Báo cáo toán học: " Gelfand–Graev characters of the finite unitary groups" pdf

Gelfand–Graev characters of the finite unitary groupsSubmitted: Aug 12, 2009; Accepted: Nov 25, 2009; Published: Nov 30, 2009 Mathematics Subject Classification: 20C33, 05E05 AbstractGel

Gelfand–Graev characters of the finite unitary groups

Submitted: Aug 12, 2009; Accepted: Nov 25, 2009; Published: Nov 30, 2009

Mathematics Subject Classification: 20C33, 05E05

AbstractGelfand–Graev characters and their degenerate counterparts have an importantrole in the representation theory of finite groups of Lie type Using a characteristicmap to translate the character theory of the finite unitary groups into the language

of symmetric functions, we study degenerate Gelfand–Graev characters of the finiteunitary group from a combinatorial point of view In particular, we give the values

of Gelfand–Graev characters at arbitrary elements, recover the decomposition plicities of degenerate Gelfand–Graev characters in terms of tableau combinatorics,and conclude with some multiplicity consequences

Gelfand–Graev modules have played an important role in the representation theory offinite groups of Lie type [4, 7, 22] In particular, if G is a finite group of Lie type, thenGelfand–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 acombinatorial correspondence between characters and symmetric functions (as described

in [23]) to examine the Gelfand–Graev character and its degenerate relatives for the finiteunitary group

Let B< be a maximal unipotent subgroup of a finite group of Lie type G Then theGelfand–Graev character Γ of G is the character obtained by inducing a generic linearcharacter from B< to G The degenerate Gelfand–Graev characters of G are obtained byinducing arbitrary linear characters In the case GL(n, Fq), Zelevinsky [27] described themultiplicities of irreducible characters in degenerate Gelfand–Graev characters by count-ing multi-tableaux of specified shape and weight It is the goal of this paper to describethe degenerate Gelfand–Graev characters of the finite unitary groups in a similar mannerusing tableau combinatorics In [27], Zelevinsky obtained the result that every irreducible

character of GL(n, Fq) appears with multiplicity one in some degenerate Gelfand–Graevcharacter It is known that this multiplicity one result is not true in a general finitegroup of Lie type, and in fact there are characters which do not appear in any degenerateGelfand–Graev character in the general case This result was illustrated by Srinivasan [20]

in the case of the symplectic group Sp(4, Fq), and the work of Kotlar [11] gives a geometricdescription of the irreducible characters which appear in some degenerate Gelfand–Graevcharacter in general type In the finite unitary case, we give a combinatorial descrip-tion 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 withmultiplicity one

In Section 2, 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 developmentgiven in [23] This map translates the Deligne-Lusztig theory of the finite unitary groupinto symmetric functions, which thus translates calculations in representation theory intoalgebraic combinatorics Some of the results in this paper could be obtained, albeit in

a different formulation, by applying Harish-Chandra induction and the representationtheory of Weyl groups However, this approach would not lead us to some of the com-binatorics which we study here For example, we naturally arrive at battery tableaux,which are interesting combinatorial objects in their own right Also, our more classicalapproach gives rise to useful identities in symmetric function theory, such as our Lemma4.2

Section 3 examines the (non-degenerate) Gelfand–Graev character We use a able formula for the character values of the Gelfand–Graev character of GL(n, Fq), given

remark-in Theorem 3.2 (for an elementary proof see [9]), to obtaremark-in the correspondremark-ing formulafor U(n, Fq 2) in Corollary 3.1, which states that if Γ(n) is the Gelfand–Graev character ofU(n, Fq2), and g ∈ U(n, Fq2), then

Section 4 computes the decomposition of degenerate Gelfand–Graev characters in afashion analogous to [27], using tableau combinatorics The main result is Theorem 4.4,which may be summarized as saying that the degenerate Gelfand–Graev character Γ(k,ν)

of U(n, Fq 2) decomposes as

Γ(k,ν) =X

λ

mλχλ,where λ is a multipartition and mλis a nonnegative integer obtained by counting ‘battery

tableaux’ of a given weight and shape In the process of proving Theorem 4.4, we obtainsome combinatorial Pieri-type formulas (Lemma 4.2), decompositions of induced charac-ters from GL(n, Fq 2) to U(2n, Fq 2) (Theorem 4.1 and Theorem 4.2), and a description ofall of the cuspidal characters of the finite unitary groups (Theorem 4.3).

