≥ n − k G -module corresponds to either a Levi subgroup oftype A n − or a Levi subgroup of type BC r for some r ≤ n . In particular, if Λ is a symbol ofdefect D = 1 then S [Λ] is a cuspidal k G -module whenever e e i Λ = 0 for all i .2.2.1. The B -series submatrix. Theorem 2.2.
Let λ be a bipartition such that [ λ ] B belongs to the principal Φ n -block. If λ =2 n − . then S [ λ ] B is cuspidal and so Ψ [ λ ] B cannot be obtained by Harish-Chandra induction.Otherwise, using truncated induction we obtain the unipotent parts of projective indecomposablecharacters in the B -series as follows: • If λ = 2 k n − − k . ( n − − k ) for some ≤ k < n − , then Ψ [ λ ] B = [ λ ] B + [2 k +1 n − − k . ( n − − k )] B . • If λ = 2 n − . then Ψ [ λ ] B = [ λ ] B + [2 n − . B . • If λ = 2 n − − k .k for some ≤ k ≤ n − then Ψ [ λ ] B = [ λ ] B + [2 n − − k .k ( k − B + [2 n − k . ( k − k − B + [2 n +1 − k . ( k − ] B . • If λ = 2 i j − i − . ( n − i − n − j ) with < i + 1 < j < n then Ψ [ λ ] B = [ λ ] B + [2 i +1 j − i − . ( n − i − n − j )] B + [2 i j − i . ( n − i − n − j − B + [2 i +1 j − i − . ( n − i − n − j − B . Proof. If λ = 2 n − . then ˜ e i λ = 0 for all i ∈ Z / n Z . Since S [ λ ] B does not have cuspidalsupport on a type A parabolic, it follows that S [ λ ] B is cuspidal. For the unipotent part of thecharacters of the projective covers of non-cuspidal simple representations in the B -series, wewill find Ψ [ λ ] B by taking the i -induction of Ψ [ µ ] B for some i ∈ Z / n Z such that ˜ f i µ = λ . Wefind that in this case, f i Ψ [ µ ] B is indecomposable and equals Ψ [ λ ] B . Note that the ℓ -blocks of G n − have either trivial or cyclic defect groups, so that the PIMs can be easily obtained from[20].Let λ = 2 k n − − k . ( n − − k ) for some 0 ≤ k < n −
1. Set µ = 2 k n − − k . ( n − − k ). Then λ = ˜ f n − k − µ . We have Ψ [ µ ] B = [ µ ] B by [20]. (Note that this implies the cuspidal supportof S [ µ ] B , and thus also of S [ λ ] B , is different from [ B ], and is labeled by some non-emptybipartition in the B -series, see [20].) Then Ψ [ λ ] B = Ψ [ ˜ f n − k − µ ] B is a summand of f n − k − Ψ [ µ ] B = f n − k − [ µ ] B = [ λ ] B + [2 k +1 n − − k . ( n − − k )] B . Thus either f n − k − Ψ [ µ ] B is the unipotent part of an indecomposable projective character andequals Ψ [ λ ] B , or it is the sum of the unipotent parts of two indecomposable projective char-acters, each of which would have to contain a single unipotent character. If k = n − [2 k +1 n − − k . ( n − − k )] B = Ψ [2 n − . ] B does not appear as a summand of any induced projectivecharacter by cuspidality. Therefore Ψ [ λ ] B = [ λ ] B + [2 k +1 n − − k . ( n − − k )] B when k = n − k , f n − − k Ψ [ µ ] B = Ψ [ λ ] B for all 0 ≤ k < n − λ = 2 n − . . Set µ = 2 n − .
1. Then λ = ˜ f n µ . We have Ψ [ µ ] B = [ µ ] B by [20].Therefore Ψ [ λ ] B = Ψ [ ˜ f n µ ] B is a summand of f n Ψ [ µ ] B = f n [ µ ] B = [ λ ] B + [2 n − . B , thus either Ψ [ λ ] B = [ λ ] B and [2 n − . B = Ψ [2 n − . B , or Ψ [ λ ] B = f n Ψ [ µ ] B . We showedin the previous paragraph that Ψ [2 n − . B = [2 n − . B + [2 n − . ] B . Therefore Ψ [ λ ] B = f n Ψ [ µ ] B and the desired character formula holds.In the case that λ = 2 n − − k .k for 2 ≤ k ≤ n −
1, we have λ = ˜ f n + k − n − − k .k ( k − λ = 2 i j − i − . ( n − i − n − j ) for 1 < j − i < n , we have λ = ˜ f n − i − i j − i − . ( n − i − n − j ).By [20] the following characters are the unipotent part of PIMs of G n − ( q ):Ψ [2 n − − k .k ( k − B = [2 n − k − .k ( k − B + [2 n − k . ( k − ] B , Ψ [2 i j − i − . ( n − i − n − j )] B = [2 i j − i − . ( n − i − n − j )] B + [2 i j − i . ( n − i − n − j − B . In either case, the cuspidal support of S [ λ ] B is then equal to [ B ], see [20] (and the sourcevertex of λ in the b sl n -crystal is the empty bipartition). This means that if ν is one of thebipartitions of the first two types listed in the theorem or ν = 2 n − . , then Ψ [ ν ] B cannot appearas a summand of f n + k − Ψ [2 n − − k .k ( k − B or f n − i − Ψ [2 i j − i − . ( n − i − n − j )] B . ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 15 We compute that f n + k − Ψ [2 n − − k .k ( k − B = [2 n − − k .k ] B + [2 n − − k . ( k )( k − B + [2 n − k . ( k − k − B + [2 n +1 − k . ( k − ] B ,f n − i − Ψ [2 i j − i − . ( n − i − n − j )] B = [2 i j − i − . ( n − i − n − j )] B + [2 i +1 j − i − . ( n − i − n − j )] B + [2 i j − i . ( n − i − n − j − B + [2 i +1 j − i − . ( n − i − n − j − B . By induction on the number of boxes in λ , together with the fact that no Ψ [ ν ] B as above can bea summand of these induced projective characters, it follows that for none of the bipartitions ρ appearing on the right-hand side of these formulas can Ψ [ ρ ] B be a summand except for ρ = λ . That is, f n + k − Ψ [2 n − − k .k ( k − B = Ψ [2 n − − k .k ] B and f n − i − Ψ [2 i j − i − . ( n − i − n − j )] B =Ψ [2 i j − i − . ( n − i − n − j )] B . We thus get the desired character formulas for Ψ [ λ ] B in the final twocases listed in the theorem. This concludes the proof. (cid:3) The B -series submatrix. The proof of Theorem 2.3 is identical to the proof of Theorem2.2, except that the roles of λ and λ are switched. Theorem 2.3.
