Real representations of finite symplectic groups over fields of characteristic two
aa r X i v : . [ m a t h . R T ] A ug REAL REPRESENTATIONS OF FINITE SYMPLECTICGROUPS OVER FIELDS OF CHARACTERISTIC TWO
C. RYAN VINROOT
Abstract.
We prove that when q is a power of 2, every complex irre-ducible representation of Sp(2 n, F q ) may be defined over the real num-bers, that is, all Frobenius-Schur indicators are 1. We also obtain agenerating function for the sum of the degrees of the unipotent charac-ters of Sp(2 n, F q ), or of SO(2 n + 1 , F q ), for any prime power q .2010 AMS Mathematics Subject Classification : 20C33, 05A15 Introduction
It was proved by Gow [12, Theorem 1] that if q is the power of an oddprime, and χ is an irreducible complex character of the finite symplecticgroup Sp(2 n, F q ) which is real-valued, then the complex representation π which affords χ is a real representation (or defined over R ) if and onlyif π acts trivially on the center of Sp(2 n, F q ). That is, Gow computes theFrobenius-Schur indicators of the real-valued characters of Sp(2 n, F q ) when q is odd. As it turns out, when q ≡ n, F q ) are real-valued, but this is not the case when q ≡ q , there is a certain twisted Frobenius-Schur indicator which is 1 for allirreducible complex characters of Sp(2 n, F q ).The technique used to prove the results just stated is not amenable forthe case when q is even, although several partial results have been proved.It follows from results of Gow [11] and Ellers and Nolte [7] that when q is even, all complex irreducible characters of Sp(2 n, F q ) are real-valued. Aresult of Prasad [23, Theorem 3] implies that any complex irreducible rep-resentation of Sp(2 n, F q ) with q even, which appears in the Gelfand-Graevrepresentation, can be defined over the real numbers. It has also been showncomputationally [24, Theorem 5.2] than when q is even and n ≤
8, all com-plex irreducible representations of Sp(2 n, F q ) are real representations. Inthis paper, we finally settle the general case, and we show in Theorem 4.2that when q is even every complex irreducible representation of Sp(2 n, F q ),for any n , is a real representation. In the process of proving this statement,we also obtain a new combinatorial identity in Theorem 4.1, which statesthat for any prime power q , the sum of the degrees of the unipotent charac-ters of Sp(2 n, F q ) (or of SO(2 n + 1 , F q )) is Q ni =1 ( q i −
1) times the coefficient
C. RYAN VINROOT of u n in the expansion of Y i ≥ u/q i − u/q i − Y ≤ i Acknowledgements. The author was supported in part by a grantfrom the Simons Foundation, Award Preliminaries Frobenius-Schur indicators and involutions. If G is a finite group,we let Irr( G ) denote the set of irreducible complex characters of G . Given χ ∈ Irr( G ), the Frobenius-Schur indicator of χ , which we denote by ε ( χ ), isdefined by the formula ε ( χ ) = 1 | G | X g ∈ G χ ( g ) . Let ( π, V ) be an irreducible complex representation afforded by the character χ . We say that ( π, V ) is a real representation or defined over R if there is abasis for V such that all images of the corresponding matrix representationhave all real entries. By classical results of Frobenius and Schur, ε ( χ ) hasthe following property: ε ( χ ) = π, V ) is defined over R , − χ = ¯ χ but ( π, V ) is not