Section 5 concludes with a discussion of the multiplicity implications of Section 4 Inparticular, in Theorem 5.2 we give combinatorial conditions on multipartitions λ whichguarantee that the irreducible character χλ appears with multiplicity one in some degen-erate Gelfand–Graev character Our Theorem 5.2 improves a multiplicity one result ofOhmori [18]

Another question one might ask is how the generalized Gelfand–Graev representations

of the finite unitary group decompose Generalized Gelfand–Graev representations, whichwere defined by Kawanaka in [10], are obtained by inducing certain irreducible represen-tations (not necessarily one dimensional) from a unipotent subgroup Rainbolt studiesthe generalized Gelfand–Graev representations of U(3, Fq2) in [19], but in the generalcase they seem to be significantly more complicated than the degenerate Gelfand–Graevrepresentations

Acknowledgements We would like to thank G Malle for suggesting the questions thatled to the results in Section 5, S Assaf for a helpful discussion regarding Section 5.1,

T Lam for helping us connect Lemma 4.2 to the literature, and anonymous referees forhelpful comments

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>1, let kν = (kν1, kν2, ), and if everypart of ν is divisible by k, then we let ν/k = (ν1/k, ν2/k, ) A partition ν is even if νi

is even for 1 6 i 6 ℓ(ν)

2.2 The ring of symmetric functions

Let X = {X1, X2, } be an infinite set of variables and let

Λ(X) = C[p1(X), p2(X), ], where pk(X) = X1k+ X2k+ · · · ,

be the graded C-algebra of symmetric functions in the variables {X1, X2, } For apartition ν = (ν1, ν2, , νℓ) ∈ P, the power-sum symmetric function pν(X) is

pν(X) = pν 1(X)pν 2(X) · · · pν ℓ(X)

The irreducible characters ωλ of Sn are indexed by λ ∈ Pn Let ωλ(ν) be the value of

ωλ on a permutation with cycle type ν

The Schur function sλ(X) is given by

We will also use several product formulas in the ring of symmetric functions Theusual product on Schur functions

sνsµ =X

λ∈P

gives us the Littlewood-Richardson coefficients cλ

νµ The plethysm of pν with pk is

pν ◦ pk= pkν

Thus, we can consider the nonnegative integers cγλ given by

RU(2n,Fq2 ) GL(n,Fq2)(χλ˜) = X

|γ|=2|˜ λ|

cγ˜λχγ,

where RG

H is Harish-Chandra induction

Let ¯Gn = GL(n, ¯Fq) be the general linear group with entries in the algebraic closure ofthe finite field Fq with q elements

For the Frobenius automorphisms ˜F , F, F′ : ¯Gn→ ¯Gn given by

n and Un are conjugate subgroups

Tη = T(η 1 )× T(η2)× · · · × T(ηℓ)

˜

Tη = ˜T(η 1 )× ˜T(η 2 )× · · · × ˜T(η ℓ ).Every maximal torus of Gnis isomorphic to ˜Tη for some η ∈ Pn, and every maximal torus

of Un is isomorphic to Tη for some η ∈ Pn

Φ = {F -orbits of ¯F×

q},and note that ¯G1 =Sf ∈Φf = SkT(k) In particular, we may view ¯G1 as a direct limit ofthe T(k) with respect to inclusion We also have norm maps, Nm,k, whenever k|m,

Nm,k : T(m) −→ T(k)

α 7→ Q(m/k)−1i=0 α(−q) ki , where m, k ∈ Z>1, k|m (2.8)When k|m, denote by N∗

m,k the transpose of the map Nm,k, which embeds T∗

(k) into T∗

(m)

as follows:

N∗ m,k : T∗

(k) −→ T∗

(m)

ξ 7→ ξ ◦ Nm,k

(2.9)Now, define L to be the direct limit of the groups T∗

(k) with respect to the maps N∗

m,k:

L = lim

−→T(m)∗ Since the map F acts naturally on each T∗

(m), it acts on their direct limit L Note that wemay identify the fixed points LF m

with the character group T∗

(m) Let Θ be the collection

The semisimple part λs of an X -partition λ is the X -partition given by

λ(x)s = (1|λ(x)|), for x ∈ X (2.10)For λ ∈ PX, define the set Pλ

x 6= {1}

Note that we can think of “normal” partitions as X -partitions λ that satisfy λu = λ

By a slight abuse of notation, we will sometimes interchange the multipartition λu andthe partition λ({1})u For example, Tλ u will denote the torus corresponding to the partition

λ({1})u

Given the torus Tη, η = (η1, η2, , ηℓ) ∈ Pn, there is a natural surjection

τΘ: {θ = θ1⊗ θ2⊗ · · · ⊗ θℓ ∈ Hom(Tη, C×)} −→ {ν ∈ PΘ | ν({1})u = η}

θ = θ1⊗ θ2⊗ · · · ⊗ θℓ 7→ τΘ(θ), (2.12)where

ϕ∈Θ(mi/|ϕ|(ν(ϕ)))!. (2.13)The conjugacy classes Kµ of Un are parametrized by µ ∈ PΦ

n, a fact on which weelaborate in Section 2.5 We have another natural surjection,

τΦ : Tη → {ν ∈ PΦ | ν({1})u = η}

t = (t1, t2, , tℓ) 7→ τΦ(t1) ∪ τΦ(t2) ∪ · · · ∪ τΦ(tℓ), (2.14)where

τΦ(ti) = µ′, if ti ∈ Kµin Uη i

2.5 The characteristic map

For every f ∈ Φ, let X(f ) = {X1(f ), X2(f ), } be an infinite set of variables, and for every

ϕ ∈ Θ, let Y(ϕ) = {Y1(ϕ), Y2(ϕ), } be an infinite set of variables We relate symmetricfunctions in the variables X(f ) to those in the variables Y(ϕ) through the transform

The conjugacy classes Kµof Un are indexed by µ ∈ PΦ

n and the irreducible characters

κµ(g) =



1 if g ∈ Kµ

0 otherwise

We let χλ(µ) denote the value of the character χλ on any element in the conjugacy Kµ.For ν ∈ PΘ

n, let the Deligne-Lusztig character Rν = RU n

ν be given by

Rν = RUn

Tν u(θ)where θ ∈ Hom(Tν u, C×) is any homomorphism such that τΘ(θ) = ν (see (2.12))

χλ(µ)Pµ for λ ∈ PkΘ, (2.15)

sλ= X

ν∈P Θ k

λ s =ν s

Y

X

3 Gelfand–Graev characters on arbitrary elements

(k) as in(2.9), where ˜T(m)∗ is the character group of ˜T(m) We now let ˜L be the direct limit of thegroups ˜T(m) with respect to the maps ˜Nm,k∗ :

˜

Θ = { ˜F -orbits in ˜L}

The same set-up of Sections 2.4 and 2.5 gives a characteristic map for Gn= GL(n, Fq)

by replacing Φ by ˜Φ, Θ by ˜Θ, −q by q, T(k) by ˜T(k), and (−1)⌊n/2⌋+n(λ)sλby sλ With theexception of the Deligne-Lusztig characters (which follows from the parallel argument of[23, Theorem 4.2]), this can be found in [17, Chapter IV]

We will use U′

n = GL(n, ¯Fq)F ′

(see (2.7)) to give an explicit description of the Gelfand–Graev character For a more general description see [4], for example

For 1 6 i < j 6 n and t ∈ Fq, let xij(t) denote the matrix with ones on the diagonal,

t in the ith row and jth column, and zeroes elsewhere Let

uij(t) = xij(t)xn+1−j,n+1−i(−tq) for 1 6 i < j 6 ⌊n/2⌋, t ∈ Fq2,

ui,n+1−j(t) = xi,n+1−j(t)xj,n+1−i(−tq) for 1 6 i < j 6 ⌊n/2⌋, t ∈ Fq2,

and for 1 6 k 6 ⌊n/2⌋, and t, a, b ∈ Fq 2, let

uk(a) = xk,n+1−k(a) for n even, and aq+ a = 0,

uk(a, b) = x⌈n/2⌉,n+1−k(−aq)xk,n+1−k(b)xk,⌈n/2⌉(a) for n odd, and aq+1+ b + bq = 0.Examples In U4′, we have