Let λ be a bipartition such that [ λ ] B belongs to the principal Φ n -block. If λ = . n − then S [ λ ] B is cuspidal and so Ψ [ λ ] B cannot be obtained by Harish-Chandra induction.Otherwise, using truncated induction we obtain the unipotent parts of projective indecomposablecharacters in the B -series as follows: • If λ = ( n − − k ) . k n − − k for some ≤ k < n − then Ψ [ λ ] B = [ λ ] B + [( n − − k ) . k +1 n − − k ] B . • If λ = 1 . n − then Ψ [ λ ] B = [ λ ] B + [1 . n − B . • If λ = k . n − − k for some ≤ k ≤ n − then Ψ [ λ ] B = [ λ ] B + [( k )( k − . n − − k B + [( k − k − . n − − k B + [( k − . n − − k ] B . • If λ = ( n − i − n − j − . i − j − i − for some < i < j − < n − then Ψ [ λ ] B = [ λ ] B + [( n − i − n − j − . i j − i − ] B + [( n − i − n − j − . i − j − i ] B + [( n − i − n − j − . i j − i − ] B . The principal series submatrix.
We will give explicit formulas in the theorem below, butfirst, for the sake of intuition we sketch a visual and conceptual way to describe the unipotentconstituents of the projective indecomposable characters in the principal Φ n -block that belongto the principal series, inspired by [5]. One can make a simple graph with vertices the biparti-tions λ of 2 n belonging to the principal Φ n -block, and an edge between λ and µ if and onlyif µ is obtained from λ by moving either a single row or a single column of boxes preservingtheir charged contents mod d . In this case, we place µ below λ if µ ≺ s λ . We refer the readerto [25, § λ in the graph is thenthe top vertex of the closure of a cell of maximal possible dimension, or the top vertex of aunion of closures of cells sharing that top vertex. These shapes are either a diamond, a triangle,a single edge, a pair of edges (this happens only for λ = 1 n +1 . n − ), or simply the vertex itself.There turn out to be two such shapes that are points, and these are the two cuspidals. For the λ that are top vertices of 1- and 2-dimensional shapes, taking all vertices in the boundary of the shape and reading off their labels µ yields the unipotent constituents [ µ ] of Ψ [ λ ] . Moreover,those λ at the top of 2-dimensional shapes label the irreducible representations of the Heckealgebra, while the λ that are only at the top of 1-dimensional shapes label the simple moduleswith cuspidal support in some non-trivial, proper parabolic. All of this is just an interpretationof the formulas in the following theorem, which are the same formulas as for the principal blockof Category O of the rational Cherednik algebra of B n at parameters ( n , n ) [25]. Theorem 2.4.
Let λ be a bipartition of n such that [ λ ] belongs to the principal Φ n -block.If λ = 1 n . or . n then S [ λ ] is cuspidal and so Ψ [ λ ] cannot be obtained by Harish-Chandrainduction. Otherwise, the column of the decomposition matrix labeled by λ is given by theappropriate choice of formula below: • If λ = ( n + k )1 n − k . for some ≤ k ≤ n , then Ψ [( n + k )1 n − k . ] = [( n + k )1 n − k . ] + [( n + k − n − k +1 . ] + [( n − n − k .k +1] + [( n − n − k +1 .k ] . • If λ = n n . then Ψ [ n n . ] = [ n n . ] + [( n − n .
1] + [( n − n +1 . ] . • If λ = . (2 n − k )1 k for some ≤ k ≤ n − then Ψ [ . (2 n − k )1 k ] = [ . (2 n − k )1 k ] + [ n − − k. ( n +1)1 k ] + [ n − k − . ( n +1)1 k +1 ] + [ . (2 n − k − k +1 ] . • If λ = . ( n + 2)1 n − then Ψ [ . ( n +2)1 n − ] = [ . ( n +2)1 n − ] + [1 . ( n +1)1 n − ] + [ . ( n +1)1 n − ] . • If λ = 1 k +1 . ( n + 1 − k )1 n − for some ≤ k ≤ n − then Ψ [1 k +1 . ( n +1 − k )1 n − ] = [1 k +1 . ( n +1 − k )1 n − ] + [1 k +2 . ( n − k )1 n − ] + [ . ( n +1 − k )1 n − k ] + [ . ( n − k )1 n + k ] . • If λ = ( n − k )1 n . k for some ≤ k ≤ n − then Ψ [( n − k )1 n . k ] = [( n − k )1 n . k ] + [( n − k − n . k +1 ] + [( n − k )1 n + k . ] + [( n − k − n + k +1 . ] . • If λ = ( n − i )1 j . ( n − j + 1)1 i − for some ≤ i ≤ n − and some ≤ j ≤ n − then Ψ [ λ ] = [( n − i )1 j . ( n − j +1)1 i − ]+ [( n − i )1 j +1 . ( n − j )1 i − ]+ [( n − i − j . ( n − j +1)1 i ]+ [( n − i − j +1 . ( n − j )1 i ] . • If λ = 1 n +1 . n − then Ψ [1 n +1 . n − ] = [1 n +1 . n − ] + [1 n . ] + [ . n ] . • If λ = k n − k . for some ≤ k ≤ n − then Ψ [( k )1 n − k . ] = [ k n − k . ] + [( k − n − k +1 . ] . • If λ = .k n − k for some ≤ k ≤ n + 1 then Ψ [ .k n − k ] = [ .k n − k ] + [ . ( k − n − k +1 ] . Proof.