defined over R ,0 if χ = ¯ χ. By applying this fact and the orthogonality relations of characters, one ob-tains the Frobenius-Schur involution formula : X χ ∈ Irr( G ) ε ( χ ) χ (1) = { g ∈ G | g = 1 } . Define an involution in G to be an element g ∈ G such that g = 1. Itthen follows from the formula above that ε ( χ ) = 1 for every χ ∈ Irr( G ) ifand only if the sum of all character degrees of G is equal to the number ofinvolutions in G . See [14, Chapter 4] for all of the above theory regardingthe Frobenius-Schur indicator.We now consider the finite groups of interest. Throughout this paper, welet F q be a finite field with q elements with q a power of a prime p , and ¯ F q afixed algebraic closure of F q . First let G = SO(2 n +1 , F q ) be the finite specialorthogonal group over F q , where q is odd. Gow [12, Theorem 2] provedthat ε ( χ ) = 1 for every χ ∈ Irr( G ). Meanwhile, Fulman, Guralnick, andStanton have computed a generating function for the number of involutionsin G = SO(2 n + 1 , F q ) (where G = 1 if n = 0). Specifically, it followsfrom [8, Theorem 2.17 and Lemma 6.1(3)] and the fact that O(2 n + 1 , F q ) ∼ =SO(2 n + 1 , F q ) × {± } (see also [24, Theorem 7.1(2)]), that the number of C. RYAN VINROOT involutions in SO(2 n + 1 , F q ) is Q ni =1 ( q i − 1) times the coefficient of u n inthe expansion of 11 − u Y i ≥ (1 + u/q i ) − u /q i . From this result, along with the Frobenius-Schur involution formula andGow’s result that ε ( χ ) = 1 for every χ ∈ Irr(SO(2 n + 1 , F q )), we have thefollowing consequence. Lemma 2.1. For q odd, the sum of the character degrees of SO(2 n + 1 , F q ) is Q ni =1 ( q i − times the coefficient of u n in the expansion of − u Y i ≥ (1 + u/q i ) − u /q i . We now consider the case that q is even, and G = Sp(2 n, F q ), and recallthat G is isomorphic to SO(2 n + 1 , F q ) in this case (and when n = 0 wetake Sp(0 , F q ) = SO(1 , F q )). Fulman, Guralnick, and Stanton have shownthat [8, Theorems 2.14 and 5.3] when q is even, the number of involutionsin Sp(2 n, F q ) is Q ni =1 ( q i − 1) times the coefficient of u n in the expansion of11 − u Y i ≥ u/q i − u /q i . This generating function, together with the Frobenius-Schur involution for-mula, provides the following criterion for all complex irreducible represen-tations of Sp(2 n, F q ), with q even, to be defined over the real numbers. Lemma 2.2. Let q be even. Then ε ( χ ) = 1 for every χ ∈ Irr(Sp(2 n, F q )) if and only if the character degree sum of Sp(2 n, F q ) is Q ni =1 ( q i − timesthe coefficient of u n in the expansion of − u Y i ≥ u/q i − u /q i . Lemmas 2.1 and 2.2 motivate our main strategy in proving that indeed ε ( χ ) = 1 for all χ ∈ Irr(Sp(2 n, F q )) when q is even.2.2. Self-dual polynomials. Let f ( t ) ∈ F q [ t ] be a monic polynomial withnonzero constant term of degree d , say f ( t ) = t d + a d − t d − + · · · + a t + a .We define the dual polynomial of f ( t ), denoted f ∗ ( t ), as f ∗ ( t ) = a − t n f ( t − ) . We say f ( t ) is self-dual if f ( t ) = f ∗ ( t ). Equivalently, f ( t ) is self-dual if,whenever α ∈ ¯ F × q is a root of f ( t ) with multiplicity m , then α − is also aroot of f ( t ) with multiplicity m .We let N ( q ) be the set of monic irreducible self-dual polynomials withnonzero constant in F q [ t ], and we let N ∗ ( q ; d ) denote