ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS

Transcription

1 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS NATHANIEL THIEM AND C. RYAN VINROOT 1. Introduction In his seminal work [9], Green described a remarkable connection between the class functions of the finite general linear group GLn, F q ) and a generalization of the ring of symmetric functions of the symmetric group S n. In particular, Green defines a map, called the characteristic map, that takes irreducible characters to Schur-like symmetric functions, and recovers the character table of GLn, F q ) as the transition matrix between these Schur functions and Hall-Littlewood polynomials [19, Chapter IV]. Thus, we can use the combinatorics of the symmetric group S n to understand the representation theory of GLn, F q ). Some of the implications of this approach include an indexing of irreducible characters and conjugacy classes of GLn, F q ) by multi-partitions and a formula for the degrees of the irreducible characters in terms of these partitions. This paper describes the parallel story for the finite unitary group Un, F q 2) by collecting known results for this group and examining some applications of the unitary characteristic map. Inspired by Green, Ennola [5, 6] used results of Wall [22] to construct the appropriate ring of symmetric functions and characteristic map. Ennola was able to prove that the analogous Schur-like functions correspond to an orthonormal basis for the class functions, and conjectured that they corresponded to the irreducible characters. He theorized that the representation theory of Un, F q 2) should be deduced from the representation theory of GLn, F q ) by substituting q for every occurrence of q. The general phenomenon of obtaining a polynomial invariant in q for Un, F q 2) by this substitution has come to be known as Ennola duality. Roughly a decade after Ennola made his conjecture, Deligne and Lusztig [3] constructed a family of virtual characters, called Deligne-Lusztig characters, to study the representation theory of arbitrary finite reductive groups. Lusztig and Srinivasan [18] then computed an explicit decomposition of the irreducible characters of Un, F q 2) in terms of Deligne-Lusztig characters. Kawanaka [15] used this composition to demonstrate that Ennola duality applies to Green functions, thereby improving results of Hotta and Springer [12] and finally proving Ennola s conjecture. This paper begins by describing some of the combinatorics and group theory associated with the finite unitary groups. Section 2 defines the finite 1

2 2 NATHANIEL THIEM AND C. RYAN VINROOT unitary groups, outlines the combinatorics of multi-partitions, and gives a description of some of the key subgroups. Section 3 analyzes the conjugacy classes of Un, F q 2) and the Jordan decomposition of these conjugacy classes. Section 4 outlines the statement and development of the Ennola conjecture from two perspectives. Both points of view define a map from a ring of symmetric functions to the character ring C of Un, F q 2). However, the first uses the multiplication for C as defined by Ennola, and the second uses Deligne-Lusztig induction as the multiplicative structure of C. This multiplicative structure on the graded ring of characters of the unitary group was studied by Digne and Michel in [4], where the focus is that this multiplication induces a Hopf algebra structure. This structure theorem in [4] is equivalent to our Corollary 4.1, although our approach focuses on the explicit map between characters and symmetric functions. The main results are I. Theorem 4.2) The Deligne-Lusztig characters correspond to power-sum symmetric functions via the characteristic map of Ennola. II. Corollary 4.2) The multiplicative structure that Ennola defined on C is Deligne-Lusztig induction. Section 5 computes the degrees of the irreducible characters, and uses this result to evaluate various sums of character degrees see [19, IV.6, Example 5] for the GLn, F q ) analogue of this method). The main results are III. Theorem 5.1) An irreducible χ λ character of Um, F q 2) corresponds to q i 1) i ) 1 i m 1) m/2 +nλ) s λ and χ λ 1) = q nλ ) q h ) 1) h ) ), where s λ is a Schur-like function, and both nλ) and h ) are combinatorial statistics on the multi-partition λ. IV. Corollary 5.2) If Pn Θ indexes the irreducible characters χ λ of Un, F q 2), then χ λ 1) = {g Un, F q 2) g symmetric}. λ P Θ n Section 6 uses results by Ohmori [21] and Henderson [11] to adapt a model for the general linear group, found by Klyachko [17] and Inglis and Saxl [13], to the finite unitary group. The main result is V. Theorem 6.2) Let U m = Um, F q 2), where q is odd, and let Γ m be the Gelfand-Graev character of U m, 1 be the trivial character of the finite symplectic group Sp 2r = Sp2r, F q ), and RL G be the Deligne-Lusztig induction functor. Then R Um U m 2r U Γm 2r 2r Ind U 2r Sp 2r 1) ) = χ λ. 0 2r m λ λ P Θ m

3 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 3 That is, in the theorem of Klyachko, one may replace parabolic induction by Deligne-Lusztig induction to obtain a theorem for the unitary group. These results give considerable combinatorial control over the representation theory of the finite unitary group, and there are certainly more applications to these results than what we present in this paper. Furthermore, this characteristic map gives some insight as to how a characteristic map might look in general type, using the invariant rings of other Weyl groups. 2. Preliminaries 2.1. The unitary group and its underlying field K. Let K = F q be the algebraic closure of the finite field with q elements and let K m = F q m denote the finite subfield with q m elements. Let GLn, K) denote the general linear group over K, and define Frobenius maps 2.1) F :GLn, K) GLn, K) a ij ) a q ji ) 1, and F :GLn, K) GLn, K) a ij ) a q n j,n i ) 1. Then the unitary group U n = Un, K 2 ) is given by 2.2) 2.3) U n = GLn, K) F = {a GLn, K) F a) = a} = GLn, K) F = {a GLn, K) F a) = a}. In fact, it follows from the Lang-Steinberg theorem see, for example, [2]) that there exists y GLn, K 2 ) such that GLn, K) F = ygln, K) F y 1. We define the multiplicative groups T m as T m = GL1, K) F m = {x K x qm 1) m = 1}. Note that T m = K m only if m is even. We identify K with the inverse limit lim T m with respect to the norm maps N mr : T m T r x xx q x q)m/r 1, where m, r Z 1 with r m. If T m is the group of characters of T m, then the direct limit K = lim T m gives the group of characters of K. Let Θ = {F -orbits of K }. A polynomial ft) K 2 [t] is F -irreducible if there exists an F -orbit {x, x q,..., x q)d } of K such that Let ft) = t x)t x q ) t x q)d ). 2.4) Φ = {f K 2 [t] f is F -irreducible} 1 1 {F -orbits of K }.

