Gelfand-Graev characters of the finite unitary groups

Transcription

1 Gelfand-Graev characters of the finite unitary groups Nathaniel Thiem and C. Ryan Vinroot Abstract Gelfand-Graev characters and their degenerate counterparts have an important role in the representation theory of finite groups of Lie type. Using a characteristic map to translate the character theory of the finite unitary groups into the language of symmetric functions we study degenerate Gelfand-Graev characters of the finite unitary group from a combinatorial point of view. In particular we give the values of Gelfand-Graev characters at arbitrary elements recover the decomposition multiplicities of degenerate Gelfand-Graev characters in terms of tableau combinatorics and conclude with some multiplicity consequences. Introduction Gelfand-Graev modules have played an important role in the representation theory of finite groups of Lie type [4 7 ]. In particular if G is a finite group of Lie type then Gelfand- Graev modules of G both contain cuspidal representations of G as submodules and have a multiplicity free decomposition into irreducible G-modules. Thus Gelfand-Graev modules can give constructions for some cuspidal G-modules. This paper uses a combinatorial correspondence between characters and symmetric functions as described in [] to examine the Gelfand-Graev character and its degenerate relatives for the finite unitary group. Let B < be a maximal unipotent subgroup of a finite group of Lie type G. Then the Gelfand- Graev character Γ of G is the character obtained by inducing a generic linear character from B < to G. The degenerate Gelfand-Graev characters of G are obtained by inducing arbitrary linear characters. In the case GLn F q Zelevinsky [6] described the multiplicities of irreducible characters in degenerate Gelfand-Graev characters by counting multi-tableaux of specified shape and weight. It is the goal of this paper to describe the degenerate Gelfand-Graev characters of the finite unitary groups in a similar manner using tableau combinatorics. In [6] Zelevinsky obtained the result that every irreducible appears with multiplicity one in some degenerate Gelfand-Graev character. It is known that this multiplicity one result is not true in a general finite group of Lie type and in fact there are characters which do not appear in any degenerate Gelfand-Graev character in the general case. This result was illustrated by Srinivasan [9] in the case of the symplectic group Sp4 F q and the work of Kotlar [] gives a geometric description of the irreducible characters which appear in some degenerate Gelfand-Graev character in general type. In the finite unitary case we give a combinatorial description of which irreducible characters appear in some degenerate Gelfand-Graev character as well as a combinatorial description of a large family of characters which appear with multiplicity one. University of Colorado at Boulder: thiemn@colorado.edu College of William and Mary: vinroot@math.wm.edu MSC 000: 0C33 05E05 Keywords: finite unitary group degenerate Gelfand-Graev multiplicity domino tableaux characteristic map

2 In Section we describe the main combinatorial tool which we use for calculations which is the characteristic map of the finite unitary group and we follow the development given in []. This map translates the Deligne-Lusztig theory of the finite unitary group into symmetric functions which thus translates calculations in representation theory into algebraic combinatorics. The results of this paper could certainly be obtained in a more representation theoretic manner by pulling the combinatorial calculations here back through the characteristic map. However our main goal is to view these characters as purely combinatorial objects and to understand the Gelfand-Graev characters as such. This point of view also gives rise to interesting identities in symmetric function theory such as our Lemma 4.. Section 3 examines the non-degenerate Gelfand-Graev character and uses a remarkable formula for the character values of the Gelfand-Graev character of GLn F q for an elementary proof see [9] to obtain the corresponding formula for Un F q. This formula is applied in another paper to obtain [3 Theorem 4.4]. The following is the main result of this section. I. Corollary 3. If Γ n is the Gelfand-Graev character of Un F q and g Un F q then { Γ n g = n/ + l q l l q + if g is unipotent block type µ µ... µ l 0 otherwise. Section 4 computes the decomposition of degenerate Gelfand-Graev characters in a fashion analogous to [6] with the main result as follows. II. Theorem 4.3 The degenerate Gelfand-Graev character Γ kν decomposes as Γ kν = λ m λ χ λ where λ is a multi-partition and m λ is a nonnegative integer obtained by counting battery tableaux of a given weight and shape. In the process of proving Theorem 4.3 we obtain some combinatorial Pieri-type formulas Lemma 4. and decompositions of induced characters from GLn F q to Un F q Theorem 4. and Theorem 4.. Section 5 concludes with a discussion of the multiplicity implications of Section 4. In particular we improve a multiplicity one result by Ohmori [7]. III. Theorem 5. We give combinatorial conditions on multipartitions λ that guarantee that the irreducible character χ λ appears with multiplicity one in some degenerate Gelfand-Graev character. Another question one might ask is how the generalized Gelfand-Graev representations of the finite unitary group decompose. Generalized Gelfand-Graev representations which were defined by Kawanaka in [0] are obtained by inducing certain irreducible representations not necessarily one dimensional from a unipotent subgroup. Rainbolt studies the generalized Gelfand-Graev representations of U3 F q in [8] but in the general case they seem to be significantly more complicated than the degenerate Gelfand-Graev representations. Acknowledgements. We would like to thank G. Malle for suggesting the questions that led to the results in Section 5 S. Assaf for a helpful discussion regarding Section 5. and T. Lam for helping us connect Lemma 4. to the literature.

3 Preliminaries. Partitions Let P = n 0 P n where P n = {partitions of n}. For ν = ν ν... ν l P n the length lν of ν is the number of parts l and the size ν of ν is the sum of the parts n. Let ν denote the conjugate of the partition ν. We also write ν = m ν m ν where m i ν = {j Z ν j = i}. Define nν to be nν = i i ν i. If µ ν P we define µ ν P to be the partition of size µ + ν whose set of parts is the union of the parts of µ and ν. For k Z let kν = kν kν... and if every part of ν is divisible by k then we let ν/k = ν /k ν /k.... A partition ν is even if ν i is even for i lν.. The ring of symmetric functions Let X = {X X...} be an infinite set of variables and let ΛX = C[p X p X...] where p k X = X k + X k + be the graded C-algebra of symmetric functions in the variables {X X...}. For a partition ν = ν ν... ν l P the power-sum symmetric function p ν X is p ν X = p ν Xp ν X p νl X. The irreducible characters ω λ of S n are indexed by λ P n. Let ω λ ν be the value of ω λ on a permutation with cycle type ν. The Schur function s λ X is given by s λ X = ω λ νzν p ν X where z ν = i m i m i!. ν P λ i is the order of the centralizer in S n of the conjugacy class corresponding to ν = m m P. Fix t C. For µ P the Hall-Littlewood symmetric function P µ X; t is given by s λ X = µ P λ K λµ tp µ X; t. where K λµ t is the Kostka-Foulkes polynomial as in [6 III.6]. For ν µ P n the classical Green function Q µ ν t is given by p ν X = µ P ν Q µ ν t t nµ P µ X; t..3 3