The first seven cases deal with λ such that λ labels an irreducible representation of theHecke algebra. The formulas are then given by [15]. They may easily be checked to coincidewith F i Ψ [ µ ] for µ and i such that ˜ f i µ = λ .Let L be the standard Levi subgroup of G of type A n − . If λ = 1 n +1 . n − , we find thatΨ [ λ ] occurs as a summand of R GL Ψ n by maximality of the family of λ among the families ofthe three unipotent characters occurring in [ λ ] + [1 n . ] + [ . n ], which equals R GL Ψ n cut to theprincipal Φ n -block. WriteΨ [1 n +1 . n − ] = [1 n +1 . n − ] + α [1 n . ] + β [ . n ] ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 17 for some α, β ∈ { , } . We calculate that e n Ψ [1 n +1 . n − ] = [1 n . n − ] + β [ . n − ]which is only a projective character if β ≥
1. Then we calculate that e Ψ [1 n +1 . n − ] = [1 n +1 . n − ] + α [1 n − . ]which is only a projective character if α ≥
1. We conclude that α = β = 1.Now we are ready to show that S [1 n . ] and S [ . n ] are cuspidal. Since their projective coversare not summands of R GL Ψ n , these two simple modules are cuspidal if their Harish-Chandrarestriction to a standard Levi subgroup of type BC n − is 0. This is confirmed by checkingthat ˜ e i (1 n . ) = 0 = ˜ e i ( . n ) for all i ∈ Z / n Z . Indeed, for either bipartition, there is only oneremovable box, and it is canceled by the addable box of the same residue in the first row of thesame component.It remains to verify the formulas in the last two cases listed in the theorem. Consider first thecase λ = k n − k . for some 2 ≤ k ≤ n −
1. We find that ˜ e k − λ = ( k − n − k . , which belongsto a defect 1 block. By [20], Ψ [( k − n − k . ] = [( k − n − k . ]. Therefore Ψ [ λ ] is a summand of f k − Ψ [( k − n − k . ] = f k − [( k − n − k . ] = [ k n − k . ]+[( k − n − k +1 . ]. Now we argue by inductionon k that this is a projective indecomposable character. The base case is k = 2: since S [1 n . ] iscuspidal, Ψ [1 n . ] is not a summand of the projective character f Ψ [1 n − . ] = [21 n − . ] + [1 n . ].The induction step says that Ψ [ k n − k . ] has the desired formula, and therefore is not a summandof the projective character f k Ψ [ k n − k − ] = [( k + 1)1 n − k − . ] + [ k n − k . ]. Finally, the argumentfor the case λ = .k n − k for some 2 ≤ k ≤ n + 1 is similar. (cid:3) Cuspidal columns.
In the previous section we have accounted for all the columns of thedecomposition matrix of the principal Φ n -block except the ones corresponding by unitriangu-larity to the unipotent characters (cid:2) . n (cid:3) , (cid:2) n . (cid:3) , (cid:2) n − . (cid:3) B , (cid:2) . n − (cid:3) B . The corresponding projective indecomposable modules (PIMs) are the projective covers of cus-pidal simple modules. The purpose of this section is to determine those remaining columnsexplicitly. As before, we will denote byΨ [ . n ] , Ψ [1 n . ] , Ψ [2 n − . ] B , Ψ [ . n − ] B the unipotent part of the characters of the corresponding PIMs. By unitriangularity of thedecomposition matrix and maximality of . n in the partial order (by a -value, or by the reversedominance order on families), it holds that Ψ [ . n ] = (cid:2) . n (cid:3) , the Steinberg character. Forthe remaining three columns we determine the decomposition numbers using arguments fromDeligne–Lusztig theory and from Kac–Moody categorification (truncated induction). The lattertechnique allows us to study the image of the corresponding PIMs under certain sequences of i -induction functors, providing us with an argument that all or all but one of the decompositionnumbers (excepting the one on the diagonal) are zero. Then, two different Deligne–Lusztigcharacters give an upper and lower bound for the unique non-zero decomposition number,which turn out to coincide. Before treating each PIM in turn, we start with the followinglemma. Lemma 2.5.
Let Λ be a symbol corresponding to a unipotent character (cid:2) Λ (cid:3) of G . Let S be aset of symbols Λ ′ < Λ such that Ψ [Λ] ∈ [Λ] + X Λ ′ ∈S N [Λ ′ ] . Let i = ( i , i , . . . , i r ) be a tuple of elements of Z / n and let i ∗ = ( i r , i r − , . . . , i ) be the reversetuple. Assume that (i) f i Λ = e f i Λ = Θ for some symbol Θ ; (ii) if Λ ′ ∈ S and Θ ′ occurs in f i Λ ′ with Θ ′ < Θ then e i ∗ Θ ′ = Λ ′ .Then Ψ [Λ] contains only [Λ] and the characters [Λ ′ ] for symbols Λ ′ ∈ S such that there exists Θ ′ occurring in f i Λ ′ with Θ ′ < Θ .Proof. Recall that P [Λ] denotes the PIM corresponding to the unipotent character [Λ] by uni-triangularity, and that Ψ [Λ] is the unipotent part of its character. Since i -induction functorsare exact, the module f i P [Λ] is projective, and by (i) it contains P [Θ] as a direct summand.Therefore any unipotent character [Θ ′ ] = [Θ] occurring in Ψ [Θ] corresponds to a symbol Θ ′ which satisfies the following two conditions: • there exists Λ ′ ∈ S such that Θ ′ occurs in f i Λ ′ ; • Θ ′ < Θ (by unitriangularity of the decomposition matrix).Now, by (ii), such a symbol satisfies e i ∗ Θ ′ = Λ ′ . The result follows from the fact that P [Λ] is adirect summand of e i ∗ P [Θ] by (i). (cid:3) Remark 2.6. If Λ has a unique addable i -box (i.e. f i Λ is a symbol) and no removable i -box(i.e. e i Λ = 0 ) then e f i Λ = f i Λ by § The column corresponding to (cid:2) n . (cid:3) . The symbol corresponding to this unipotent char-acter is (cid:18) − . . . \ − n +1 . . . − . . . . . . . . . (cid:19) so that the composition of the associated family is (1 , , − , − , − , − , . . ., \ − n +1 , . . . ). Itdominates (0 , , − , − , − , − , . . . , d − n, . . . ), which is the family of the Steinberg character (cid:2) . n ], and does not dominate any other family of unipotent characters of G . Therefore Ψ [1 n . ] = (cid:2) n . (cid:3) + α (cid:2) . n (cid:3) for some α ≥
0, by unitriangularity. We shall prove that α = 0. Proposition 2.7.
We have Ψ [1 n . ] = (cid:2) n . (cid:3) . Proof.