4 4 NATHANIEL THIEM AND C. RYAN VINROOT The set Φ has an alternate description, as given in [5]. For f = t d + a 1 t d a d K 2 [t] with a d 0, let f = t d + a q d aq d 1 td a q 1 t + 1), which has the effect of applying F to the roots of f in K. Then a polynomial f K 2 [t] is F -irreducible if and only if either a) f is irreducible in K 2 [t] and f = f d must be odd in this case), or b) ft) = ht) ht) where h is irreducible in K 2 [t] and ht) ht) Combinatorics of Φ-partitions and Θ-partitions. Fix an ordering of Φ and Θ, and let P = {partitions} and P n = {ν P ν = n}. Let X be either Φ or Θ. An X -partition ν = νx 1 ), νx 2 ),...) is a sequence of partitions indexed by X. The size of an X -partition ν is 2.5) ν = { x if X = Θ, x νx), where x = dx) if X = Φ, x X x is the size of the orbit x Θ, and dx) is the degree of the polynomial x Φ. Let 2.6) Pn X = {X -partitions ν ν = n}, and P X = Pn X. For ν P X, let 2.7) nν) = x X lν) n=1 x nνx)), where nν) = i 1)ν i. The conjugate ν of ν is the X -partition ν = νx 1 ), νx 2 ),...), where ν is the usual conjugate partition for ν P. The semisimple part ν s of ν = νx 1 ), νx 2 ),...) Pn X is i=1 2.8) ν s = 1 νx 1) ), 1 νx 2) ),...) P X n, and the unipotent part ν u of ν P X n is given by 2.9) ν u 1) has parts { x νx) i x X, i = 1,..., lνx))} where 1 = { {1} if X = Θ, t 1 if X = Φ, 1 is the trivial character in K, and ν u x) = for x 1. Example. If µ P Φ is given by ) f) g) h) µ =,,, where df) = 1, dg) = dh) = 2,

5 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 5 then µ = = 18, nµ) = = 4, f) h) t 1) g) µ s =,, and µu = Levi subgroups and maximal tori. Let X be either Φ or Θ as in Section 2.2. For ν P X n, let 2.10) L ν = x X ν L ν x), where X ν = {x X νx) }, and for x X ν, { U νx), K2 x ) if x is odd, 2.11) L ν x) = GL νx), K x ) if x is even. Then L ν is a Levi subgroup of U n = Un, K 2 ) though not uniquely determined by ν). The Weyl group 2.12) W ν = x X ν S νx), of L ν has conjugacy classes indexed by 2.13) P ν s = {γ P X γ s = ν s }, and the size of the conjugacy class c γ is 2.14) c γ = W γ, where z γ = z z γx) and z γ = γ x X lγ) i m i m i!, for γ = 1 m 1 2 m2 ) P. For every ν = ν 1, ν 2,..., ν l ) P n there exists a maximal torus unique up to isomorphism) T ν of U n such that T ν = Tν1 T ν2 T νl. For every γ P µ s, there exists a maximal torus unique up to isomorphism) T γ L ν such that 2.15) T γ = i=1 x X ν T γ x), where T γ x) = T x γx)1 T x γx)l. Note that as a maximal torus of U n, the torus T γ = Tγu 1).

6 6 NATHANIEL THIEM AND C. RYAN VINROOT 3. Conjugacy classes and Jordan decomposition Let U n = Un, K 2 ) as in 2.2). For r Z 0, let r 3.1) ψ r x) = 1 x i ). i=1 In the following proposition, i) is due to Ennola in [6], and the order of the centralizer in ii) was obtained by Wall in [22] in a slightly different form, while the version in the proposition below appears in [6]. Proposition 3.1 Ennola, Wall). a) The conjugacy classes c µ of U n are indexed by µ P Φ n. b) Let g c µ. The order a µ of the centralizer g in U n is a µ = 1) µ f Φ for µ = 1 m 1 2 m 2 3 m3 ) P. a µf) q) df) ), where a µ x) = x µ +2nµ) j For µ P Φ, let L µ be as in 2.10). Note that L µ = a µs. ψ mj x 1 ), Lemma 3.1. Suppose g c µ with Jordan decomposition g = su. Then a) s c µs and u c µu, where µ s and µ u are as in 2.8) and 2.9), b) the centralizer C Un s) of s in U n is isomorphic to L µ. Proof. a) Suppose f = t d a d 1 t d 1 a 1 t a 0 Φ is irreducible so d is odd). Define ) Jf) = GLd, K 2 ) a 0 a 1 a 2 a d 1 For ν = ν 1, ν 2,..., ν l ) P, let Jf) Id d 0. 0 Jf).. J ν f) = J ν1 f) J νl f), where J m f) = Idd 0 0 Jf) is an md md matrix and Id d is the d d identity matrix. Note that Id d Jf) 1 0. J m f) = J 1 m )f)u m) f), where u m) f) = Id.. d... Jf) 1 0 Id d is the Jordan decomposition of J m f) in GLd, K 2 ).