Similarly, let

˜

Bn< = hxij(t) | 1 6 i < j 6 n, t ∈ Fqi ⊆ Gn

be the subgroup of unipotent upper-triangular matrices in Gn

Fix a homomorphism ψ : F+q2 → C× of the additive group of the field such that for all

1 6 k 6 ⌊n/2⌋, ψ is nontrivial on Xk/[Xk, Xk] Define the homomorphism ψ(n) : B<

n → Cby

where ˜ψ(n): ˜Bn<→ C is given by

˜

ψ(n)(xij(t)) =

ψ(t) if j = i + 1,

1 otherwise

It is well-known that the Gelfand–Graev character has a multiplicity free position into irreducible characters [22, 25, 26] The following explicit decompositionsessentially follow from [3] Specific proofs are given in [27] in the Gn case and in [18] inthe Un case

decom-Theorem 3.1 Let ht(λ) = max{ℓ(λ(ϕ))} Then

Γ(n) = X

λ∈P Θ n

ht(λ)=1

χλ and Γ˜(n)= X

λ∈P Θ˜n

ht(λ)=1

χλ

A unipotent conjugacy class Kµof Un or Gn is a conjugacy class that satisfies

ht(λ)61

X

ν∈P Θ n

ν s =λ s

Y

Since ht(λ) 6 1, ωλ is the trivial character for all ϕ ∈ Θ Thus, the summand isindependent of λ, and

ch(Γ(n)) = (−1)⌊n/2⌋ X

ν∈P Θ n

Y

By the orthogonality of characters of Tν, the inner-most sum is equal to zero for all t 6= 1

If t = 1, then τΦ(1, 1, , 1)(f ) = ∅ for f 6= {1} and τΦ(1, 1, , 1)({1}) = ν Thus,

ch(Γ(n)) = (−1)⌊n/2⌋ X

µ∈P Φ n

(b) The proof is similar to (a), just using the Gn characteristic map

Remark In the proof of Lemma 3.1, one may skip to (3.1) by using 10.7.3 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]

Apply this last identity to Lemma 3.1 (a) to obtain the desired result.

4.1 Gn = GL(n, Fq 2) notation (different from Section 3)

In this Section 4, let Gn = GL(n, Fq 2), and define

Θ = {F2-orbits in ˜L}

The same set-up of Sections 2.4 and 2.5 gives a characteristic map for Gn by replacing

Φ by ˜Φ, Θ by ˜Θ, −q by q, T(k) by T(2k), and (−1)⌊n/2⌋+n(λ)sλ by sλ

Let (k, ν) be a pair such that ν ⊢ n−k2 ∈ Z>0, and let

ν6= (ν61, ν62, , ν6ℓ), where ν6j = ν1+ ν2+ · · · + νj.Then the map ψ(k,ν) : B<

n → C×, given by

ψ(k,ν)

ny−1 = Un In particular, the Gelfand–Graevcharacter is Γ(n,∅)

Let

L′(k,ν) = hLk, L(1)ν , L(2)ν , · · · , L(ℓ)ν i,where

Lk = hXij, Xi,n+1−j, Xr | |ν| < i < j 6 |ν| + k, |ν| 6 r 6 |ν| + ki ∼= U(k, Fq 2)

L(r)ν = hXij | ν6r−1 6i < j 6 ν6ri ∼= GL(νr, Fq 2)