We apply Lemma 2.5 to the symbol Λ associated to (cid:2) n . (cid:3) and i = (0). We haveΘ := f Λ = f (cid:18) − . . . \ − n +1 . . . − . . . . . . . . . (cid:19) = (cid:18) − . . . d − n . . . − . . . . . . . . . (cid:19) . By Remark 2.6 it is also equal to e f Λ, therefore condition (i) of Lemma 2.5 is satisfied. Onthe other hand, the symbol Λ ′ associated to (cid:2) . n (cid:3) is the unique symbol smaller than Λ and itsatisfies f (cid:18) − . . . . . . . . . − . . . d − n . . . (cid:19) = (cid:18) − . . . . . . . . . − . . . d − n . . . (cid:19) . Since f Λ and f Λ ′ lie in the same family, condition (ii) of Lemma 2.5 is empty hence auto-matically satisfied. As a consequence (cid:2) . n (cid:3) is not a constituent of Ψ [1 n . ] and the propositionis proved. (cid:3) ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 19 The column corresponding to (cid:2) n − . (cid:3) B . The unipotent character (cid:2) n − . (cid:3) B of the B -series corresponds to the following symbol of defect −
3Λ = (cid:18) − . . . d − n +1 c − n . . . − . . . . . . . . . . . . (cid:19) . The associated composition is (1 , , , − , − , . . . , − n +1 , d − n +1 , c − n, − n, . . . ). From the descriptionin § n -block lying in smallerfamilies are (cid:2) n . (cid:3) , (cid:2) . n − (cid:3) and (cid:2) . n (cid:3) . Consequently there exist non-negative integers α , β ,and γ such that Ψ [2 n − . ] B = (cid:2) n − . (cid:3) B + α (cid:2) n . (cid:3) + β (cid:2) . n − (cid:3) + γ (cid:2) . n (cid:3) . The following theorem explicitly determines these integers whenever the ℓ -part of Φ n ( q ) is nottoo small. Theorem 2.8.
There exists γ ≤ such that Ψ [2 n − . ] B = (cid:2) n − . (cid:3) B + γ (cid:2) . n (cid:3) . Furthermore, if Φ n ( q ) ℓ > n , then γ = 2 . Note that by a result of Feit, the condition Φ n ( q ) ℓ > n is satisfied for at least one primenumber ℓ except for finitely many pairs ( q, n ) [16]. Proof.
Step 1 . The first step of the proof establishes that (cid:2) . n (cid:3) is the only unipotent characterdifferent from (cid:2) n − . (cid:3) that can occur in Ψ [2 n − . ] B with a nonzero coefficient. For that purposewe use Lemma 2.5 with Λ being the symbol attached to [2 n − . ] B and i = (1 , , . . . , n − f i Λ = f f · · · f n − Λ = (cid:18) . . . d − n +1 . . . \ − n +1 . . . . . . . . . . . . . . . . . . (cid:19) which also equals e f e f · · · e f n − Λ by Remark 2.6 so that condition (i) of Lemma 2.5 holds. Welist below the symbols obtained by inducing the ones associated to the unipotent characters (cid:2) n . (cid:3) , (cid:2) . n − (cid:3) , and (cid:2) . n (cid:3) .[ µ ] Λ ′ f f · · · f n − Λ ′ (cid:2) n . (cid:3) (cid:18) . . . \ − n +1 . . . − . . . . . . (cid:19) . . . . . . \ − n +1 . . . . . . d − n +1 . . . . . . ! + . . . . . . \ − n +1 . . . . . . d − n +2 . . . . . . !(cid:2) . n − (cid:3) (cid:18) . . . . . . . . . . . . − . . . \ − n +1 . . . (cid:19) (cid:18) . . . . . . . . . . . . . . . . . . d − n +1 . . . \ − n +1 . . . (cid:19)(cid:2) . n (cid:3) (cid:18) . . . . . . . . . . . . d − n . . . (cid:19) (cid:18) . . . . . . . . . . . . \ − n +1 . . . (cid:19) The only symbol in the last column which is smaller than f i Λ corresponds to the inductionof the symbol Λ ′ of [ . n ] (the last row in the table). One checks that e i ∗ f i Λ ′ = Λ ′ for thatsymbol, so that assumption (ii) of Lemma 2.5 is satisfied. We deduce that [ . n ] is indeed theonly unipotent constituent of Ψ [2 n − . ] B apart from (cid:2) n − . (cid:3) B . Step 2.
We now use the method in [8] to show that γ ≤
2. This requires to know how todecompose certain Deligne–Lusztig characters on the basis of PIMs, or at least to know thecoefficient of the PIMs corresponding to cuspidal modules. The following lemma will be usefulto deal with PIMs that can be obtained by induction from proper Levi subgroups of G . Lemma 2.9.
Let L (resp. M ) be a -split Levi of G of type BC n − (resp. A n − ). We have U := bR GL (cid:16) n − X i =1 ( − i − (cid:2) i n − i − . (cid:3)(cid:17) = (cid:2) n . (cid:3) + (cid:2) n. (cid:3) + ( − n (cid:2) ( n − n . (cid:3) ,U := bR GL (cid:16) n − X j =1 ( − j + n (cid:2) ( n − j )1 n . j − (cid:3)(cid:17) = ( − n (cid:2) ( n − n . (cid:3) − (cid:2) n +1 . n − (cid:3) ,U := bR GL (cid:16) n − X k =0 ( − k + n (cid:2) k n − k − .n − k − (cid:3) B (cid:17) = ( − n (cid:2) n − .n − (cid:3) B + (cid:2) n − . (cid:3) B ,U := bR GM (cid:0)(cid:2) n (cid:3)(cid:1) = (cid:2) n . (cid:3) + (cid:2) . n (cid:3) + (cid:2) n +1 . n − (cid:3) . In particular the character (cid:2) n. (cid:3) − (cid:2) . n ] = U − U − U is a combination of characters inducedfrom proper Levi subgroups of G .Proof. The Harish-Chandra induction from L to G of a unipotent character associated to abipartition λ = λ .λ is described by adding one box to λ or λ in all possible ways so thatthe result is a bipartition. The formulas for U , U and U are easily deduced from that ruleand the description of the principal Φ n -block given in § R GM (cid:0)(cid:2) n (cid:3)(cid:1) = P i (cid:2) i . n − i (cid:3) from which we deduce the value of U from § (cid:3) With the notation in § c = s s · · · s n ∈ W n . Thedecomposition of the Deligne–Lusztig character of G attached to c can be deduced from [18,(3.2)]. Most of its constituents do not belong to the principal Φ n -block and we have bR c = (cid:2) n. (cid:3) + (cid:2) . n (cid:3) + ( − n − (cid:2) n − .n − (cid:3) B . Using the combination of unipotent characters defined in Lemma 2.9, it can be rewritten as bR c = U − U − U − U + (cid:2) n − . (cid:3) B + 2 (cid:2) . n ]= U − U − U − U + Ψ [2 n − . ] B + (2 − γ )Ψ [ . n ] . Since the U i ’s are obtained by induction from proper Levi subgroups of G , they cannot involveΨ [ . n ] since it corresponds to a cuspidal module. Therefore the coefficient of Ψ [ . n ] in R c isexactly 2 − γ . By [8, Prop. 1.5], we must have 2 − γ ≥ γ ≤ Step 3.