the number of poly-nomials in N ( q ) of degree d . Denote by M ( q ) the set of unordered pairs EAL REPRESENTATIONS OF FINITE SYMPLECTIC GROUPS 5 { g, g ∗ } of monic irreducible polynomials with nonzero constant in F q [ t ] suchthat g ( t ) = g ∗ ( t ), and let M ∗ ( q ; d ) be the number of unordered pairs { g, g ∗ } in M ( q ) such that g has degree d . By [9, Lemma 1.3.16], we have that if d isodd and d > N ∗ ( q ; d ) = 0. That is, other than t ± 1, all polynomialsin N ( q ) have even degree.Define, throughout this paper, the number e ( q ) = e to be e = 1 if q is evenand e = 2 if q is odd. We recall the following identities involving self-dualpolynomials which we will need. These results are in [9, Lemma 1.3.17(a,d)]. Lemma 2.3. We have the following formal identities of power series in anindeterminate w : (a) Y d ≥ (1 − w d ) − N ∗ ( q ;2 d ) (1 − w d ) − M ∗ ( q ; d ) = (1 − w ) e − qw . (b) Y d ≥ (1 + w d ) − N ∗ ( q ;2 d ) (1 − w d ) − M ∗ ( q ; d ) = 1 − w. Partitions and Schur functions. Let P denote the set of all par-titions of non-negative integers, and let P n denote the set of partitions of n , where P consists of only the empty partition. That is, if λ ∈ P n with n ≥ 1, then λ = ( λ , λ , . . . , λ l ) such that λ i ≥ λ i +1 and λ i > i ,and | λ | = l X i =1 λ i = n. Given a partition λ = ( λ , λ , . . . , λ ℓ ) ∈ P , we will need the statistic a ( λ )defined as a ( λ ) = l X i =1 ( i − λ i . Given λ ∈ P , we let λ ′ denote the conjugate partition, which is obtained bytransposing the Young diagram of λ . That is, λ ′ i is the number of j such that λ j ≥ i . If we identify λ with its Young diagram, let y ∈ λ denote a positionin the Young diagram. We then let h ( y ) denote the hook-length of thatposition in λ , so if y is in position ( i, j ) of λ , then h ( y ) = λ i + λ ′ j − i − j + 1(see [19, I.1, Example 1]). From [19, I.1, Example 2], we have the identity(2.1) X y ∈ λ h ( y ) = a ( λ ) + a ( λ ′ ) + | λ | . Given λ ∈ P and a countable set of variables { x , x , . . . } , we let s λ = s λ ( x , x , . . . )denote the Schur function corresponding to λ in these variables, see [19, I.3]for a definition. In particular, s λ is a homogeneous symmetric function ofdegree | λ | , so for any b we have(2.2) b | λ | s λ ( x , x , . . . ) = s λ ( bx , bx , . . . ) . C. RYAN VINROOT We will need to apply the following identities for Schur functions, which are[19, I.5, Examples 4 and 6]. Lemma 2.4. We have the identities (1) X λ ∈P s λ ( x , x , . . . ) = Y i ≥ (1 − x i ) − Y ≤ i Let G be a con-nected reductive group over ¯ F q , defined over F q through a Frobenius auto-morphism F . In this section, we recall facts about the irreducible complexcharacters of the finite reductive group G = G F .If T is any maximal F -stable torus of G , write T = T F , and let θ beany linear complex character of T . Deligne and Lusztig [5] defined a virtualcharacter R GT ( θ ) of G corresponding to the pair ( T, θ ). A unipotent characterof G is any irreducible character of G which appears as a constituent of R GT ( ) for some maximal torus T , where is the trivial character. We needthe following basic property of unipotent characters. Lemma 2.5. Let G , G , and G be connected reductive groups over ¯ F q suchthat G ∼ = G × G . Suppose G is defined over F q by Frobenius F , and G i