4 As a graded ring ΛX = C-span{p ν X ν P} = C-span{s λ X λ P} = C-span{P µ X; t µ P} with change of bases given in.. and.3. We will also use several products formulas in the ring of symmetric functions. The usual product on Schur functions s ν s µ = c λ νµs λ.4 λ P gives us the Littlewood-Richardson coefficients c λ νµ. The plethysm of p ν with p k is p ν p k = p kν. Thus we can consider the nonnegative integers c γ λ given by s λ p k = ωλ ν p kν = z ν ν P λ γ P k λ c γ λ s γ..5 Chen Garsia and Remmel [] give a combinatorial algorithm for computing the coefficients c γ λ. We will use the case k = in Section 4.4. Remark. The unipotent characters χ λ of GLn F q are indexed by partitions λ of n and the unipotent characters χ γ of Un F q are indexed by partitions γ of n. It will follow from Theorem 4. that R UnF q χ λ GLnF q = c γ λχ γ where R G H is Harish-Chandra induction..3 The finite unitary groups γ = λ Let Ḡn = GLn F q be the general linear group with entries in the algebraic closure of the finite field F q with q elements. For the Frobenius automorphisms F F F : Ḡn Ḡn given by let F a ij = a q ij F a ij = a q ji.6 F a ij = a q n jn i where a ij Ḡn = Ḡ F n = {a Ḡn F a = a} U n = ḠF n = {a Ḡn F a = a} U n = ḠF n = {a Ḡn F a = a}..7 Then = GLn F q and U n = U n are isomorphic to the finite unitary group Un F q. In fact if follows from the Lang-Steinberg theorem that U n and U n are conjugate subgroups of Ḡn. 4

5 For k Z 0 let T k = Ḡ F k = F q k and T k = ḠF k = { F q k if k is even {t F q t qk + = } if k is odd. For every partition η = η η... η l P n let T η = T η T η T ηl T η = T η T η T ηl. Every maximal torus of is isomorphic to T η for some η P n and every maximal torus of U n is isomorphic to T η for some η P n..4 Multipartitions Let F : Ḡn Ḡn be as in.6 and let T k = {ξ : T k C } be the group of multiplicative complex-valued characters of T k = ḠF k. We identify F q with Ḡ = GL F q. Consider Φ = {F -orbits of F q } and note that Φ = k T k. In particular we may view Ḡ as a direct limit of the T k with respect to inclusion. We also have norm maps N mk whenever k m N mk : T m T k α m/k i=0 α qki where m k Z k m..8 When k m denote by N mk the transpose of the map N mk which embeds T k into T m as follows: N mk : T k T m ξ ξ N mk.9 Now define L to be the direct limit of the groups T k with respect to the maps N mk : L = lim T m. Since the map F acts naturally on each Tm it acts on their direct limit L. Note that we may identify the fixed points L F m with the character group Tm. Let Θ be the collection of F -orbits on L: Θ = {F -orbits of L}. For X {Φ Θ} an X -partition λ = λ x λ x... is a sequence of partitions indexed by X. The size of λ is λ = x λ x x X where x is the size of the orbit x. Let P X = n 0 P X n where P X n = {X -partitions of size n}. For λ P X let lλ = x X lλ x and nλ = x X x nλ x. 5

6 The conjugate of λ P X is the X -partition λ defined by λ x = λ x and if µ λ P X then µ λ P X is defined by µ λ x = µ x λ x. The semisimple part λ s of an X -partition λ is the X -partition given by For λ P X define the set P λ s λ x s = λx for x X..0 by P λ s = {µ P X µ s = λ s }. The unipotent part λ u of λ is the X -partition given by λ {} u has parts { x λ x i x X i =... lλx}. where {} is the orbit containing in Φ or the trivial character in Θ and λ x u = when x {}. Note that we can think of normal partitions as X -partitions λ that satisfy λ u = λ. By a slight abuse of notation we will sometimes interchange the mulipartition λ u and the partition λ u {}. For example T λu will denote the torus corresponding to the partition λ u {}. Given the torus T η η = η η... η l P n there is a natural surjection τ Θ : {θ = θ θ θ l HomT η C } {ν P Θ ν {} u = η} θ = θ θ θ l τ Θ θ. where τ Θ θ ϕ = η i / ϕ η i / ϕ... η ir / ϕ where θ i θ i... θ ir ϕ. It follows from a short calculation that if ν P Θ has support {ϕ ϕ... ϕ r } then the preimage τ Θ ν has size r ϕ j lνϕ j m i ν {} u m j= i i/ ϕ ν ϕ m i/ ϕ ν ϕ m i/ ϕr ν ϕr = ϕ lνϕ mi ν u {}! i m i/ ϕ ν ϕ.3! The conjugacy classes K µ of U n are parameterized by µ P Φ n a fact on which we elaborate in Section.5. We have another natural surjection τ Φ : T η {ν P Φ ν {} u = η} t = t t... t l τ Φ t τ Φ t τ Φ t l.4 where τ Φ t i = µ if t i K µ in U ηi..5 The characteristic map For every f Φ let X f = {X f X f...} be an infinite set of variables and for every ϕ Θ let Y ϕ = {Y ϕ Y ϕ...} be an infinite set of characters. We relate symmetric functions in the variables X f to those in the variables Y ϕ through the transform p k Y ϕ = k φ ξxp k ϕ / fx X fx where ξ ϕ x f x. x T k ϕ 6

7 The ring of symmetric functions Λ is Λ = f Φ ΛX f = ΛY ϕ. For µ P Φ the Hall-Littlewood polynomial P µ is P µ = q nµ f Φ P µ fx f ; q f and for λ P Θ the power-sum symmetric function p λ and the Schur function s λ are p λ = p λ ϕy ϕ and s λ = s λ ϕy ϕ. For µ ν P Φ the Green function is Q µ ν q = Q µf ν q f f f Φ µ where Φ µ = {f Φ µ f }. As a graded rings Λ = C-span{p ν ν P Θ } = C-span{s λ λ P Θ } = C-span{P µ µ P Φ }. The conjugacy classes K µ of U n are indexed by µ Pn Φ and the irreducible characters χ λ of U n are indexed by λ Pn Θ [5 6]. Thus the ring of class functions C n of U n is given by where κ µ : U n C is given by C n = C-span{χ λ λ P Θ n } = C-span{κ µ µ P Θ n } { κ µ if g K µ g = 0 otherwise. We let χ λ µ denote the value of the character χ λ on any element in the conjugacy K µ. For ν Pn Θ let the Deligne-Lusztig character R ν = Rν Un be given by R ν = R Un T νu θ where θ HomT νu C is any homomorphism such that τ Θ θ = ν see.. Let C = n C n so that C = C-span{χ λ λ P Θ } = C-span{κ µ µ P Φ } = C-span{R ν ν P Θ } is a ring with multiplication given by R λ R η = R λ η. The next theorem follows from the results of [ ]. A summary of the relevant results in these papers and how they imply the following theorem is given in []. 7