In this final step of the proof, we show that γ ≥ n ( q ) ℓ > n . We consider the element w := c = ( s s · · · s n ) . By [34, § n -regular element. Therefore any torus T w of type w is the centraliser of a Φ n -Sylow subgroup of G and the constituents of the Deligne–Lusztig character R w are exactly the unipotent charactersin the principal Φ n -block, see [2, Thm. 5.24]. The exact decomposition of R w can be deducedfor example from [18, (3.2)]. We find in particular (cid:10) R w ; (cid:2) n − . (cid:3) B (cid:11) G = − (cid:10) R w ; (cid:2) . n (cid:3)(cid:11) G = 1 . On the other hand, if Φ n ( q ) ℓ > n then there exists by [11, Ex. 1.17] an ℓ -character of T w ingeneral position. By [11, Lem. 1.13] this forces (cid:10) R w ; Ψ [2 n − . ] B i G ≥
0, which gives0 ≤ (cid:10) R w ; Ψ [2 n − . ] B i G = (cid:10) R w ; (cid:2) n − . (cid:3) B i G + γ (cid:10) R w ; (cid:2) . n (cid:3)(cid:11) G = − γ and therefore proves that γ ≥
2. Note that it also shows that Ψ [ . n ] does not occur in R c . (cid:3) ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 21 The column corresponding to (cid:2) . n − (cid:3) B . We now focus on the last column, correspondingto the unipotent character (cid:2) . n − (cid:3) B of the B -series. It corresponds to the following symbol ofdefect 5 Λ = (cid:18) − . . . . . . . . . . . . − . . . d − n +2 d − n +1 . . . (cid:19) whose associated composition is (2 , , , − , − , . . . , d − n +2 , d − n +1 , . . . ). From the description in § B -series lying in theprincipal Φ n -block. There are 3 unipotent characters in the principal Φ n -block belonging tothe B -series which are smaller for the order on families. We list these characters with theirassociated symbol: (cid:2) n − . (cid:3) B ←→ (cid:18) . . . d − n +1 c − n . . . . . . . . . . . . . . . (cid:19) , (cid:2) n − . (cid:3) B ←→ (cid:18) . . . d − n +2 − n +1 c − n . . . . . . . . . . . . . . . . . . (cid:19) , (cid:2) n − . (cid:3) B ←→ (cid:18) . . . d − n +2 d − n +1 . . . − . . . . . . . . . (cid:19) . Finally, there are 8 unipotent characters in the principal series belonging to the principal Φ n -block which are smaller than (cid:2) . n − (cid:3) B : (cid:2) n − . (cid:3) ←→ (cid:18) . . . \ − n +2 − . . . (cid:19) , (cid:2) . n − (cid:3) ←→ (cid:18) . . . . . . . . . − . . . \ − n +2 (cid:19) , (cid:2) . n − (cid:3) ←→ (cid:18) . . . . . . . . . − . . . \ − n +1 (cid:19) , (cid:2) . n (cid:3) ←→ (cid:18) . . . . . . . . . d − n (cid:19) , (cid:2) n . (cid:3) ←→ (cid:18) . . . \ − n +1 − . . . (cid:19) , (cid:2) n . n − (cid:3) ←→ (cid:18) . . . . . . c − n . . . d − n +2 . . . (cid:19) , (cid:2) n . n − (cid:3) ←→ (cid:18) . . . . . . d − n +11 − . . . d − n +1 (cid:19) , (cid:2) n +1 . n − (cid:3) ←→ (cid:18) . . . . . . . . . c − n . . . d − n +1 . . . (cid:19) . We will show that none of these characters contribute to Ψ [ . n − ] B with the exception of (cid:2) n . (cid:3) Theorem 2.10.
There exists β ≥ such that Ψ [ . n − ] B = (cid:2) . n − (cid:3) B + β (cid:2) n . (cid:3) . Furthermore, if Φ n ( q ) ℓ > n then β = 2 .Proof. The proof follows that of Theorem 2.8, but more computations are needed since morecharacters are involved.
Step 1.