is F -stable for i = 1 , , so that G F ∼ = G F × G F . Then the collection ofunipotent characters of G = G F is the collection of characters of the form ψ ⊗ ψ where ψ i is a unipotent character of G i = G Fi for i = 1 , .Proof. First, every maximal torus T of G is of the form T ∼ = T × T , where T i is a maximal torus of G i for i = 1 , 2. So if T = T F is a maximal F -stabletorus of G , we can write T = T × T , where T i = T Fi with T i a maximal F -stable torus of G i for i = 1 , 2. Then we have R GT ( ) = R G × G T × T ( ⊗ ) ∼ = R G T ( ) ⊗ R G T ( ), by [13, Lemma 2.6(ii)] for example. The result follows. (cid:3) Now let G ∗ be the group dual to G , with dual Frobenius map F ∗ , andwrite G ∗ = G ∗ F ∗ (see [4, Secs. 4.2 and 4.3]). We now assume that G has connected center. If s ∈ G ∗ is any semisimple element of G ∗ , then thecentralizer C G ∗ ( s ) is a connected reductive group since we are assuming thecenter Z ( G ) is connected [4, Theorem 4.5.9]. So we may consider unipotentcharacters of the group C G ∗ ( s ) F ∗ .With the assumption that Z ( G ) is connected, we have a parametrizationof Irr( G ) through the Jordan decomposition of characters, originally devel-oped by Lusztig [18]. Given any χ ∈ Irr( G ), χ bijectively corresponds to a G ∗ -conjugacy class of pairs ( s, ψ ), where s ∈ G ∗ is a semisimple element,and ψ is a unipotent character of C G ∗ ( s ) F ∗ . We refer to [2, Chapter 15] fora thorough treatment of this bijection, which has many useful properties.The main information we need from this parametrization of characters for G is the description of character degrees, which follows from the main resultof Lusztig on the Jordan decomposition of characters [18, Theorem 4.23] EAL REPRESENTATIONS OF FINITE SYMPLECTIC GROUPS 7 (for relevant discussion see [4, Section 12.9] and [6, Remark 13.24]). Sup-pose that χ ∈ Irr( G ) corresponds to the G ∗ -class of pairs ( s, ψ ), and write χ = χ ( s,ψ ) . The degree of χ ( s,ψ ) is then given by(2.3) χ ( s,ψ ) (1) = [ G ∗ : C G ∗ ( s )] p ′ ψ (1) , where p = char( F q ), and the subscript p ′ denotes the prime-to- p part. Weremark that, at least when G is a simple algebraic group, for any semisimple s ∈ G ∗ we have C G ∗ ( s ) = C G ∗ ( s ) F ∗ (see [3, pg. 491]).3. Character degrees of SO(2 n + 1 , F q )3.1. Semisimple classes and centralizers. We now apply the ideas ofSection 2.4 to the specific case G = G F = SO(2 n + 1 , F q ), and note that G = SO(2 n + 1 , ¯ F q ) has connected center (since the center is trivial). In thiscase, G ∗ = G ∗ F ∗ = Sp(2 n, F q ) (see [4, pg. 120]). We describe the semisimpleclasses of G ∗ = Sp(2 n, F q ), and unipotent characters of their centralizers.To describe the semisimple classes of Sp(2 n, F q ) and their centralizers, weapply the results of Wall [26]. Specifically, when q is odd this descriptionmay be concluded from [26, Sec. 2.6 Case (B)], and when q is even this isfrom [26, Sec. 3.7]. For details and more general results, see [21, Sec. 2].Recall the notation from Section 2.2 that N ( q ) denotes the set of monicirreducible self-dual polynomials with nonzero constant in F q [ t ], and M ( q )is the set of unordered pairs { g ( t ) , g ∗ ( t ) } such that g = g ∗ , where g ( t ) ismonic irreducible with nonzero constant in F q [ t ]. Let us further denote N ′ ( q ) = N ( q ) \ { t + 1 , t − } , and so if f ( t ) ∈ N ′ ( q ) then deg( f ) is even. Asemisimple class ( s ) in Sp(2 n, F q ) depends only on its elementary divisors,which must each be of the form f ( t ) m f for f ( t ) ∈ N ′ ( q ), or g ( t ) n g and( g ( t ) ∗ ) n g for { g ( t ) , g ∗ ( t ) } ∈ M ( q ), or ( t − m + or ( t + 1) m − (and when q is even we only have ( t − m + ). In other words, a semisimple class ( s ) ofSp(2 n, F q ) is parametrized by a function(3.1) Φ : N ( q ) ∪ M ( q ) → Z ≥ , such that, if we write Φ( f ) = m f for f ∈ N ( q ) ′ , Φ( { g, g ∗ } ) = m g for { g, g ∗ } ∈ M ( q ), and Φ( t ± 1) = m ± , we have(3.2) | Φ | := X f ∈N ′ ( q ) m f deg( f ) / X { g,g ∗ }∈M ( q ) m g deg( g ) + m + + m − = n, where we leave off m − in the case that q is even. Given a semisimple class( s ) of G ∗ = Sp(2 n, F q ) parametrized by Φ as above, the centralizer C G ∗ ( s )has structure given by C G ∗ ( s ) ∼ = Y f ∈N ′ ( q ) U( m f , F q deg( f ) / ) × Y { g,g ∗ }∈M ( q ) GL( m g , F q deg( g ) )(3.3) × Sp(2 m + , F q ) × Sp(2 m − , F q ) , where the last factor is left off in the case that q is even, and U( n, F q ) denotesthe full unitary group defined over F q . In reference to the character degree C. RYAN VINROOT formula given in (2.3), if p = char( F q ), then using the orders of the finiteunitary, general linear, and symplectic groups from (3.3) we have(3.4) [ G ∗ : C G ∗ ( s )] p ′ = Q ni =1 ( q i − P ( s ) , where P ( s ) is given by Y f ∈N ′ ( q ) m f Y i =1 ( q i deg( f ) / − ( − i ) Y { g,g ∗ }∈M ( q ) m g Y i =1 ( q i deg( g ) − m + Y i =1 ( q i − m − Y i =1 ( q i − , and the last product is left off when q is even.3.2. Unipotent characters. We now must describe the unipotent charac-ters of C G ∗ ( s ), which by Lemma 2.5 are products of unipotent characters ofthe factors of C G ∗ ( s ) given in (3.3).Unipotent characters of GL( n, F q ) and U( n, F q ) are each parametrized by P n , the partitions of n . We use the notation of Section 2.3 in the following.Given λ ∈ P n , the unipotent character ψ GL ,λ of GL( n, F q ) parametrized by λ has degree which can be written as ψ GL ,λ (1) = n Y i =1 ( q i − · q a ( λ ′ ) Q y ∈ λ ( q h ( y ) − , a form which can be found in [22, Equation (21)], and can also be obtainedfrom [19, Chapter 4, Equation (6.7)]. From [16, Remark 9.5], the unipotentcharacter ψ U ,λ of U( n, F q ) parametrized by λ has degree ψ U ,λ (1) = n Y i =1 ( q i − ( − i ) · q a ( λ ′ ) Q y ∈ λ ( q h ( y ) − ( − h ( y ) ) . Using (2.1), we have q a ( λ ′ ) Q y ∈ λ ( q h ( y ) − ( ± h ( y ) ) = q a ( λ ′ ) q P y ∈ λ h ( y ) Q y ∈ λ (1 − ( ± /q ) h ( y ) )= 1 q | λ | q a ( λ ) Q y ∈ λ (1 − ( ± /q ) h ( y ) )= 1 q n ( ± a ( λ ) ( ± /q ) a ( λ ) Q y ∈ λ (1 − ( ± /q ) h ( y ) ) . Similar to the calculation in [19, pg. 286], we now apply [19, I.3, Example2] to write(3.5) ψ U ,λ (1) = n Y i =1 ( q i − ( − i ) · q n ( − a ( λ ) s λ (cid:0) , − /q, ( − /q ) , . . . (cid:1) , and(3.6) ψ GL ,λ (1) = n Y i =1 ( q i − · q n s λ (cid:0) , /q, /q , . . . (cid:1) . EAL REPRESENTATIONS OF FINITE SYMPLECTIC GROUPS 9 In order to describe the unipotent characters of Sp(2 n, F q ), we need todefine symbols , originally introduced in [16]. Following [4, Sec. 13.8], asymbol is an ordered pair of finite sets of non-negative integers, say Λ =( µ, ν ), where we write µ = ( µ < µ < . . . < µ r ), ν = ( ν < ν < . . . < ν k ),such that µ and ν cannot both be 0, and such that r − k > 0. The number r − k is called the defect of the symbol Λ. The rank of a symbol Λ, whichwe denote by | Λ | , is defined as | Λ | = r X i =1 µ i + k X i =1 ν i + $(cid:18) r + k − (cid:19) % . Note that the symbol (0 , ∅ ) is the only symbol of rank 0, and has defect1. We let S denote the collection of all symbols of odd defect, and we let S n denote the set of all symbols of odd defect with rank n . It was provedby Lusztig [16] (see also [1, 17]) that the set S n parametrizes the unipotentcharacters of Sp(2 n, F q ) as well as those of SO(2 n + 1 , F q ) (where we defineSp(0 , F q ) = SO(1 , F q ) = 1 for any q ). If ψ Λ is the unipotent character ofSp(2 n, F q ) (or of SO(2 n + 1 , F q )) corresponding to Λ ∈ S n , then the degreeof ψ Λ is given by(3.7) ψ Λ (1) = n Y i =1 ( q i − · δ (Λ) , where δ (Λ) is given by (following [4, Sec. 13.8]) δ (Λ) = Q i With G = SO(2 n + 1 , F q ), we now consideran arbitrary χ ∈ Irr( G ), so χ = χ ( s,ψ ) corresponds to some G ∗ -class of pairs( s, ψ ) in the Jordan decomposition. Then s ∈ G ∗ = Sp(2 n, F q ) correspondsto some Φ satisfying (3.1) and (3.2), where C G ∗ ( s ) has structure given by(3.3), and there is no m − term when q is even. As in Section 3.2, by Lemma2.5 the unipotent character ψ then has structure given by(3.8) ψ = O f ∈N ′ ( q ) ψ U ,λ f ⊗ O { g,g ∗ }∈M ( q ) ψ GL ,λ g ⊗ ψ Λ + ⊗ ψ Λ − , where ψ U ,λ f is a unipotent character of U( m f , F q deg( f ) / ) with | λ f | = m f , ψ GL ,λ g is a unipotent of GL( m g , F q deg( g ) ) with | λ g | = m g , and ψ Λ ± is a unipotent of Sp(2 m ± , F q ) with | Λ ± | = m ± but there is no m − or Λ − when q is even.Because of the expressions in (3.5) and (3.6), we define δ U ( λ, d ) = 1 q d | λ | / ( − a ( λ ) s λ (1 , − /q d/ , ( − /q d/ ) , ( − /q d/ ) , . . . ) , and δ GL ( λ, d ) = 1 q d | λ | s λ (1 , /q d , /q d , /q d , . . . ) . Then the degree of ψ in (3.8) is given by ψ (1) = P ( s ) Y f ∈N ′ ( q ) δ U ( λ f , deg( f )) Y { g,g ∗ }∈M ( q ) δ GL ( λ g , deg( g )) · δ (Λ + ) δ (Λ − ) , where P ( s ) is exactly as in (3.4), and there are no m − or Λ − factors when q iseven. Finally, by (2.3) and (3.4), the degree of χ ( s,ψ ) is given by Q ni =1 ( q i − Y f ∈N ′ ( q ) δ U ( λ f , deg( f )) Y { g,g ∗ }∈M ( q ) δ GL ( λ g , deg( g )) · δ (Λ + ) δ (Λ − ) , again with no δ (Λ − ) factor when q is even.We next give a generating function for the sum of the degrees of all irre-ducible characters of SO(2 n + 1 , F q ). For this purpose, we define the powerseries W ( u ) as follows, which is a generating function for the degrees of theunipotent characters of Sp(2 n, F q ) for any n ≥ W ( u ) = X Λ ∈S δ (Λ) u | Λ | . The following result gives us the fundamental tool for our main argumentsin the next section. Proposition 3.1. The character degree sum of SO(2 n + 1 , F q ) is Q ni =1 ( q i − times the coefficient of u n in the expansion of Y d ≥ X λ ∈P δ U ( λ, d ) u | λ | d ! N ∗ ( q ;2 d ) Y d ≥ X λ ∈P δ GL ( λ, d ) u | λ | d ! M ∗ ( q ; d ) W ( u ) e , where e = 2 if q is odd and e = 1 if q is even.Proof. To obtain such a generating function, we need the coefficient of u n to be the sum of the expressions in (3.9), over all choices for ( s, ψ ) whichparametrize irreducible