8 Theorem. Characteristic Map. The map ch : C Λ χ λ λ / +nλ s λ κ µ P µ R ν ν lν p ν is an isometric ring isomorphism with respect to the natural inner products χ λ χ η = δ λη and s λ s η = δ λη. In the following change of basis equations.5 follows from Theorem..6 follows from. and.7 follows from [ Theorem 4.]. k/ +nλ s λ = µ P Φ k s λ = ν P k Θ λs=νs χ λ µp µ for λ P Θ k.5 k lν p ν = µ P Φ k ω λϕ ν ϕ p ν for λ Pk Θ z.6 ν ϕ µ θtq t Tνu τ Φ ts=µ s τ Φ t q P µ for ν P Θ k τ Θθ = ν..7 3 Gelfand-Graev characters on arbitrary elements 3. = GLn F q notation In this Section 3 let Φ = { F -orbits in F q }. Define norm maps Ñmk : T m T k whenever k m the same as in.8 except by replacing q by q and define the corresponding transpose maps Ñ mk : T m T k as in.9 where T m is the character group of T m. We now let L be the direct limit of the groups T m with respect to the maps Ñ mk : L = lim T m and since F acts on L we may consider the corresponding orbits and we define Θ = { F -orbits in L}. The same set-up of sections.4 and.5 gives a characteristic map for = GLn F q by replacing Φ by Φ Θ by Θ q by q T k by T k and n/ +nλ s λ by s λ. With the exception of the Deligne-Lusztig characters which follows from the parallel argument of [ Theorem 4.] this can be found in [6 Chapter IV]. 3. The Gelfand-Graev character We will use U n = GLn F q F see.7 to give an explicit description of the Gelfand-Graev character. For a more general description see [4] for example. 8

9 For i < j n and t F q let x ij t denote the matrix with ones on the diagonal t in the ith row and jth column and zeroes elsewhere. Let u ij t = x ij tx n+ jn+ i t q for i < j n/ t F q u in+ j t = x in+ j tx jn+ i t q for i < j n/ t F q and for k n/ and t a b F q let u k a = x kn+ k a for n even and a q + a = 0 u k a b = x n/ n+ k a q x kn+ k bx k n/ a for n odd and a q+ + b + b q = 0. Examples. In U 4 we have u t = t t q u 3 t = where a q + a = 0. In U 5 we have u t = u a b = 0 t t q u a = t u t q 4 t = a 0 b a q u a b = 0 0 a u a = 0 0 t t q a b a q where a q+ + b + b q = 0. For i < j n/ and k n/ define the one-parameter subgroups so that X ij = {u ij t t F q } = F + q X in+ j = {u in+ j t t F q } = F + q { {uk t t F X k = q t q + t = 0} if n is even {u k a b a b F q a q+ + b + b q = 0} if n is odd. B < n = X ij X in j X k i < j n/ k n/ U n a is the subgroup of U n of upper-triangular matrices with ones on the diagonal. Noting that a direct calculation gives { X k /[X k X k ] F + q if n is even = if n is odd { B n < /[B n < B n < ] = X X 3... X n/ n/ X n/ F + n/ F + = q q if n is even F + n/ if n is odd. q F + q Similarly let B < n = x ij t i < j l t F q be the subgroup of unipotent upper-triangular matrices in. 9

10 Fix a homomorphism ψ : F + C of the additive group of the field such that for all q k n/ ψ is nontrivial on X k /[X k X k ]. Define the homomorphism ψ n : B n < C by { ψ if α = i i + i < n/ or if α = n/ ψ n = Xα/[X αx α] otherwise. The Gelfand-Graev character of U n is Γ n = Ind U n ψ B n < n. Recall that U n is conjugate to U n in Ḡn. If U n = yu n y then let Γ n = Ind Un y B < n y y ψ n y. Similarly the Gelfand-Graev character Γ n of is where ψ n : B < n C is given by Γ n = Ind Gn ψ B n < n. ψ n x ij t = δ ji+ ψt. It is well-known that the Gelfand-Graev character has a multiplicity free decomposition into irreducible characters [ 4 5]. The following explicit decompositions essentially follow from [3]. Specific proofs are given in [6] in the case and in [7] in the U n case. Theorem 3.. Let htλ = max{lλ ϕ }. Then Γ n = χ λ and Γn = χ λ. λ P Θ n htλ= λ P Θ n htλ= 3.3 The character values of the Gelfand-Graev character A unipotent conjugacy class K µ of U n or is a conjugacy class that satisfies µ u = µ. The unipotent conjugacy classes of U n and are thus parameterized by partitions µ of n. Lemma 3.. a Let µ P Θ n µ {} u b Let µ P Θ n µ {} u = µ. Then { n+ n/ lν Γ n µ = ν P n z ν T ν Q µ ν q if µ is unipotent 0 otherwise. = µ. Then { n lν Γ n µ = ν P n z ν T ν Q µ ν q if µ is unipotent 0 otherwise. 0

11 Proof. Note that if htλ then nλ = 0. Thus by applying the characteristic map and.6 to Theorem 3. chγ n = n/ λ P Θ n htλ ν P Θ n νs=λs ω λϕ ν ϕ p ν. z ν ϕ Since htλ ω λϕ is the trivial character for all ϕ Θ. Thus the summand is independent of λ and chγ n = n/ z p ν ϕ ν. 3. By.3 chγ n = n/ ϕ lνϕ i ν P Θ n = n/ ν P Θ n = n/ θ HomTνu C τ Θ θ=ν ν P n θ HomT νc The change of basis.7 gives chγ n = n/ = n/ z ν u p ν zν p τ Θ θ. ν P n θ HomT νc µ P Φ n ν P Θ n mi ν {} u! m i/ ϕ ν ϕ! n lν z ν ν P n n lν z ν t Tν τ Φ ts=µ s µ P Φ n θ HomTνu C τ Θ θ=ν θtq t Tν τ Φ ts=µ s θ HomT νc z ν ϕ p ν µ τ Φ t qp µ θtq µ τ Φ t qp µ. By the orthogonality of characters of T ν the inner-most sum is equal to zero for all t. If t = then τ Φ... f = for f {} and τ Φ... {} = ν. Thus chγ n = n/ and in particular if µ {} u = µ µ P Φ n µ u =µ n lν z ν ν P n T ν Q µ u ν qp µ { n+ n/ lν Γ n µ = ν P n z ν T ν Q µ ν q if µ is unipotent 0 otherwise. b The proof is similar to a just using the characteristic map. Remark. In the proof of Lemma 3. one may skip to 3. by using in [3]. The values of the Gelfand-Graev character of the finite general linear group are well-known. An elementary proof of the following Theorem is given in [9].