7 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 7 Similarly, if f = h h Φ or df) is even), then define Jf) = Jh) J h), J m f) = J m h) J m h), J ν f) = J ν h) J ν h), so that J m f) = J 1 m )h) J 1 m ) h) ) u m) h) u m) h) ) is the Jordan decomposition of J m f). If g c µ, then g is conjugate to J µ = f Φ J µf) f) in GL µ, K 2 ). Claim a) follows from the uniqueness of the Jordan decomposition. b) In GLn, K 2 ), { C GLn,K2 )J 1 m )f)) GLm, K2df) ) if df) is odd, = GLm, K df) ) GLm, K df) ) if df) is even. If df) is odd and s = xj 1 m )f)x 1 for some x GLn, K 2 ), then xglm, K 2df) )x 1 ) F = GLm, K 2df) ) x F = GLm, K2df) ) F = Um, K 2df) ). Suppose df) is even so that f = h h, and let y GLn, K 2 ) such that GLn, K) F = ygln, K) F y 1 see the comment after 2.2)). The element Jf) is conjugate to whose centralizer in GLn, K 2 ) F Jh) F Jh))) GLn, K 2 ) F, is {g F g) g GLm, K df) )} = GLm, K df) ). 4. The Ennola Conjecture 4.1. The characteristic map. Let X = {X 1, X 2,...} be an infinite set of variables and let ΛX) be the graded C-algebra of symmetric functions in the variables {X 1, X 2,...}. For ν = ν 1, ν 2,..., ν l ) P, the power-sum symmetric function p ν X) is p ν X) = p ν1 X)p ν2 X) p νl X), where p m X) = X m 1 + X m 2 +. 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. For λ P, the Schur function s λ X) is given by 4.1) s λ X) = ω λ ν)zν 1 p ν X), where z ν = i m i m i! ν P λ i 1 is the order of the centralizer in S n of the conjugacy class corresponding to ν = 1 m 1 2 m2 ) P. Let t C. For µ P, the Hall-Littlewood symmetric function P µ X; t) is given by 4.2) s λ X) = µ P λ K λµ t)p µ X; t),

8 8 NATHANIEL THIEM AND C. RYAN VINROOT where K λµ t) is the Kostka-Foulkes polynomial as in [19, III.6]). For ν, µ P n, the classical Green function Q µ ν t) is given by 4.3) p ν X) = µ P ν Q µ ν t 1 )t nµ) P µ X; t). 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}. For every f Φ, fix a set of independent variables X f) = {X f) 1, X f) 2,...}, and for any symmetric function h, we let hf) = hx f) ) denote the symmetric function in the variables X f). Let Λ = C-span{P µ µ P Φ }, where P µ = q) nµ) f Φ P µf) f; q) df) ). Then Λ = n 0 Λ n, where Λ n = C-span{P µ µ = n}, makes Λ a graded C-algebra. Define a Hermitian inner product on Λ by P µ, P ν = a 1 µ δ µν. For each ϕ Θ let Y ϕ) = {Y ϕ) 1, Y ϕ) 2,...} be an infinite variable set, and for a symmetric function h, let hϕ) = hy ϕ) ). Relate symmetric functions in the X variables to symmetric functions in the Y variables via the transform 4.4) p n ϕ) = 1) n ϕ 1 ξx)p n ϕ /dfx)f x ), x T n ϕ where ϕ Θ, ξ ϕ, and f x Φ satisfies f x x) = 0. Then 4.5) Λ = C-span{s λ λ P Θ }, where s λ = ϕ Θ s λϕ) ϕ). Let C n denote the set of complex-valued class functions of the group U n, and for µ = n, let π µ : U n C be given by { 1 if u cµ, π µ u) = where u U 0 otherwise, n. Then the π µ form a C-basis for C n. By Proposition 3.1, the usual inner product on class functions of finite groups,, : C n C n C, satisfies π µ, π λ = a 1 µ δ µλ.

9 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 9 For α i C ni, Ennola [6] defined a product α 1 α 2 C n1 +n 2, which takes the following value on the conjugacy class c λ : α 1 α 2 c λ ) = gµ λ 1 µ 2 α 1 c µ1 )α 2 c µ2 ), µ i =n i where g λ µ 1 µ 2 is the product of Hall polynomials see [19, Chapter II]) g λ µ 1 µ 2 = f Φ Extend the inner product to g λf) µ 1 f)µ 2 f) q)df) ). C = n 0 C n, by requiring the components C n and C m to be orthogonal for n m. This gives C a graded C-algebra structure. The characteristic map is ch : C Λ π µ P µ, for µ P Φ. The following is implicit in [6], and a proof quickly follows from [19, III.3.6]. Proposition 4.1. Let multiplication in the character ring C of U n be given by. Then the characteristic map ch : C Λ is an isometric isomorphism of graded C-algebras. Following the work of Green [9] on the general linear group, Ennola was able to obtain the following result. One may obtain a proof from the characteristic map point of view by following Macdonald [19, IV.4] on the general linear group case. Proposition 4.2 Ennola). The set {s λ λ P Θ } is an orthonormal basis for Λ. Now let χ λ C be class functions so that χ λ 1) > 0 and chχ λ ) = ±s λ. Ennola conjectured that {χ λ λ P Θ n } is the set of irreducible characters of U n. He pointed out that if one could show that the product takes virtual characters to virtual characters, then the conjecture would follow. There is no known direct proof of this fact, however. Significant progress on Ennola s conjecture was only made after the work of Deligne and Lusztig [3] Deligne-Lusztig Induction. Let T ν = Tν1 T νl be a maximal torus of U n. If t T ν, then t is conjugate to J 1 m 1 ) f 1 ) J 1 m l) f l ), where f i Φ, m i df i ) = ν i. Define γ t P Φ by 4.6) γ t f) has parts {m i f i = f}. Note that γ t ) u t 1) = ν, but in general t / c γt.

10 10 NATHANIEL THIEM AND C. RYAN VINROOT Example. If t T 4,4,2,2,1) and t is conjugate to J t 1) J t 2 + 1) J t 1) J t 2 + 1) J t 1), then γ t = t 1), t 2 +1) ). For µ P Φ, let L µ, γ P s µ and T γ be as in Section 2.3. Let θ be a character of T ν. The Deligne-Lusztig character R ν θ) = R Un T ν θ) is the virtual character of U n given by Rν θ) ) g) = θt)q Lµ T γt u), t Tν γ t P s µ where g c µ has Jordan decomposition g = su thus, by Lemma 3.1 C Un s) = L µ ), and Q Lµ T γt u) is a Green function for the unitary group see, for example, [2]). It is proven by Lusztig and Srinivasan [18] that C n = {class functions of U n } = C-span{R ν θ) ν P n, θ HomT ν, C )}, so we may define Deligne-Lusztig induction by 4.7) R U m+n U m U n : C m C n C m+n R Um α θ α ) R Un β θ β) R U m+n T α T β θ α θ β ), for α P m, β P n, θ α HomT α, C), and θ β HomT β, C). Let Λ and C be as in Section 4.1, except we now give C a graded C- algebra structure using Deligne-Lusztig induction. That is, we define a multiplication on C by χ η = R U m+n U m U n χ η), for χ C m and η C n. We recall the characteristic map defined in Section 4.1, ch : C Λ π µ P µ for µ P Φ. As noted in Proposition 4.1, it is immediate that ch is an isometric isomorphism of vector spaces, but it is not yet clear if ch is also a ring homomorphism when C has multiplication given by Deligne-Lusztig induction The Ennola conjecture. To prove the Ennola conjecture we require two further ingredients: 1) Theorem 4.2 and Corollary 4.1 establish that this new characteristic map is also a ring isomorphism by using a key result by Kawanaka on Green functions to evaluate chr ν θ)).