Then

L′(k,ν) ∼= U(k, Fq 2) ⊕ GL(ν1, Fq2) ⊕ · · · ⊕ GL(νℓ, Fq2)

is a maximally split Levi subgroup of U′

n For example, if n = 9, k = 3, and ν = (2, 1),then

Note that since L(i)ν ⊆ U′



RU2ν1

Gν1 (˜Γ(ν 1 ))

ch

This proposition is a consequence of Theorem 2.1 and the following lemma

Lemma 4.1 Let (k, ν) be such that ν ⊢ n−k2 ∈ Z>0 Then

Γ(k,ν) ∼= RUn

U(k,ν) Γ(k)⊗ RU2ν1

L 1 (˜Γ(ν1)) ⊗ · · · ⊗ RU2νℓ

L ℓ (˜Γ(νℓ))
.Proof Since L′

(k,ν) is maximally split,IndUn

where IndfGL is Harish-Chandra induction However,

IndfUn′

L ′ (k,ν)(Γ(k)⊗ ˜Γ(ν 1 )⊗ · · · ⊗ ˜Γ(ν ℓ )) = RUn′

L ′ (k,ν)(Γ(k)⊗ ˜Γ(ν 1 )⊗ · · · ⊗ ˜Γ(ν ℓ )),

Augment the nonnegative integers by symbols {¯i | i ∈ Z>0}, so that we have

{0, ¯1, 1, ¯2, 2, ¯3, 3, },and order this set by i−1 < ¯i < i < i + 1 Alternatively, one could identify this augmentedset with 12Z>0 by ¯i = i − 12

Let λ = (λ1, λ2, , λr) be a partition of n and (m0, m1, m2, , mℓ) be a sequence

of nonnegative integers that sum to n with m0 6 λ1 A symplectic tableau Q of shapeλ/(m0) and weight (m0, m1, , mℓ) is a column strict filling of the boxes of λ by symbols

{0, ¯1, 1, ¯2, 2, , ¯ℓ, ℓ},

such that

mi =

number of 0’s in Q if i = 0,number of ¯i’s + number of i’s in Q if i > 0

We write sh(Q) = λ/(m0) and wt(Q) = (m0, m1, , mℓ) For example, if

Q = 0 0 ¯1 ¯ 2 ¯1 1 ¯2 4

¯ 3

, then sh(Q) = and wt(Q) = (2, 3, 2, 2, 1)

Let

Tλ (m 0 ,m 1 , ,m ℓ ) =

symplectic tableaux of shape λ/(m0)and weight (m0, m1, , mℓ)

 (4.1)

A tiling of λ by dominoes is a partition of the boxes of λ into pairs of adjacent boxes.For example, if

λ = , then

is a tiling of λ by dominoes

Let (m0, m1, , mℓ) be a sequence of nonnegative integers such that m0 6 λ1 and

|λ| = m0 + 2(m1+ · · · + mℓ) A domino tableau Q of shape λ/(m0) = sh(Q) and weight(m0, m1, , mℓ) = wt(Q) is a column strict filling of a tiling of the shape λ/(m0) bydominoes, where if a domino is filled with a number, then that number occupies bothboxes covered by that domino We put 0’s in the non-tiled boxes of λ, and mi is thenumber of i’s which appear For example, if



In the following Lemma, (a) is a straightforward use of the usual Pieri rule, and (b) isboth similar to (and perhaps a special case of) [14, Theorem 6.3], and also related to aPieri formula in [12]

Lemma 4.2 Let (m0, m1, , mℓ) be an ℓ + 1-tuple of nonnegative integers which sum

Proof (a) Note that

(−1)Number of barred entries in Q



sλ

(4.3)

We therefore need to determine the cancellations for a given shape λ

Fix r ∈ {1, 2, , ℓ} and λ ∈ P such that Tλ

(m 0 ,2m 1 , ,2m ℓ ) 6= ∅ For a tableau Q ∈

Tλ

(m 0 ,2m 1 , ,2m ℓ ), let

Qr = skew tableaux consisting of the boxes in Q containing ¯r or r,

SQ(r) = {column strict fillings of sh(Qr) by elements in {¯r, r}}

0 otherwise

such that

mi =

number of 0’s in Q if i = 0,number of ¯i’s + number of i’s... integers which sum

Proof (a) Note that

(−1)Number of barred entries in Q

...

 (4.1)

A tiling of λ by dominoes is a partition of the boxes of λ into pairs of adjacent boxes.For example, if

λ = , then

is a tiling of λ by dominoes

Let (m0,