12 Theorem 3.. Let µ P Φ n with µ = µ {} u. Then { Γ n µ = n lµ lµ i= q i if µ is unipotent 0 otherwise. We may now apply Theorem 3. and Lemma 3. to give the values of the Gelfand-Graev character of U n. Corollary 3.. Let µ Pn Φ with µ = µ {} u. Then { Γ n µ = n/ lµ lµ i= q i if µ is unipotent 0 otherwise. Proof. Combine Lemma 3. b with Theorem 3. to get lµ lµ i= q i = which implies lµ q i = z i= ν ν P n Make the substitution q q to get which implies lµ i= lµ n/ +lµ q i = ν P n z ν i= q i = lν z ν ν P n lν T ν Q µ ν q q ν i Q µ ν q. i= lν q ν i Q µ ν q i= n/ + ν lν z ν ν P n Apply this last identity to Lemma 3. a to obtain the desired result. T ν Q µ ν q. 4 Degenerate Gelfand-Graev characters 4. = GLn F q notation different from Section 3 In this Section 4 let = GLn F q and define Φ = {F -orbits of F q }. Note that now G F n = U n and G F n = U n and also that T m = T m. Through the norm maps Ñ mk : T m T k where k m defined in.8 and the corresponding transpose maps Ñmk : T m T k defined in.9 we let L be the direct limit of the groups Tm with respect to the maps Nmk : L = lim T m. Since F = F acts on L we may consider the corresponding orbits and define Θ = {F -orbits in L}. The same set-up of sections.4 and.5 gives a characteristic map for by replacing Φ by Φ Θ by Θ q by q T k by T k and n/ +nλ s λ by s λ.

13 4. Degenerate Gelfand-Graev characters Let k ν be a pair such that ν n k Z 0 and let ν = ν ν... ν l where ν j = ν + ν + + ν j. Then the map ψ kν : B < n C given by ψ if α = i i + i < n/ and i / ν ψ kν = ψ if α = n/ and n/ / ν Xα/[X αx α] otherwise is a linear character of U n. Note that ψ n/ n/ n/ is the trivial character and ψ n = ψ n of Section 3. The degenerate Gelfand-Graev character Γ kν is Γ kν = Ind U n ψ B n < kν = Ind Un yψ yb n < y kν y where y is an element of Ḡn such that yu ny = U n. In particular the Gelfand-Graev character is Γ n. Let L kν = L k L ν L ν L l ν where L k = X ij X in+ j X r ν < i < j ν + k ν r ν + k = Uk F q L r ν = X ij ν r i < j ν r = GLν r F q. Then L kν = Uk F q GLν F q GLν l F q is a maximally split Levi subgroup of U n. For example if n = 9 k = 3 and ν = then L kν = Note that since L i ν A B C F B F C U ν i = Uνi the Levi subgroup A GL F q B GL F q C U3 F q. U kν = U k U ν U ν U νl U n contains a Levi subgroup L = U k L L l with L i U νi such that L = L kν. Proposition 4.. Let k ν be such that ν n k Z 0. Then chγ kν = ch Γ k ch R U ν G ν Γ ν ch R U ν G ν Γ ν ch R U ν l G νl Γ νl where Γ m is the Gelfand-Graev character of G m = GLm F q. 3

14 This proposition is a consequence of Theorem. and the following lemma. Lemma 4.. Let k ν be such that ν n k Z 0. Then Γ kν = R U n U Γk kν R U ν Γ ν R U ν l L l Γ νl. L Proof. Since L kν is maximally split Ind Un yψ yb n < y kν y = Ind U n ψ B n < kν = Indf U n L kνγ k Γ ν Γ νl. where Indf G L is Harish-Chandra induction. However Indf U n Γ L k Γ ν Γ νl = R U n L kν kνγ k Γ ν Γ νl = R Un L Γ k Γ ν Γ νl. By transitivity of Deligne-Lusztig induction we now have Ind Un yb < n y yψ kν y = R Un U kν Γk R U ν L Γ ν R U ν l L l Γ νl. 4.3 Symplectic tableaux and domino tableaux combinatorics Augment the nonnegative integers by symbols {ī i Z >0 } so that we have { } and order this set by i < ī < i < i +. Alternatively one could identify this augmented set with Z 0 by ī = i. Let λ = λ λ... λ r be a partition of n and m 0 m m... m l be a sequence of nonnegative integers that sum to n with m 0 λ. A symplectic tableau Q of shape λ/m 0 and weight m 0 m... m l is a column strict filling of the boxes of λ by symbols such that m i = {0... l l} { number of 0 s in Q if i = 0 number of ī s + number of i s in Q if i > 0. We write shq = λ/m 0 and wtq = m 0 m... m l. For example if Let Q = then shq = and wtq = 3 T λ m 0 m...m l = { symplectic tableaux of shape λ/m0 and weight m 0 m... m l }. 4. A tiling of λ by dominoes is a partition of the boxes of λ into pairs of adjacent boxes. For example if λ = then 4

15 is a tiling of λ by dominoes. Let m 0 m... m l be a sequence of nonnegative integers such that m 0 λ and λ = m 0 + m + + m l. A domino tableau Q of shape λ/m 0 = shq and weight m 0 m... m l = wtq is a column strict filling of a tiling of the shape λ/m 0 by dominoes where we put 0 s in the non-tiled boxes of λ and where m i is the number of i s which appear. For example if Q = then shq = and wtq =. Let D λ m 0 m...m l = { domino tableaux of shape λ/m0 and weight m 0 m... m l }. 4. In the following Lemma a is a straightforward use of the usual Pieri rule and b is both similar to and perhaps a special case of [4 Theorem 6.3] and also related to a Pieri formula in []. Lemma 4.. Let m 0 m... m l be an l + -tuple of nonnegative integers which sum to n. Then a s m0 b s m0 l m r s i s mr i = Tm λ 0 m...m l s λ λ P n r= i=0 l m r i s i s mr i = r= i=0 Proof. a Note that λ P n m0 nλ D λ m 0 m...m l s λ. s m0 r= i=0 l m r s i s mr i = 0 ir mr r l s m0 r= l s irs mr ir. Now repeated applications of Pieri s rule implies the result. b Note that s m0 r= i=0 l m r i s i s mr i = 0 ir mr r l i + +i l s m0 l s irs mr ir. r= By Pieri s rule 0 ir mr r l s m0 r= l s irs mr ir = λ P n m0 Number of column strict fillings of λ using m 0 0 s and for r =... l using i r r s and m r i r r s. s λ. By observing that the sign counts the number of barred entries s m0 r= i=0 l m r i s i s mr i = λ P n m0 Q T λ m 0 m...m l Number of barred entries in Q s λ