11 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 11 2) Corollary 4.3 uses an explicit decomposition by Lusztig and Srinivasan of irreducible characters of U n into Deligne-Lusztig characters to complete the reproof of the Ennola conjecture. To compute chr ν θ)), we need to write the Green functions Q Lµ T γ u) as polynomials in q. These Green functions turn out to be those of the general linear group, except with q replaced by q, which is the essence of Ennola s original idea. This fact was proven by Hotta and Springer [12] for the case that p = charf q ) is large compared to n, and was finally proven in full generality by Kawanaka [15]. Theorem 4.1 Hotta-Springer, Kawanaka). The Green functions for the unitary group are given by Q Lµ T γ u) = Q µ γ q), where Q µ γ q) = γf) q)df) ), f Φ µ Q µf) and Q µ γq) is the classical Green function as in 4.3). For ν = ν 1, ν 2,..., ν l ) P and θ = θ 1 θ 2 θ l a character of T ν, define p µνθ = ϕ Θ p µνθ ϕ)ϕ), where µ νθ ϕ) = ν i / ϕ θ i ϕ). Theorem 4.2. Let ν = ν 1, ν 2,..., ν l ) P, θ = θ 1 θ 2 θ l be a character of T ν, and ν = µ νθ P Θ. Then ch R ν θ) ) = 1) ν lν) p ν. Proof. By Theorem 4.1, and since chπ µ ) = P µ, it suffices to show that the coefficient of P µ in the expansion of 1) ν lν) p ν is θ ν t)q µ γ t q). Since t Tν γ t P µ s p ν = p ν1 / ϕ 1 ϕ 1 ) p νl / ϕ l ϕ l ), where θ i ϕ i, the transform 4.4) implies 1) ν lν) p ν = l l ) θ i t i )p νi /d i f i ) = θt) p νi /d i f i ), t i k ν t T ν i=1 i i=1 where f i Φ satisfies f i t i ) = 0 and d i = df i ). By definition 4.6), 4.8) 1) ν lν) p ν = ) θt) p γt f)f). t T ν f Φ

12 12 NATHANIEL THIEM AND C. RYAN VINROOT Change the basis from power-sums to Hall-Littlewood polynomials 4.3), and we obtain 1) ν lν) p ν = θt) t T ν = t T ν θt) f Φ µf) P γt f) µ P Φ γ t P µ s = µ P Φ as desired. t Tν γ t P µ s Q µ γ t q)p µ ) θt)q µ γ t q) P µ, ) Q µf) γ t f) q)df) ) q) df)nµf)) P µf) f; q) df) ) The following result immediately follows from the definition of the Deligne- Lusztig product 4.7) and Theorem 4.2. This result is equivalent to the Hopf algebra structure theorem obtained in [4]. Corollary 4.1. Let multiplication in the character ring C of U n be given by. Then the characteristic map ch : C Λ is an isometric isomorphism of graded C-algebras. An immediate consequence is that the graded multiplication that Ennola originally defined on C is exactly Deligne-Lusztig induction, or Corollary 4.2. Let χ C m and η C n. Then χ η = χ η. We therefore have the advantage of using either product as convenience demands. For λ P Θ, let L λ, W λ, and T γ, γ P λ s, be as in Section 2.3. Note that the combinatorics of γ almost specifies character θ γ of T γ in the sense that In fact, we may define θ γ T γ ϕ)) = θ ϕ T γ ϕ)), for some θ ϕ ϕ. 4.9) R γ = R Un T γ θ γ ) = ch 1 1) γ lγ) p γ ), where θ γ is any choice of the θ ϕ s. For every λ P Θ there exists a character ω λ of W λ defined by ω λ γ) = ϕ Θ ω λϕ) γϕ)), where ω λ γ) is the value of ω λ on the conjugacy class c γ corresponding to γ P Θ s. In [18], Lusztig and Srinivasan decomposed the irreducible characters of U n as linear combinations of Deligne-Lusztig characters, as follows.

13 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 13 Theorem 4.3 Lusztig-Srinivasan). Let λ P Θ n. Then there exists τ λ) Z 0 such that the class function Rλ) = 1) τ λ)+ n/2 + P ϕ Θ λϕ) + ϕ λϕ) /2 is an irreducible character of U n z γ is as in 2.14)). Remark. The sign γ P λ s ω λ γ) R γ z γ 1) n/2 +P ϕ Θ λϕ) + ϕ λϕ) /2 = 1) Fq-rank of Un + Fq-rank of L λ, and Theorem 5.1 will show that τ λ) = nλ ) + λ ϕ Θ ϕ λϕ) /2 = nλ ) + ϕ Θ ϕ λϕ) /2. Corollary 4.3 Ennola Conjecture). For λ P Θ, there exists τλ) Z 0 such that ) } {ch 1) 1 τλ) s λ λ Pn Θ is the set of irreducible characters of U n. Proof. By Theorem 4.3 and Theorem 4.2, chrλ)) = 1) τ λ)+ n/2 + P ϕ Θ λϕ) + ϕ λϕ) /2 = 1) τ λ)+ n/2 +n+ P ϕ Θ ϕ λϕ) /2 Note that the sign character ω λs γ P λ s of W λ acts by γ P λ s ω λs γ) = 1) P ϕ Θ γϕ) lγ), and that ω λ ω λs = ω λ, so since γ P λ s, chrλ)) = 1) τ λ)+ n/2 +n+ P ϕ Θ ϕ λϕ) /2 ω λ γ) 1) n lγ) p γ z γ ω λ γ) z γ 1) P ϕ Θ λϕ) lγ) p γ. γ P λ s = 1) τ λ)+ n/2 +n+ P ϕ Θ ϕ λϕ) /2 and by applying 4.1) to a product over Θ, γ P λ s ω λ ω λs )γ) p γ z γ ω λ γ) z γ p γ, = 1) τ λ)+ n/2 +n+ P ϕ Θ ϕ λϕ) /2 s λ. Remark. There are at least two natural ways to index the irreducible characters of U n by Θ-partitions: Theorem 4.3 gives a natural indexing by Θ-partitions, but Corollary 4.3 indicates that the conjugate choice is equally