We start by computing the image of the various symbols involved under the operator f i for i = (1 , B -series wehave(2.11) f i Λ = (cid:18) − . . . . . . . . . . . . − . . . d − n +1 c − n . . . (cid:19) which also equals e f i Λ by Remark 2.6. For the symbols corresponding to characters in the B -series we obtain [Λ ′ ] f i Λ ′ (cid:2) n − . (cid:3) B (cid:18) − . . . d − n +1 c − n . . . − . . . . . . . . . . . . (cid:19)(cid:2) n − . (cid:3) B (cid:18) − . . . d − n +2 − n +1 c − n . . . − . . . . . . . . . . . . . . . (cid:19)(cid:2) n − . (cid:3) B (cid:18) − . . . d − n +2 d − n +1 . . . − . . . . . . . . . . . . (cid:19) None of these symbol is smaller than f i Λ. The case of the principal series characters is givenin the following table:[Λ ′ ] f i Λ ′ [Λ ′ ] f i Λ ′ (cid:2) n − . (cid:3) (cid:18) − . . . \ − n +1 − . . . . . . (cid:19) (cid:2) . n − (cid:3) (cid:18) − . . . . . . − . . . \ − n +2 (cid:19)(cid:2) . n − (cid:3) (cid:18) − . . . . . . − . . . \ − n +1 (cid:19) (cid:2) . n (cid:3) (cid:18) − . . . . . . . . . . . . d − n (cid:19)(cid:2) n . (cid:3) (cid:18) . . . d − n . . . − . . . . . . (cid:19) (cid:2) n . n − (cid:3) (cid:18) − . . . . . . c − n . . . . . . d − n +1 . . . (cid:19)(cid:2) n . n − (cid:3) (cid:18) . . . . . . d − n +11 − . . . c − n (cid:19) (cid:2) n +1 . n − (cid:3) (cid:18) . . . . . . . . . c − n . . . c − n . . . (cid:19) The induction of the symbols corresponding to the characters (cid:2) n . n − (cid:3) , (cid:2) . n − (cid:3) , (cid:2) n . n − (cid:3) and (cid:2) n +1 . n − (cid:3) are not strictly dominated by the symbol f i Λ computed in (2.11). The othersymbols Λ ′ satisfy e i ∗ f i Λ ′ = Λ ′ , therefore Lemma 2.5 shows that there exist non-negativeintegers β , β , β and β such thatΨ [ . n − ] B = (cid:2) . n − (cid:3) B + β (cid:2) n − . (cid:3) + β (cid:2) . n − (cid:3) + β (cid:2) n . (cid:3) + β (cid:2) . n (cid:3) . We now consider the induction with respect to the sequence i = ( − , − , , − , , e f i Λ = f i Λ = (cid:18) − . . . . . . . . . . . . − . . . d − n − d − n − . . . (cid:19) using Remark 2.6. For the four remaining principal series characters we obtain the following:[Λ ′ ] Λ ′ f i Λ ′ (cid:2) n − . (cid:3) (cid:18) . . . \ − n +2 − . . . (cid:19) (cid:18) − − . . . \ − n − − . . . . . . . . . (cid:19)(cid:2) . n − (cid:3) (cid:18) . . . . . . . . . − . . . \ − n +1 (cid:19) (cid:18) − . . . − . . . \ − n +1 (cid:19)(cid:2) n . (cid:3) (cid:18) . . . \ − n +1 − . . . (cid:19) (cid:18) − . . . \ − n − − . . . . . . . . . (cid:19)(cid:2) . n (cid:3) (cid:18) . . . . . . . . . d − n (cid:19) (cid:18) − . . . . . . . . . . . . d − n (cid:19) Note that for the computations we can use that f i and f j commute whenever i ≇ j ± n so that f i = ( f − f f )( f − f − f ). The induction of the symbols correspondingto the unipotent characters (cid:2) . n − (cid:3) and (cid:2) . n (cid:3) are not strictly dominated by the symbol f i Λcomputed in (2.12). The other symbols satisfy the conditions of Lemma 2.5 which shows that
ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 23 β = β = 0. It remains to show that β = 0. For that purpose we use again Lemma 2.5 for Λand i = ( − n + 2 , . . . , − , − , ′ ] Λ ′ f i Λ ′ (cid:2) . n − (cid:3) B (cid:18) . . . . . . . . . . . . − . . . d − n +2 d − n +1 (cid:19) (cid:18) . . . . . . . . . . . . . . . − . . . d − n +2 . . . \ − n +2 (cid:19)(cid:2) n − . (cid:3) (cid:18) . . . \ − n +2 − . . . (cid:19) (cid:18) . . . d − n +2 . . . \ − n +2 − . . . . . . . . . (cid:19)(cid:2) n . (cid:3) (cid:18) . . . \ − n +1 − . . . (cid:19) (cid:18) . . . \ − n +2 − . . . (cid:19) The induced symbols corresponding to the characters (cid:2) . n − (cid:3) B and (cid:2) n − . (cid:3) lie in the samefamily. In addition, if Λ ′ is the symbol attached to (cid:2) n . (cid:3) then e i ∗ f i Λ ′ = Λ ′ . Finally, one cancheck that e f i Λ = f i Λ using § β = 0 (note thatRemark 2.6 does not apply to the last step of the computation of e f i Λ). For the remainder ofthe proof we will write β := β . Step 2.
Assume now that Φ n ( q ) ℓ > n . Then Ψ [2 n − . ] B = (cid:2) n − . (cid:3) B +2 (cid:2) . n ] by Theorem 2.8.In addition, it was observed in the proof of that theorem that the Deligne–Lusztig characterassociated to a Coxeter element c decomposes as R c = U − U − U − U + Ψ [2 n − . ] B . This shows that apart from Ψ [2 n − . ] B , none of the PIMs corresponding to cuspidal modulesappear in R c . In order to decompose the next Deligne–Lusztig characters we shall use thefollowing identities, whose proofs are identical to those for Lemma 2.9. Lemma 2.13.
Let L be a -split Levi of G of type B n − . We have U := bR GL (cid:16)(cid:2) n − . (cid:3)(cid:17) = (cid:2) n. (cid:3) + (cid:2) (2 n − . (cid:3) ,U := bR GL (cid:16)(cid:2) n − .n − (cid:3) B + (cid:2) n − .n − (cid:3) B (cid:17) = 2 (cid:2) n − .n − (cid:3) B + (cid:2) n − .n − (cid:3) B + (cid:2) n − . ( n − (cid:3) B ,U := bR GL (cid:16)(cid:2) (2 n − . (cid:3)(cid:17) = (cid:2) (2 n − . (cid:3) + (cid:2) (2 n − . (cid:3) ,U := bR GL (cid:16) n X j =1 ( − j + n (cid:2) j . ( n +1 − j )1 n − (cid:3)(cid:17) = ( − n +1 (cid:2) . ( n +1)1 n − . (cid:3) + (cid:2) n +1 . n − (cid:3) ,U := bR GL (cid:16)(cid:2) n − . ( n − (cid:3) B + (cid:2) n − . ( n − (cid:3) B + (cid:2) n − .n − (cid:3) B − (cid:2) n − .n − (cid:3) B − (cid:2) n − .n − (cid:3) B (cid:17) , = (cid:2) n − . ( n − (cid:3) B + (cid:2) n − . ( n − (cid:3) B + (cid:2) n − .n − (cid:3) B − (cid:2) n − .n − (cid:3) B ,U := bR GL (cid:16) n − X r =0 ( − r + n (cid:2) n − r − . r n − r − (cid:3) B (cid:17) = ( − n (cid:2) n − . n − (cid:3) B + (cid:2) . n − (cid:3) B . Since we want to use the result in [8] to get information on the PIMs with cuspidal head, it isenough to consider the Deligne–Lusztig characters R w for elements w ∈ W n whose conjugacyclass does not meet any proper parabolic subgroup. Such classes are called cuspidal (or elliptic)and are described in [23, Prop. 3.4.6]. With the notation in [23, § c = w − (2 n ) , v n = s s c = w − (1 , n − ,and w n := s s s s c = w (2 , n − . Let us first consider the Deligne–Lusztig character associatedto v n . From the decomposition of R v n given in Lemma A.15 and the characters defined in Lemma 2.9 and 2.13 we have bR v n = 2( U − U − U − U ) − U − D G ( U ) + ( − n U + 2 (cid:2) n − . (cid:3) B + 4 (cid:2) . n ]= 2( U − U − U − U ) − U − D G ( U ) + ( − n U + 2Ψ [2 n − . ] B . Again, apart from Ψ [2 n − . ] B , none of the PIMs corresponding to cuspidal modules appear in R v n . We move on to the element w n := s s s s c . Using Lemma A.17 and Lemma 2.9 we candecompose R w n as bR w n = 3 U − U − U − U − U + U + U + D G ( U ) − D G ( U ) + D G ( U )+ ( − n − U − U + (cid:2) . n − (cid:3) B + 3 (cid:2) n − . (cid:3) B + 2 (cid:2) n . (cid:3) + 3 (cid:2) . n (cid:3) = 3 U − U − U − U − U + U + U + D G ( U ) − D G ( U ) + D G ( U )+ ( − n − U − U + Ψ [ . n − ] B + 3Ψ [2 n − . ] B + (2 − β )Ψ [1 n . ] . Here we have also used Proposition 2.7 and Theorem 2.8 which give the decomposition ofΨ [2 n − . ] B and Ψ [1 n . ] . It follows from [8, Prop. 1.5] that 2 − β ≥ β ≤ Step 3.