16 We therefore need to determine the cancellations for a given shape λ. Fix r {... l} and λ P such that Tm λ 0 m...m l. For a tableau Q Tm λ 0 m...m l let For example if Q r = skew tableaux consisting of the boxes in Q containing r or r S r Q = {column strict fillings of shq r by elements in { r r}}. Q = then Q = and S Q. In fact S Q = 8. In light of 4.3 b is equivalent to { Number of r s in nshq r if shq Q = r has a domino tiling Q S r Q 0 otherwise. Note that in row j Q S r Q is of the form dj d j+ d j {}}{{}}{ r r?? { }} { r r?? r r?? r r row j row j row j + Thus we have d j + choices for the values in row j. If the total number of choices is even then exactly half of these choices give a positive sign and half give a negative sign. So we have Number of r s in Q = 0 Q S r Q unless d j is even for all rows j that is shq r has a tiling by dominoes. In this case the signs of all but one of the possible tableaux will cancel each other out so the only tableau that we have to count has row j of the form d j d j {}}{ d j+ {}}{ r r r r {}}{ r r r r r r r r r r row j row j row j + which can clearly be tiled by dominoes of the form r and r. For this tableau we have Number of r s = nshqr. Thus as desired. s m0 r= i=0 l m r i s i s mr i = λ P n m0 nλ D λ m 0 m...m l s λ 6

17 Let λ P Θ and γ P Θ be such that htγ and γ ϕ λ ϕ for all ϕ Θ. A battery Θ-tableau Q of shape λ/γ is a sequence of tableaux indexed by Θ such that { Q ϕ a domino tableau of shape λ ϕ /γ = ϕ if ϕ is odd a symplectic tableau of shape λ ϕ /γ ϕ if ϕ is even. The weight of Q is wtq = wtq wtq... where Let wtq i = ϕ odd ϕ wtq ϕ i + ϕ even ϕ wtqϕ i. Bkν λ = {Q battery tableaux shq = λ/γ γ PΘ k htγ wtq = ν}. 4.4 Example. If λ = ϕ ϕ ϕ 3 where ϕ i = i then B λ 54 contains ϕ ϕ ϕ ϕ ϕ 3 ϕ 0 0 ϕ ϕ 3 ϕ 0 0 ϕ 0 0 ϕ ϕ ϕ 3 ϕ 0 0 ϕ ϕ 3 ϕ 0 0 ϕ ϕ 3. ϕ 3 ϕ 3 Some intuition. If λ P Θ we can think of the boxes in λ ϕ as being ϕ deep so in the above example ϕ λ = ϕ ϕ 3. A battery Θ-tableau is a way of stuffing the slots by numbered batteries where front and back are distinguished by i and ī but the sides look generically like i so ī i i Then a battery Θ-tableau might look like: ϕ 0 0 i i i ϕ ϕ 3 so the weight of the tableau counts the number of batteries of a given type get used regardless of the cardinality of ϕ. 7

18 4.4 Inducing from to U n Note that any maximal torus T ν of U n becomes the maximal torus T ν of U n which gives rise to the map i : Pairs T ν θ ν with T ν a maximal torus of θ ν Hom T ν C Pairs T ν θ ν with T ν a maximal torus of U n θ ν HomT ν C T ν θ ν T ν θ ν. To translate the combinatorics between and U n we define the map { ι : P Θ n Pn Θ where for ϕ Θ ι λ ϕ λ ϕ = λ ι λ λ ϕ F ϕ λ if if ϕ = ϕ ϕ = ϕ F ϕ which has the property that τ Θ i = ι τ Θ see.. The map ι is neither surjective nor injective. We note that F ϕ = ϕ implies that ϕ is odd and if F ϕ ϕ then ϕ = ϕ F ϕ implies ϕ is even see [5]. Thus the image of ι is the set of even Θ-partitions Imageι = {λ P Θ n ϕ λ ϕ is even for ϕ Θ}. Theorem 4.. R U n Γ n = λ P Θ n htλ B λ 0n χλ. Proof. Note that by Theorem 3..6 and the characteristic map for Γ n = λ P Θ n ht λ= χ λ = λ P Θ n htλ= ν P λ s n l ν z ν R Gn ν. By transitivity of induction and the fact that τ Θ i = ι τ Θ we have R U n R Gn ν = RU n ι ν and so R U n Γ n = n l ν R U n ι ν. λ P Θ n ht λ= ν P λ s We now change the second sum to a sum over ν = ι ν P ι λ s R U n Γ n = λ P Θ n ht λ= = ν P Θ n ν even ν P ι λ s ν P Θ n ι ν=ν ν P λ s ι ν=ν z ν z ν z ν and we obtain n lν R U n ν n lν R U n ν. 8

19 Recall that F ϕ = ϕ implies that ϕ is odd and F ϕ ϕ implies that ϕ = ϕ F ϕ where ϕ is even. Apply the characteristic map factor and then reindex to obtain chr U n Γ n = n p ν Note that by. ν P Θ n ν even = n ν P Θ n ν even = n ηµ P η + µ = γ γ P Θ n htγ= γ even z η z µ p η µ = ν P Θ ι ν=ν ϕ odd ϕ odd z ν z ν ϕ / γ i=0 ν P ν = γ ϕ ν even η =i p ν ϕy ϕ A computation similar to [6 I..4] shows that Thus ν = γ ν even chr U n Γ n = n z ν/ p ν = γ P Θ n htγ= γ even ϕ even z ν/ p ν Y ϕ z η p η Lemma 4. a and b respectively imply that γ µ = γ i νµ P η µ=ν ϕ ϕ even z µ p µ = i s i s γ i. i=0 γ ϕ z η z µ ηµ P η + µ = γ ϕ γ p ν ϕy ϕ s i s γ i. i=0 p η µ Y ϕ. z η z µ ϕ i s i Y ϕ s γ ϕ i Y ϕ. 4.5 i=0 k s i s k i = T0k λ s λ and i=0 λ P k k i s i s k i = nλ D0k/ λ s λ. λ P k i=0 Since D0k/ λ = T 0k λ = 0 unless htλ chr U n Γ n = n = λ P Θ n htλ γ P Θ n htγ= γ even ϕ odd λ ϕ = γ ϕ n+nλ B λ 0n s λ. Apply ch to get the result. nλϕ D λϕ 0 γ ϕ / s λ ϕy ϕ ϕ even T λϕ 0 γ ϕ λ ϕ = γ ϕ s λ ϕy ϕ 9