14 14 NATHANIEL THIEM AND C. RYAN VINROOT natural. Following Macdonald [19], we have chosen the latter indexing. However, several references, including Ennola [6], Ohmori [21], and Henderson [11], make use of the former. 5. Characters degrees 5.1. A formula for character degrees. Let λ P Θ, and suppose λ is in position i, j) in λϕ) for some ϕ Θ. The hook length h ) of is h ) = ϕ h ), where h ) = λϕ) i λϕ) j i j + 1, is the usual hook length for partitions. Example. For θ = 1, ϕ = 3 and ) ϕ) θ) λ =,, the hook lengths are ) ϕ) θ) 12 3, 6. 3 Example 2 in [19, I.1] implies that 5.1) h ) = λ + nλ) + nλ ). λ For λ P Θ, let η λ = ch 1 s λ ). Theorem 5.1. Let λ P Θ and let 1 be the identity in U λ. Then q i 1) i ) 1 i λ η λ 1) = 1) τλ) q nλ ) q h ) 1) h ) ), λ where τλ) = λ λ + 3)/2 + nλ) λ /2 + nλ) mod 2). So for each λ, we have χ λ = 1) τλ) η λ. Proof. We follow the computations in [19, IV.6]. Let ηµ λ be the value of η λ on the conjugacy class c µ. Then s λ = ηµp λ µ, µ P Φ λ implies ηµ λ = s λ, a µ P µ. Since the s λ are orthonormal, 5.2) a µ P µ = ηµ s λ λ where s λ = λ η λ µp µ µ P Φ λ is obtained by taking the complex conjugates of the coefficients of the P µ.

15 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 15 If c µ1 is the conjugacy class corresponding to the identity element 1 of U m, then µ 1 t 1) = 1 m ) and µ 1 f) = 0 for f t 1. From the definition of a µ, and the fact that P 1 m )x; t) = e m x) see [19, III.8]), we have a µ1 P µ1 = 1) m q) m+mm 1) ψ m q 1 ) q) mm 1)/2 P 1 m )X t 1) ; q) 1 ) = 1) m q) mm+1)/2 ψ m q) 1 )P 1 m )X t 1) ; q) 1 ) = ψ m q)e m t 1). Therefore, by 5.2), 5.3) ψ m q)e m t 1) = λ =m η λ 1) s λ. Let δ : Λ C be the C-algebra homomorphism defined by { 1) m 1 /q m 1) m ) if f = t 1, δp m f)) = 0 if f t 1. It follows from [19, I.4.3] and the argument from [19, p. 279] that 5.4) log s λ s λ = 1 q ndf) 1) ndf) )p n f) p n f). n λ P Θ n 1 f Φ Apply δ 1 to both sides of Equation 5.4) to obtain ) log δs λ ) s λ = 1) m 1 m p mt 1) = log λ P Θ m 1 i By exponentiating and expanding the product to e m s, we have δs λ ) s λ, which gives m 0 e m t 1) = λ e m t 1) = λ =m Compare coefficients with 5.3) to deduce δs λ ) s λ. 1 + X t 1) i ). 5.5) η λ 1) = ψ m q)δs λ ), where δs λ ) = ϕ Θ δs λϕ) ϕ)). By the definitions of δ and p m ϕ), ϕ Θ, δp m ϕ)) = 1) ϕ m q) ϕ m 1 ) 1 = 1) ϕ m i 1 q) i ϕ m, so δp m Y ϕ) 1, Y ϕ) 2,...)) = 1) ϕ m p m q) ϕ, q) 2 ϕ,...).

16 16 NATHANIEL THIEM AND C. RYAN VINROOT That is, applying δ to a homogeneous symmetric function in Y ϕ) of degree m replaces each Y ϕ) i by q) i ϕ and multiplies by 1) ϕ m. In particular, for λ P, ϕ Θ, 5.6) δs λ ϕ)) = 1) λ ϕ s λ q) ϕ, q) 2 ϕ,... ). Example 2 in [19, I.3] implies 5.7) δs λ ϕ)) = 1) ϕ λ q) ϕ λ +nλ)) 1 q) ϕ h ) ) 1. λ Combine 5.5) and 5.7) to get δs λ ) = 1) λ q) λ nλ) λ 1 q) h ) ) 1 = 1) λ q) λ nλ)+p λ h ) λ q) h ) 1 ) 1 = 1) λ q) nλ ) λ q) h ) 1 ) 1, by 5.1)) = 1) λ +nλ )+ P λ h ) q nλ ) λ q h ) 1) h )) 1. Since λ + nλ ) + λ h ) = 2 λ + 2nλ ) + nλ) nλ) mod 2), we have δs λ ) = 1) nλ) q nλ ) λ q h ) 1) h )) 1. Since m m ψ m q) = 1 q) i ) = 1) m+mm+1)/2 q i 1) i ), i=1 i=1 we have η λ 1) = 1) τλ) q nλ ) 1 i λ q i 1) i ) q h ) 1) h ) ) 1. λ By the Littlewood-Richardson rule, for any µ, ν P, we have s µ s ν = λ c λ µνs λ, where c λ µν is the number of tableaux T of shape λ µ and weight ν such that the word wt ) is a lattice permutation [19, I.9]. So we have 5.8) s µ s ν = λ c λ µνs λ where c λ µν = ϕ Θ c λϕ) µϕ)νϕ).