The argument is entirely similar to that given in the proof of Theorem 2.8 with theexception that one uses the multiplicities (cid:10) R w ; (cid:2) . n − (cid:3) B (cid:11) G = − (cid:10) R w ; (cid:2) n . (cid:3)(cid:11) G = 1 . This shows that β ≥
2, hence β = 2. (cid:3) Appendix: Computation of Deligne–Lusztig characters
We fix an integer n . As in § W n the Weyl group of type B n with gen-erators s , . . . , s n . We explain here how to compute the Deligne–Lusztig characters associatedwith the elements v n := ( s s )( s s s · · · s n − s n )and w n := ( s s s s )( s s s · · · s n − s n )of respective lengths 2 n + 2 and 2 n + 4.A.1. The Deligne–Lusztig character associated to v n . Using the notation in [23, § v n = w − (1 , n − = s b − , n − = b − , b − , n − and one can compute thevalues of irreducible characters of W n at the element v n using the Murnaghan–Nakayama rule.More precisely, if λ is a bipartition of 2 n and χ λ is the corresponding irreducible character of W n then by [23, Thm. 10.3.1] we have(A.14) χ λ ( v n ) = X γ ( − h ( γ ) χ λ r γ ( s )where γ runs over all the (2 n − λ , and h ( γ ) equals the leg length of γ plus 1 if γ isin the second component. From the description of the principal Φ n -block B given in § λ which have a (2 n − B . They are listed in Table 1, together with the value of h ( γ ), the (2 n − λ r γ , the value of χ λ r γ ( s s ) and that of χ λ ( w n ) using (A.14). Note that the second half ofthe table can be obtained from the first using the relation χ λ ∗ ( v n ) = ( − l ( v n ) χ λ ( v n ) = χ λ ( v n ),where l ( w ) is the Coxeter length of w ∈ W n .The families corresponding to the bipartitions 2 n. and . n contain only one unipotent char-acter each, the trivial and the Steinberg character respectively. The other families have 4elements and are listed in Table 2. By convention the special character is the first one in each ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 25 λ h ( γ ) λ r γ χ λ r γ ( s ) χ λ ( v n )2 n. . n n − . n − − n − n . n − . ( n +1)1 n − n − − n − . n − n − − . n n − . − .n n − n − − n − . n n − n . n ( − n − n − . − Table 1.
Bipartitions λ with a (2 n − n -block λ F n n − . (cid:2) n. n − (cid:3) , (cid:2) n n − . (cid:3) , (cid:2) . ( n +1)1 n − (cid:3) , (cid:2) n − . ( n − (cid:3) B n . (cid:2) . n − (cid:3) , (cid:2) n . (cid:3) , (cid:2) . n − (cid:3) , [ . n − (cid:3) B . ( n +1)1 n − (cid:2) n. n − (cid:3) , (cid:2) n n − . (cid:3) , (cid:2) . ( n +1)1 n − (cid:3) , (cid:2) n − . ( n − (cid:3) B . n − (cid:2) . n − (cid:3) , (cid:2) n . (cid:3) , (cid:2) . n − (cid:3) , [ . n − (cid:3) B .n n − (cid:2) ( n − . n (cid:3) , (cid:2) .n n − (cid:3) , (cid:2) ( n − n . (cid:3) , (cid:2) n − .n − (cid:3) B . n (cid:2) n − . (cid:3) , (cid:2) . n (cid:3) , [(2 n − . (cid:3) , (cid:2) n − . (cid:3) B ( n − n . (cid:2) ( n − . n (cid:3) , (cid:2) .n n − (cid:3) , (cid:2) ( n − n . (cid:3) , (cid:2) n − .n − (cid:3) B n − . (cid:2) n − . (cid:3) , (cid:2) . n (cid:3) , [(2 n − . (cid:3) , (cid:2) n − . (cid:3) B Table 2.
Families F with 4 elements occurring in bR v n list. The corresponding almost characters can be computed from [30, § R w = P λ χ λ ( w ) R χ λ , we obtainthe explicit decomposition of R v n on the block. Lemma A.15.
The decomposition of the Deligne–Lusztig character R v n of G n associated to v n = s s s s · · · s n on the principal Φ n -block is given by bR v n = (cid:2) n. (cid:3) + (cid:2) . n (cid:3) − (cid:2) (2 n − . (cid:3) − (cid:2) . n − (cid:3) + ( − n (cid:16)(cid:2) n − . ( n − (cid:3) B + (cid:2) n − .n − (cid:3) B (cid:17) . A.2.
The Deligne–Lusztig character associated to w n . Using again the notation in [23, § w n as w n = w − (2 , n − = b − , b − , n − = s s b − , n − . Let λ be a bipartitionof 2 n . By [23, Thm. 10.3.1] we have(A.16) χ λ ( w n ) = X γ ( − h ( γ ) χ λ r γ ( s s )where γ runs over all the (2 n − λ , and h ( γ ) equals the leg length of γ plus 1 if γ is inthe second component. We list in Table 4 those bipartitions λ which have a (2 n − γ , the λ bR χ λ n n − . − (cid:16)(cid:2) . ( n +1)1 n − (cid:3) + (cid:2) n − . ( n − (cid:3) B (cid:17) n . (cid:16)(cid:2) n . (cid:3) − (cid:2) . n − (cid:3)(cid:17) . ( n +1)1 n − (cid:16)(cid:2) . ( n +1)1 n − (cid:3) − (cid:2) n − . ( n − (cid:3) B (cid:17) . n − (cid:16)(cid:2) n . (cid:3) + (cid:2) . n − (cid:3)(cid:17) .n n − − (cid:16)(cid:2) ( n − n . (cid:3) + (cid:2) n − .n − (cid:3) B (cid:17) . n (cid:16)(cid:2) . n (cid:3) − (cid:2) (2 n − . (cid:3)(cid:17) ( n − n . (cid:16)(cid:2) ( n − n . (cid:3) − (cid:2) n − .n − (cid:3) B (cid:17) n − . (cid:16)(cid:2) (2 n − . (cid:3) − (cid:2) . n (cid:3)(cid:17) Table 3.
Some almost characters occurring in bR v n λ h ( γ ) λ r γ χ λ r γ ( s s ) χ λ ( w n )2 n. . i n − i − . n − i − − i n − . n − − .j n − − j n − − j + 1 ( − j − n . n − . − k n − k − . n − i − − i − (2 n − . − .l n − − l n − − j + 1 ( − j Table 4.
Bipartitions λ with (2 n − . or 1 . value of h ( γ ), the (2 n − λ r γ , the value of χ λ r γ ( s s ) and that of χ λ ( w n ) using (A.16).We do not list the ones with (2 n − . B vanisheson the element s s . In addition, since χ λ ∗ ( w n ) = ( − ℓ ( w n ) χ λ ( w n ) = χ λ ( w n ) it is enough todeal with the cores 2 . and 1 . .For each bipartition λ in Table 4 one can easily compute the family associated to each ofthese bipartitions and check using § i = l = n , j = 1 or n + 1 and k = n − n -block B . The family associated to the bipartition 2 n. has size one and contains only the trivialcharacter. We give in Table 5 the families with 4 elements, starting with the special characterin the family.The two remaining bipartitions ( n − n − . and 1 .n n − correspond to unipotent characterslying in a family with 16 elements. Using [30, §
4] one can deduce from Table 5 the almostcharacter corresponding to each bipartition. These are listed in Table 6. For the family with
ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 27 λ F n n − . (cid:2) n. n − (cid:3) , (cid:2) n n − . (cid:3) , (cid:2) . ( n +1)1 n − (cid:3) , (cid:2) n − . ( n − (cid:3) B n − . (cid:2) . n − (cid:3) , [ . n − (cid:3) , (cid:2) n − . (cid:3) , (cid:2) n − . (cid:3) B . ( n +1)1 n − (cid:2) n. n − (cid:3) , (cid:2) n n − . (cid:3) , (cid:2) . ( n +1)1 n − (cid:3) , (cid:2) n − . ( n − (cid:3) B . n − (cid:2) . n − (cid:3) , [ . n − (cid:3) , (cid:2) n − . (cid:3) , (cid:2) n − . (cid:3) B n . (cid:2) . n − (cid:3) , (cid:2) n . (cid:3) , (cid:2) . n − (cid:3) , [1 n − . (cid:3) B (2 n − . (cid:2) n − . (cid:3) , (cid:2) . n (cid:3) , [(2 n − . (cid:3) , (cid:2) . n − (cid:3) B Table 5.
Families F with 4 elements occurring in bR w n (up to Alvis-Curtis duality) λ bR χ λ n n − . − (cid:16)(cid:2) . ( n +1)1 n − (cid:3) + (cid:2) n − . ( n − (cid:3) B (cid:17) n − . (cid:16)(cid:2) n − . (cid:3) − (cid:2) . n − (cid:3)(cid:17) . ( n +1)1 n − (cid:16)(cid:2) . ( n +1)1 n − (cid:3) − (cid:2) n − . ( n − (cid:3) B (cid:17) . n − (cid:16)(cid:2) n − . (cid:3) + (cid:2) . n − (cid:3)(cid:17) n . (cid:16)(cid:2) n . (cid:3) − (cid:2) . n − (cid:3)(cid:17) (2 n − . (cid:16)(cid:2) (2 n − . (cid:3) − (cid:2) . n (cid:3)(cid:17) Table 6.
Some almost characters occurring in bR w n
16 elements we get bR χ ( n − n − . = − (cid:16)(cid:2) .n n − (cid:3) + (cid:2) ( n − n − . (cid:3) + (cid:2) n − . ( n − (cid:3) B + (cid:2) n − . n − (cid:3) B (cid:17) ,bR χ .n n − = 14 (cid:16)(cid:2) .n n − (cid:3) + (cid:2) ( n − n − . (cid:3) − (cid:2) n − . ( n − (cid:3) B − (cid:2) n − . n − (cid:3) B (cid:17) . Putting this all together we obtain the decomposition of R w n on the block. Lemma A.17.
The decomposition of the Deligne–Lusztig character R w n of G n associated to w n = s s s s s s · · · s n on the principal Φ n -block is given by bR w n = (cid:2) n. (cid:3) − (cid:2) (2 n − . (cid:3) + (cid:2) (2 n − . (cid:3) + (cid:2) n . (cid:3) + (cid:2) . n (cid:3) − (cid:2) . n − (cid:3) + (cid:2) . n − (cid:3) + (cid:2) . n (cid:3) + ( − n − (cid:16)(cid:2) n − . ( n − (cid:3) B + (cid:2) n − .n − (cid:3) B + (cid:2) n − . ( n − (cid:3) B + (cid:2) n − . n − (cid:3) B (cid:17) . References [1]
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ECOMPOSITION NUMBERS FOR THE PRINCIPAL Φ n -BLOCK OF Sp n ( q ) AND SO n +1 ( q ) 29 (O.D.) Universit´e de Paris and Sorbonne Universit´e, CNRS, IMJ-PRG, F-75006 Paris, France
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