20 Corollary 4.. For n Z chr U n Γ n = n ν P Θ n htν= ν even Proof. This is 4.5 in the proof of Theorem 4.. ν ϕ i ϕ s i Y ϕ s ν ϕ i Y ϕ. i=0 Using similar techniques we can prove a result for arbitrary irreducible characters of. For λ P Θ and γ P ι λ s let c γ λ = c γ λϕ where c γ λϕ = c γϕ λ ϕ c γϕ λ ϕ λf ϕ if ϕ = ϕ Θ if ϕ = ϕ F ϕ and F ϕ ϕ Θ where c λ νµ is as in.4 and c γ λ is as in.5. Theorem 4.. Let λ P Θ n. Then R U n χ λ = γ P ι λ s Proof. By.6 and the characteristic map for nγ c γ λχ γ. χ λ = ν P λ s ϕ Θ ω λ ϕ ν ϕ z ν ϕ n l ν R Gn ν. By transitivity of induction and the fact that τ Θ i = ι τ Θ we have R U n R Gn ν = RU n ι ν and so R U n χ λ = ω λ ϕ ν ϕ n l ν R U n ι ν. ν P λ s ϕ Θ We now change the sum to a sum over ν = ι ν P ι λ s and using the image of the map ι we obtain R U n χ λ = ν P ι λ s ν even ν P λ s ι ν=ν z ν ϕ ω λ ϕ ν ϕ z ν ϕ n lν R U n ν. Apply the characteristic map and rewrite the inner sum and product to get ch R U n χ λ = n ν P ι λ s ν even = n ν P ι λ s ν even ν P λ s ι ν=ν ϕ Θ F ϕ= ϕ ϕ Θ ω λ ϕ ν ϕ p ν z ν ϕ ω λ ϕ ν ϕ / z ν ϕ / ϕ Θ F ϕ ϕ γ = λ ϕ µ = λ F ϕ γ µ=ν ϕ F ϕ ω λ ϕ γω λ F ϕ µ p ν. z γ z µ 0

21 Recall that F ϕ = ϕ implies that ϕ is odd and F ϕ ϕ implies that ϕ = ϕ F ϕ where ϕ is even. Thus for every ϕ Θ such that ν ϕ if ϕ is odd then ϕ = ϕ for some ϕ Θ and if ϕ is even then ϕ = ϕ F ϕ for some ϕ Θ. Factor our expression accordingly as ch R U n χ λ = n = n ϕ= ϕ ν = λ ϕ ν P ι λ s ν even ω λ ϕ ν z ν γ = λ µ = η γ µ=ν ϕ= ϕ ω λ ϕ ν ϕ / p z ν ϕ ν ϕ / p ν Y ϕ ϕ= ϕ F ϕ ϕ= ϕ F ϕ ν = λ ϕ + λ F ϕ γ = λ ϕ µ = λ F ϕ γ µ=ν ϕ γ = λ ϕ µ = λ F ϕ γ µ=ν ω λ ϕ γω λ F ϕ µ p z γ z ν ϕ µ ω λ ϕ γω λ F ϕ µ p ν Y ϕ. z γ z µ The first product is the case that ϕ is odd and the second product is the case that ϕ is even. For the sum in the first product note that ω λ γω η µ ω λ γ ω η µ p ν = p γ p µ z γ z µ z γ z µ ν = λ + η γ = λ µ = η For the sum in the product for ϕ even we have ν = λ = s λ s η. ω λ ν p ν = ω λ ν p ν p z ν z ν ν = λ = s λ p where is the plethysm product.5. Thus from the definition of the coefficients c γ λ we have as desired. ch R U n χ λ = n γ P ι λ s γ c λ s γ It is perhaps worth noting that since we know the Harish-Chandra induction R U n χ λ gives a character then the sign of the coefficient c γ λ must be nγ. 4.5 Degenerate Gelfand-Graev characters The following theorem is our main theorem of Section 4. Theorem 4.3. Let n Z and let k ν satisfy ν n k Z 0. Then Γ kν = Bkν λ χλ. λ P Θ n Proof. Recall that by Proposition 4. chγ kν = ch Γ k ch R U ν G ν Γ ν ch R U ν G ν Γ ν ch R U ν l G νl Γ νl.

22 By Theorem 3. and Corollary 4. Thus chγ kν = n/ chγ k = k/ chr U r G r Γ r = r γ 0 P Θ k htγ 0 = r lν γ r P Θ νr htγ r = γ r even γ P Θ r htγ= γ even γ P Θ k htγ= s γ ϕy ϕ and s γ ϕ 0 γ ϕ i ϕ s i Y ϕ s γ ϕ i Y ϕ. i=0 lν r= γ ϕ r Y ϕ Fix ϕ Θ and let m r = γ ϕ r. If ϕ is even then Lemma 4. a implies lν m r s i s mr i = s m0 r= i=0 If ϕ is odd then Lemma 4. b implies Therefore lν m r i s i s mr i = s m0 r= i=0 chγ kν = n/ γ 0 P Θ k htγ 0 = = n/ λ P Θ n r lν γ r P Θ νr htγ r = γ r even γ 0 P Θ k htγ 0 = ϕ odd r lν γ r P Θ νr htγ r = γ r even = n/ +nλ Bkν λ s λ. λ P Θ n The result follows by applying ch. λ =m 0 + +m lν i=0 T λ λ =m 0 + +m lν nλ λ ϕ nλϕ ϕ odd 5 Some multiplicity consequences D λϕ γ ϕ 0 γ ϕ ϕ even λ ϕ i ϕ s i Y ϕ s γ ϕ r i Y ϕ. m 0...m lν s λ. D λ m0 m /...m lν / sλ. /... T λϕ nλϕ D λϕ γ ϕ 0 γ ϕ s λ ϕy ϕ γ ϕ 0 γ ϕ... /... s λ ϕy ϕ T λϕ γ ϕ 0 γ ϕ ϕ even In this section we explore some of the multiplicity implications of Theorem s λ