17 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 17 Corollary 5.1. Let µ, ν P Θ. Then χ µ χ ν is a character if and only if every λ P Θ such that c λ µν > 0 satisfies nµ) + nν) nλ) + µ ν mod 2) Proof. Since chχ µ χ ν ) = 1) τµ)+τν) c λ µνs λ, λ the class function χ µ χ ν is a character if and only if every λ such that c λ µν > 0 satisfies τµ) + τν) τλ) mod 2). For each λ such that c λ µν > 0, we have, for every ϕ Θ, a tableau of shape λϕ) µϕ) and weight νϕ). In particular, we have µϕ) + νϕ) = λϕ) for every ϕ Θ, so that µ + ν = λ. The result follows from the definition of τ. It follows that the operation does not always take a pair of characters to a character. For example, let ϕ 1 be the orbit of the trivial character, and define µ = ν = ϕ 1) ). Then one λ such that c λ µν > 0 is λ = ϕ1 ) ). Then we have nµ) + nν) = 0 while nλ) + µ ν = 1, and so χ µ χ ν is not a character by Corollary Character degree sums. Let d r denote the number of F -orbits in Θ of size r. Since the F -action preserves T m and T m = q m 1) m, we have 5.9) q m 1) m = r m rd r, for m Z >0. Theorem 5.2. The sum of the degrees of the complex irreducible characters of U m is given by χ λ 1) = q + 1)q 2 q 3 + 1)q 4 q 5 + 1) q m + 1 1)m )). 2 λ =m Proof. Following a similar approach to [19, IV.6, Example 5], we consider the coefficient of t m in the series S = λ P Θ 1) nλ)+ λ δs λ )t λ, where nλ) is as in 2.7). Note that S = S o S e, where S o = 1) nλ) δs λ ϕ)) t) λ ϕ and S e = δs λ ϕ))t λ ϕ. ϕ λ ϕ λ odd even Combine the following identity from Example 6 in [19, I.5], 1) nλ) s λ = 1 x i ) x i x j ) 1 i i<j λ

18 18 NATHANIEL THIEM AND C. RYAN VINROOT with 5.6) to obtain 1) nλ) 1) ϕ λ δs λ ϕ))t λ ϕ λ P 1 t q) i ) ϕ ) t 2 q) i j ) ϕ ) 1. m odd = i 1 1 i<j Taking the logarithm, log S o = t q) i ) mr d m + t 2 q) i j ) m) r ) r r m odd i 1 r 1 1 i<j r 1 = t mr d m q) imr + 1) ) r t 2mr q) i+j)mr r m odd r 1 i 1 1 i<j = t mr d m r q) mr 1 + 1) ) r t mr q) 2imr 1) r 1 i 1 = m odd d m r 1 t mr rq mr 1) mr ) 1) mr + t mr q ), 2imr i 1 where d m is as in 5.9). Similarly for ϕ even, by Example 4 in [19, I.5], s λ = 1 x i ) 1 1 x i x j ) 1 i i<j so λ δs λ ϕ))t λ ϕ = 1 tq i ) ϕ ) 1 i 1 λ P r 1 1 i<j 1 t 2 q i j ) ϕ ) 1. Taking logarithms, log S e = tq d m i ) mr + t 2 q i j ) mr ) r r m even i 1 r 1 1 i<j r 1 = t mr d m rq mr 1) mr 1) mr + t mr q ). 2imr ) m even i 1 Let N = mr, and by 5.9), log S = log S o + log S e = t) N N + N 1 i 1 t 2N q 2iN ) N = log1 + t) 1 + i 1 log1 t 2 q 2i ) 1. By exponentiating, S = t 1 t 2 q 2i = 1 t) 1 1 t 2 q 2i = 1 t) i 1 i 0 l 0 t 2l ψ l q 2 ),

19 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 19 where the last equality comes from Example 4 in [19, I.2]. Multiply the coefficient of t m by 1) m ψ m q) = 1) m m i=1 qi 1) i ) to obtain χ λ 1) = q + 1)q 2 q 3 + 1)q 4 q 5 + 1) q m + 1 1)m )). 2 λ =m Write f Um q) = λ =m χλ 1). The polynomial f Gm q) expressing the sum of the degrees of the complex irreducible characters of G m = GLm, F q ) was computed in [7] for odd q and in [17] and Example 6 of [19, IV.6] for general q. From these results we see that f Um q) = 1) mm+1)/2 f Gm q), giving another example of Ennola duality. Gow [7] and Klyachko [17] proved that the sum of the degrees of the complex irreducible characters of G n is equal to the number of symmetric matrices in G n. Gow accomplished this by considering the split extension of G n by the transpose-inverse automorphism, 5.10) G + n = G n, κ κ 2 = 1, κgκ = t g 1 for every g G n, and showed that every complex irreducible representation of G + n could be realized over the real numbers. Instead, we show directly that the sum of the degrees of the complex irreducible characters of U n is equal to the number of symmetric matrices in U n. One can apply this result to determine reality properties of the characters of U n +, where 5.11) U + n = U n, κ κ 2 = 1, κuκ = t u 1 for every u U n. Corollary 5.2. The sum of the degrees of the complex irreducible characters of Un, F q 2) is equal to the number of symmetric matrices in Un, F q 2). Proof. Define S Un to be Then U n acts on the set S Un S Un = {symmetric matrices in U n }. by the action u s = t usu, s S Un, u U n. To find the number of elements in S Un, we find the stabilizers of the orbits under this action. Consider the same action of G n = GLn, F q ) on its subset of symmetric matrices. That is, let and let G n act on the set S Gn S Gn = {symmetric matrices in G n }, by the action g s = t gsg, s S Gn, g G n. Gow [8] proved, using the Lang-Steinberg theorem, that there is a one-toone correspondence between conjugacy classes of G + n 5.10) and U n ) of elements of the form gκ and uκ, for g G n and u U n. Moreover, Gow proved that this correspondence preserves orders of the elements, and

20 20 NATHANIEL THIEM AND C. RYAN VINROOT the corresponding centralizers in G n and U n are isomorphic. The conjugacy classes of order 2 elements of this form correspond to the orbits of symmetric matrices in G n and U n, and their centralizers in G n and U n to the stabilizers under the action described above. So to find the stabilizers of the orbits in the case for U n, it is enough to do this for G n. If q is odd, or when q is even and n is odd, the stabilizers of the orbits of S Gn are exactly the orthogonal groups see, for example, [10, Chapters 9, 14] for a complete discussion and orders). When q and n are both even, there are two orbits, and one orbit consists of symmetric matrices which have at least one nonzero entry on the diagonal, the order of whose stabilizer is computed in [20]. The other orbit, consisting of symmetric matrices with zero diagonal, corresponds to the unique class of alternating forms, with stabilizer the symplectic group over F q. In each of these cases, it may be easily checked that the sum of the indices of these stabilizers as subgroups of U n, which by Gow s result is the size of S Un, is exactly the polynomial obtained in Theorem 5.2. For a finite group H with order 2 automorphism ι, and irreducible complex representation π of H, the twisted Frobenius-Schur indicator of π, denoted ε ι π), was originally defined in [16], and studied further in [1]. This indicator is a generalization of the classical Frobenius-Schur indicator επ), which takes the value 1 if π is a real representation, 1 if the character of π is real-valued but π is not real, and 0 if the character of π is not realvalued. If H = U n, and ι is the inverse-transpose automorphism, it follows from Corollary 5.2 and [1, Proposition 1] that ε ι π) = 1 for every complex irreducible representation π of U n. Using this fact, along with the formula for the twisted Frobenius-Schur indicator established in [16], we obtain the following reality properties for the group U + n. Corollary 5.3. Let U n + be the split extension of U n by the transpose-inverse involution, as in 5.11). Let π be a complex irreducible representation of U n. Then we have the following: 1) If επ) = 0, then π induces to an irreducible representation ρ = Ind U + n U n π) of U + n such that ερ) = 1. 2) If επ) = 1, then π extends to two irreducible representations π and π of U + n such that επ ) = επ ) = 1. 3) If επ) = 1, then π extends to two irreducible representations π and π of U + n such that επ ) = επ ) = 0. A Θ-partition λ is even if every part of λϕ) is even for every ϕ Θ. Let Sp 2n = Sp2n, F q ) be the symplectic group over the finite field F q. The following was proven in [11].

21 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 21 Theorem 5.3 Henderson). Let q be odd. The decomposition of Ind U 2n Sp 2n 1) into irreducibles is given by Ind U 2n Sp 2n 1) = λ =2n λ even So, for q odd, Theorem 5.3 implies an identity for the sum of the degrees of characters appearing in the permutation character. We are able to calculate this degree sum for any q, suggesting that Theorem 5.3 should hold for q even as well. Theorem 5.4. The sum of the degrees of the complex irreducible characters of U 2m corresponding to λ such that λ is even is given by λ =2m λ even χ λ. χ λ 1) = q + 1)q 2 q 3 + 1) q 2m 2 q 2m 1 + 1) = Un, F q 2) Sp2n, F q ). Proof. To calculate this sum, we find the coefficient of t 2m in the series T = 1) nλ)+ λ δs λ )t λ = T o T e, where T o = ϕ odd λ even λ even 1) nλ) δs λ ϕ))t λ ϕ and T e = ϕ even λ even It follows from the computation in Example 6 of [19, I.5] that 1 + x i x j ) 1, λ even 1) nλ) s λ = i<j and applying this yields 1) nλ) δs λ ϕ))t λ ϕ = λ even 1 i<j From Example 5b) in [19, I.5], we have the identity 1 x i x j ) 1, λ even s λ = i<j from which it follows, for ϕ even, that δs λ ϕ))t λ ϕ = λ even 1 i<j δs λ ϕ))t λ ϕ. 1 + t 2 q) i j) ϕ ) 1. 1 t 2 q i j) ϕ ) 1. Proceeding similarly as in the proof of Theorem 5.2, we obtain T = 1 1 t 2 q 2i = 1 t2 ) 1 1 t 2 q 2i = 1 t2 ) i 1 i 0 l 0 t 2l ψ l q 2 ).

22 22 NATHANIEL THIEM AND C. RYAN VINROOT Multiplying the coefficient of t 2m by ψ 2m q) = 2m i=1 qi 1) i ), we obtain χ λ 1) = q + 1)q 2 q 3 + 1) q 2m 2 q 2m 1 + 1). λ =2m λ even Write g Um q) = λ =2m,λ even χλ 1), and let g Gm q) denote the corresponding sum for G m. The polynomial g Gm q) was calculated in Example 7 of [19, IV.6], and similar to the previous degree sum, we see that we have g Um q) = 1) m g Gm q). 6. A Deligne-Lusztig model A model of a finite group G is a representation ρ, which is a direct sum of representations induced from one-dimensional representations of subgroups of G, such that every irreducible representation of G appears as a component with multiplicity 1 in the decomposition of ρ. Klyachko [17] and Inglis and Saxl [13] obtained a model for GLn, F q ), where the induced representations can be written as a Harish-Chandra product of Gelfand-Graev characters and the permutation character of the finite symplectic group. In this section we show that the same result is true for the finite unitary group, except the Harish-Chandra product is replaced by Deligne-Lusztig induction. The result is therefore not a model for Un, F q 2) in the finite group character induction sense, but rather from the Deligne-Lusztig point of view. Let Γ m) be the Gelfand-Graev character of Um, F q 2) see, for example, [2] for a general definition). For λ P Θ, define htλ) = max{lλϕ)) ϕ Θ}. It is well-known that the Gelfand-Graev character has a multiplicity free decomposition see, for example, [2]), and Ohmori [21, Section 5.2] gives the following explicit decomposition. Alternatively, the theorem also follows by applying the characteristic map to [3, Theorem 10.7]. Theorem 6.1. The decomposition of Γ m) into irreducibles is given by Γ m) = χ λ. λ P Θ m htλ)=1 For a partition λ, let oλ) denote the number of odd parts of λ, and for λ P Θ, let oλ) = ϕ Θ ϕ oλϕ)). The following gives the Deligne- Lusztig model for the finite unitary group. Theorem 6.2. Let q be odd. For each r such that 0 2r m, Γ m 2r Ind U 2r Sp 2r 1) = χ λ. oλ )=m 2r