23 5. A bijection between domino tableaux and pairs of column strict tableaux The -core of a partition λ P which we denote core λ is the partition of minimal size such that the skew partition λ/core λ may be tiled by dominoes. It is not difficult to see that the -core of any partition is always of the form m m... for some nonnegative integer m where 0 is the empty partition. The -quotient of a partition λ quot λ is a pair of partitions quot λ 0 quot λ defined in [6 I. Example 8]. We define quot λ i = quot λ 0 i + quot λ i. Also define the content of a box in the ith row and jth column of a partition λ to be j i. Let λ P n with core λ {0 }. Consider the bijection Pairs of column strict D core λ λ m...m l tableaux of shape quot λ 5. and weight m m... m l Q Q 0 Q given by the following algorithm which originally appeared in [0] and is in a more general form in [3]. Each domino in Q covers two boxes of λ/core λ. Move the entries in Q to the box that has content 0 modulo. Q = Let S 0 denote the set of all dominoes that have the entry in the lower or leftmost box and S be the set of dominoes that have the entry in the upper or rightmost box. S 0 = and S = For even lλ < i < λ and j {0 } let D j i = The increasing sequence of entries whose content is i and whose domino is in S j. D 0 4 D0 D0 0 D0 D0 4 D 4 D D 0 D D 4 = 5 6 = Let Q j be the unique tableau that has increasing diagonal sequences given by the D j i for all even lλ < i < λ. Q 0 = and Q =

24 Remarks.. If the shape of the domino tableau Q is λ/core λ then the shape of Q 0 Q is quot λ.. We may apply this algorithm to a domino tableau of shape λ/m with m core λ mod by requiring that the tableau of shape λ/core λ has m/ horizontal dominoes filled with zeroes. For example Q = has shape λ/5 so apply the algorithm to Note that all of the zero dominoes are in the same set S core λ so changing m corresponds to adding or subtracting the number of zeroes in the first row of Q core λ. We will use the lexicographic total ordering on partitions given by λ µ if there exists k Z such that λ k < µ k and λ i = µ i for i < k. 5. Lemma 5.. Let λ P n be such that core λ {0 } and let 0 m λ be such that m core λ mod. Then there exists a lexicographically maximal weight µ = µ µ... µ l such that there exists exactly one domino tableau of shape λ/m and weight m µ... µ l. Proof. First suppose λ 0 /γ 0 λ /γ is a pair of skew partitions. Let µ be the maximal number of s we can put in a tableau of shape λ 0 /γ 0 λ /γ µ be the maximal number of s we can thereafter fill into λ 0 /γ 0 λ /γ and µ j be the maximal number of j s we can fill given that we have filled in a maximum number at each step up to j. Then there is exactly one tableau Q 0 Q of shape λ 0 /γ 0 λ /γ and weight µ and this weight is lexicographically maximal. The result now follows from pulling back Q 0 Q through the bijection 5. to get a domino tableau of the same weight along with the second remark preceding this Lemma. Remark. If m = core λ then µ is given by µ 0 = core λ and µ i = quot λ i for i. 5. Multiplicity results Our first consequence of Theorem 4.3 characterizes which irreducible characters of Un F q appear with nonzero multiplicity in some degenerate Gelfand-Graev character. We note that the following could also be obtained by the description of such characters given by Kotlar in [ Corollary.6] based on Harish-Chandra series. Corollary 5.. The set of all λ Pn Θ such that the character χ λ of Un F q satisfies χ λ Γ kν 0 for some degenerate Gelfand-Graev character Γ kν is {λ Pn Θ core λ ϕ {0 } whenever ϕ is odd }. Proof. By Theorem 4.3 the irreducible character χ λ appears with nonzero multiplicity in some degenerate Gelfand-Graev character if and only if there a battery tableau of shape λ/γ for some γ P Θ with htγ. If for some odd ϕ Θ we have core λ ϕ / {0 } then the -core of λ ϕ has at least two parts. But then there is no choice of γ ϕ that allows us to tile λ ϕ /γ ϕ by dominoes. On the other hand if core λ ϕ = 0 we can choose γ ϕ = 0 and if core λ ϕ = we can let γ ϕ = and λ ϕ /γ ϕ can be tiled by dominoes. 4

25 We now specify multiplicities of certain characters χ λ in degenerate Gelfand-Graev characters. Theorem 5.. Let λ Pn Θ be such that core λ ϕ {0 } whenever ϕ is odd. Then there exists ν n k such that Γ kν χ λ = ϕ λ i λ ϕ i+ +. i odd ϕ even Proof. Let k = ϕ odd ϕ core λ ϕ and define γ by { γ ϕ core λ = ϕ if ϕ is odd otherwise. Since γ = k by Theorem 4.3 and Corollary 5. it suffices to find ν n k such that there exist ϕ λ i λ ϕ i+ + ϕ even i odd battery tableaux with shape λ/γ and weight k ν. We construct the battery tableau Q as follows. For odd ϕ Θ let Q ϕ be the unique domino tableau of shape λ ϕ /core λ ϕ and weight core λ ϕ quot λ ϕ quot λ ϕ... obtained from Lemma 5. see in particular the remark after the lemma. For even ϕ Θ and for each i we fill the i st row of λ ϕ with ī s and the ith row with i s. The resulting symplectic tableau Q ϕ has weight λ ϕ + λ ϕ λϕ 3 + λ ϕ Note that we may change up to λ ϕ i λϕ i unchanged. We therefore have exactly of the ī s to i s in row i while leaving the weight λ ϕ i i odd λ ϕ i + symplectic tableaux of shape λ ϕ and weight λ ϕ + λ ϕ λϕ 3 + λ ϕ We combine these to create a battery tableau of shape λ/γ and weight ν where ν i = ϕ odd ϕ quot λ ϕ i + ϕ even ϕ ϕ λ i + λ ϕ i. Note that from this construction ν is the maximal weight under the lexicographical ordering 5. of a battery tableau of shape λ/γ while each γ ϕ is chosen minimally. It follows that the weight ν will change if we change the weight of any Q ϕ. For example if ϕ ϕ ϕ 3 λ = where ϕ i = i then k = ν = = 9 4 and every battery tableau of shape λ and weight k ν must be of the form 0 ϕ There are 3 = 6 of such tableaux. ϕ ϕ 3 where ï {ī i}. 5