23 ON THE CHARACTERISTIC MAP OF FINITE UNITARY GROUPS 23 Furthermore, 0 2r m Γ m 2r Ind U 2r Sp 2r 1) = λ =m χ λ Proof. Suppose µ, ν P Θ, such that htµ) = 1 and ν is even. From 5.8), Corollary 4.3, and Pieri s formula [19, I.5.16], 6.1) χ µ χ ν = 1) τµ)+τν) λ χ λ, where the sum is taken over all λ such that for every ϕ Θ, λϕ) νϕ) is a horizontal µϕ) -strip. We now use Corollary 5.1 to show that χ µ χ ν is a character. As λϕ) νϕ) is a horizontal µϕ) -strip, the part λϕ) i is either νϕ) i or νϕ) i + 1 for every i = 1, 2,..., lλϕ)). By assumption, ν is even, so νϕ) i is even for every ϕ Θ, and so ) ) ) νϕ) i + 1 νϕ) = νϕ) i + i νϕ) i mod 2) Thus, nλϕ)) = λϕ) ) i i 2 nνϕ)) mod 2). The assumption htµ) = 1 implies nµϕ)) = 0, and since ν is even, nµ) + nν) nλ) + µ ν mod 2). By Corollary 5.1, χ µ χ ν is a character. Use the decompositions of Theorem 5.3 and Theorem 6.1 in the product 6.1) to observe that the irreducible characters χ λ in the decomposition of Γ m 2r Ind U 2r Sp 2r 1) are indexed by λ Pm Θ such that for every ϕ, λϕ) νϕ) is a horizontal µϕ) -strip, where µ = m 2r, for some νϕ) such that νϕ) is even. Then the number of odd parts of λϕ) is exactly µϕ), and so the λ in the decomposition must satisfy ϕ Θ ϕ oλϕ) ) = µ = m 2r. Acknowledgments. The first author would like to thank Stanford University for supplying a stimulating research environment, and, in particular, P. Diaconis for acting as a sounding board for many of these ideas. The second author thanks the Tata Institute of Fundamental Research for an excellent environment for doing much of this work, and Rod Gow for pointing out that his results in [8] apply to the proof of Corollary 5.2. Both authors express gratitude to the referee for helpful comments. References 1. D. Bump and D. Ginzburg, Generalized Frobenius-Schur numbers, J. Algebra ), no. 1, R. Carter, Finite groups of Lie type: conjugacy classes and complex characters. John Wiley and Sons, P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 2) ), no. 1,

24 24 NATHANIEL THIEM AND C. RYAN VINROOT 4. F. Digne and J. Michel, Foncteurs de Lusztig et caractères des groupes linéaires et unitaires sur un corps fini, J. Algebra ), no. 1, V. Ennola, On the conjugacy classes of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I No ), 13 pages. 6. V. Ennola, On the characters of the finite unitary groups, Ann. Acad. Sci. Fenn. Ser. A I No ), 35 pages. 7. R. Gow, Properties of the characters of the finite general linear group related to the transpose-inverse involution, Proc. London Math. Soc. 3) ), no. 3, R. Gow, A correspondence for conjugacy classes in certain extensions of order 2 of finite groups of lie type, Preprint, J.A. Green, The characters of the finite general linear groups, Trans. Amer. Math. Soc ), no. 2, L.C. Grove, Classical groups and geometric algebra, Graduate Studies in Mathematics, Volume 39, American Mathematical Society, A. Henderson, Symmetric subgroup invariants in irreducible representations of G F, when G = GL n, J. Algebra ), no. 1, R. Hotta and T.A. Springer, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups, Invent. Math ), no. 2, N.F.J. Inglis and J. Saxl, An explicit model for the complex representations of the finite general linear groups, Arch. Math. Basel) ), no. 5, N. Kawanaka, On the irreducible characters of the finite unitary groups, J. Math. Soc. Japan ), no. 3, N. Kawanaka, Generalized Gel fand-graev representations and Ennola duality, In Algebraic groups and related topics Kyoto/Nagoya, 1983), , Adv. Stud. Pure Math., 6, North-Holland, Amsterdam, N. Kawanaka and H. Matsuyama, A twisted version of the Frobenius-Schur indicator and multiplicity-free representations, Hokkaido Math. J ), no. 3, A. A. Klyachko, Models for complex representations of the groups GLn, q), Mat. Sb. N.S.) ) 1983), no. 3, G. Lusztig and B. Srinivasan, The characters of the finite unitary groups, J. Algebra ), no. 1, I.G. Macdonald, Symmetric functions and Hall polynomials. Second edition. With Contributions by A. Zelevinsky. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, J. MacWilliams, Orthogonal matrices over finite fields, Amer. Math. Monthly ), Z. Ohmori, On a Zelevinsky theorem and the Schur indices of the finite unitary groups, J. Math. Sci. Univ. Tokyo ), no. 2, G. E. Wall, On the conjugacy classes in the unitary, orthogonal and symplectic groups, J. Austral. Math. Soc ), A.V. Zelevinsky, Representations of finite classical groups. A Hopf algebra approach, Lecture Notes in Mathematics 869, Springer Verlag, Berlin New York, Department of Mathematics, Stanford University, Stanford, CA address: thiem@math.stanford.edu Department of Mathematics, The University of Arizona, 617 N Santa Rita Ave, Tucson, AZ address: vinroot@math.arizona.edu