26 Theorem 5. and its proof give the following multiplicity one result. Corollary 5.. Let λ P Θ n. Suppose core λ ϕ {0 } whenever ϕ is odd and for all i > 0 the multiplicities m i λ ϕ are even for all even ϕ Θ. Then Γ kν χ λ = where k = ϕ odd ϕ core λ ϕ and ν i = ϕ odd The next theorem generalizes Corollary 5.. ϕ quot λ ϕ i + ϕ even ϕ ϕ λ i + λ ϕ i. Theorem 5.. Suppose λ P Θ n satisfies the following: a Whenever ϕ is odd core λ ϕ {0 } b Whenever ϕ is even the partition λ ϕ has at most one nonzero part with odd multiplicity c There exists an r > 0 such that for every ϕ Θ with ϕ even either lλ ϕ < r and λ ϕ has no nonzero part of odd multiplicity or λ ϕ r has odd multiplicity and λ ϕ r < λ ϕ r. Then the irreducible character χ λ of U n appears with multiplicity one in some degenerate Gelfand- Graev character of U n. Proof. Suppose λ P Θ n satisfies a b and c. Let γ P Θ be given by { i if ϕ is even and mi γ ϕ λ ϕ is odd = core λ + quot λ core λ r/ if ϕ is odd and λ = λ ϕ. For example the Θ-partition λ = ϕ ϕ ϕ 4 with ϕ i = i satisfies a b and c with r = 5. Since quot λ ϕ 0 = and core λ ϕ = as in the example for 5. we have that core λ ϕ + quot λ ϕ r/ = 3 and γ = Consider the following battery tableau Q of shape λ/γ. ϕ ϕ.. For ϕ Θ such that ϕ is even fill λ ϕ /γ ϕ with λ ϕ j j s and λϕ j j s for j lλ ϕ such that j < r and λ ϕ j j s and λ ϕ j s j+ for j + lλ ϕ such that j > r. Then all of the nonzero entries come in pairs j and the resulting weight is lexicographically j maximal.. For ϕ Θ such that ϕ is odd use Lemma 5. to fill λ ϕ /γ ϕ in a lexicographically maximal way. 6

27 In our running example we have Q = ϕ Note that by Lemma 5. Q is the only battery tableau of shape λ/γ and weight wtq. Thus it suffices to show that there is no ν λ with ν = γ and htν such that there exists a battery tableau P of shape λ/ν and weight wtq. Since ν = γ we may think of moving from Q to P by shifting zero entries between ϕ Θ in Q. If ϕ is even it is clear from the construction of Q ϕ that if we add a zero an entry < r/ is lost while if we remove a zero an entry > r/ is gained. Now consider when ϕ is odd with Q = Q ϕ and λ = λ ϕ. Apply the bijection 5. to the domino tableau Q and notice that from Remark preceding Lemma 5. our choice of γ ϕ forces Q coreλ to have exactly quot λ core λ r/ 0 ϕ ϕ 4 0 s. Now adding a pair of zero entries to or removing a pair of zero entries from Q is the same as adding a zero to or removing a zero from Q core λ. It is clear that adding a zero to Q core λ results in losing an entry < r/ which removes a domino with entry < r/ in Q and removing a zero from Q core λ results in gaining an entry > r/ which adds a domino with entry > r/ in Q. Thus no matter how we change γ to ν we are forced to change the weight of the full battery tableau to a lexicographically smaller weight. So there is no such ν which leaves the weight unchanged and uniqueness follows. Remarks. Corollary 5. follows from Theorem 5. since a and b are easily satisfied and r = max{lλ ϕ ϕ Θ ϕ even} +. Another consequence of Theorem 5. is a result by Ohmori [7]. Corollary 5.3 Ohmori. Let λ P Θ n and define the partition µ to have parts µ j = ϕ λ ϕ j. Suppose that µ = m m... is such that m i is even for all i except for the one value i = k or that m i is always even in which case we let k = 0. Define the partition ν to be ν = m / m / k m k /. Then the irreducible character χ λ appears with multiplicity one in the degenerate Gelfand-Graev character Γ kν. Proof. Note that if λ Pn Θ satisfies the hypotheses of the corollary then for any ϕ Θ the partition λ ϕ has at most one nonzero part size with odd multiplicity otherwise µ would have more parts with odd multiplicity. Thus λ satisfies condition b of Theorem 5.. Moreover the fact that µ has at most one part with odd multiplicity implies that there must be an r > 0 such that for every ϕ Θ either lλ ϕ < r or λ ϕ r has odd multiplicity in λ ϕ and λ ϕ r < λ ϕ r. In particular this holds when ϕ is even and so λ satisfies condition c of Theorem 5.. If ϕ Θ is odd and λ ϕ has a part size i with odd multiplicity where i = 0 if lλ ϕ < r then { core λ ϕ if i is odd = 0 if i is even. Thus λ satisfies a of Theorem 5.. 7

28 Now define γ by γ ϕ = i if m i λ ϕ is odd where i = 0 if lλ ϕ < r. Then γ = k. When ϕ is even fill λ ϕ /γ ϕ just as in the proof of Theorem 5.. When ϕ is odd fill λ ϕ /γ ϕ with all vertical dominoes such that there are λ ϕ j j s for j lλϕ such that j < r and λ ϕ j j s for j lλ ϕ such that j > r. This gives a battery tableau of shape λ/γ where γ = k and weight ν as defined above. Fix a ϕ such that ϕ is odd let λ = λ ϕ and let Q ϕ = Q be the domino tableau just defined and apply the bijection 5. to Q. Since Q has been filled with all vertical dominoes the resulting weight is lexicographically maximal and so by Lemma 5. the tableaux Q 0 and Q obtained from the bijection 5. also have lexicographically maximal weights. Let j = core λ and consider Q j. From our choice of γ ϕ the tableau Q j has exactly λ r / 0 s. By the bijection 5. we also have row r/ of Q j which is quot λ ϕ j r/ is exactly λ r /. This means the domino tableau Q ϕ is exactly what is constructed in the proof of Theorem 5.. Therefore Q is exactly the battery tableau obtained in the proof of Theorem 5. and so we have χ λ Γ kν =. Note that by Corollary 5. condition a of Theorem 5. is a necessary condition. following proposition shows that condition b is also necessary. The Proposition 5.. Let λ P Θ. If there exists a ϕ Θ such that ϕ is even and λ ϕ has at least two distinct part sizes with odd multiplicity then χ λ Γ kν for all degenerate Gelfand-Graev characters Γ kν. Proof. Suppose λ P Θ and ϕ is even such that λ = λ ϕ has part sizes x < y with odd multiplicity. Let Q be a symplectic tableau of shape λ/m for some m λ and suppose wtq = µ. If there exists an ī such that there is no i directly south of ī in Q then there is a second symplectic tableau P of shape λ/m and weight µ obtained by changing this ī to an i in Q. Similarly if there is an i with no ī directly north of it then there is a second tableau P with the same weight and shape as Q. Thus the only way Q is the only tableau of shape λ/m and weight µ is if λ/m can be tiled by vertical dominoes. If m < y then the yth column of λ/m has an odd number of boxes and therefore cannot be tiled by vertical dominoes. If m > y then the mth column of λ/m has an odd number of boxes. If m = y then the xth column of λ/m has an odd number of boxes. In all cases λ/m cannot be tiled by dominoes and the result follows. Remarks.. While conditions a and b of Theorem 5. are necessary condition c is not. For example the only battery tableau of weight 8 for the Θ-partition α λ = β α with α = 4 β = is Q = 0 β.. At the same time conditions a and b of Theorem 5. are not alone sufficient. For example λ = α β with α = β = 8

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