characters of SO(2 n + 1 , F q ). Since the choice in ( s )amounts to the choice of Φ satisfying (3.1) and (3.2), and the choice of ψ in (3.8) amounts to the choices of λ f , λ g ∈ P and Λ ± ∈ S , the generatingfunction we want is given by Y f ∈N ′ ( q ) X λ f ∈P δ U ( λ f , deg( f )) u | λ f | deg( f ) / EAL REPRESENTATIONS OF FINITE SYMPLECTIC GROUPS 11 · Y { g,g ∗ }∈M ( q ) X λ g ∈P δ GL ( λ g , deg( g )) u | λ g | deg( g ) · X Λ + ∈S δ (Λ + ) u | Λ + | X Λ − ∈S δ (Λ − ) u | Λ − | , where the last factor with the sum over Λ − is not included when q is even.First note that these last two factors are both just W ( u ), so these factors(or single factor in the q even case) can be replaced by W ( u ) e . In the firsttwo products, once f ∈ N ′ ( q ) or { g, g ∗ } ∈ M ( q ) is fixed, the expression inthe summations over λ f or λ g depend only on the degrees of f or g . Fromthese observations, we can rewrite the generating function as Y d ≥ X λ ∈P δ U ( λ, d ) u | λ | d/ ! N ∗ ( q ; d ) Y d ≥ X λ ∈P δ GL ( λ, d ) u | λ | d ! M ∗ ( q ; d ) W ( u ) e , where the first product is over d ≥ N ′ ( q ) contains no polynomialsof degree 1 by definition. Further, as stated in Section 2.2, we know N ′ ( q )contains no polynomials of odd degree. We can therefore replace d with 2 d in the first product above and then take the product over all d ≥ 1, givingthe generating function claimed. (cid:3) Main Results We continue all notation from Section 3.3 above. We begin with a crucialcalculation. Proposition 4.1. We have the following identity of power series: Y d ≥ X λ ∈P δ U ( λ, d ) u | λ | d ! N ∗ ( q ;2 d ) Y d ≥ X λ ∈P δ GL ( λ, d ) u | λ | d ! M ∗ ( q ; d ) = 11 − u Y i ≥ (1 − u/q i − ) e − u /q i Y ≤ i 1. By Lemma 2.4(2), we then have X λ ∈P δ U ( λ, d ) u | λ | d = X λ ∈P ( − a ( λ ) s λ ( u d /q d , − u d /q d , u d /q d , . . . )= Y i ≥ (cid:18) − ( − i − u d q id (cid:19) − Y ≤ i We are now in the position to prove the main results. Theorem 4.1. For any prime power q , the sum of the degrees of the unipo-tent characters of Sp(2 n, F q ) , or of SO(2 n + 1 , F q ) , is Q ni =1 ( q i − times the coefficient of u n in the expansion of Y i ≥ u/q i − u/q i − Y ≤ i The generating function which we are calculating is W ( u ) = X Λ ∈S δ (Λ) u | Λ | = X n ≥ X Λ ∈S n δ (Λ) ! u n , where δ (Λ) is the expression in q given in Section 3.2. By Lemma 2.1 andProposition 3.1, we have that when q is odd Y d ≥ X λ ∈P δ U ( λ, d ) u | λ | d ! N ∗ ( q ;2 d ) Y d ≥ X λ ∈P δ GL ( λ, d ) u | λ | d ! M ∗ ( q ; d ) W ( u ) = 11 − u Q i ≥ (1 + u/q i ) Q i ≥ (1 − u /q i ) . By substituting in the expression from Proposition 4.1 with q odd (so e = 2),we have W ( u ) − u Y i ≥ (1 − u/q i − ) − u /q i Y ≤ i 11 + ( − i + j u /q i + j , and from (4.2) the sum of the degrees of the unipotent characters of GL( n, F q )is Q ni =1 ( q i − 1) times the coefficient of u n in Y i ≥ − u/q i Y ≤ i Theorem 4.2. Let q be a power of , and G = Sp(2 n, F q ) . Then ε ( χ ) = 1 for every χ ∈ Irr( G ) , that is, every complex irreducible representation of G may be defined over the real numbers.Proof. When q is a power of 2, we have G = Sp(2 n, F q ) ∼ = SO(2 n + 1 , F q ),and so by Proposition 3.1 the character degree sum of G (with e = 1 since q is even) is Q ni =1 ( q i − 1) times the coefficient of u n in the power series Y d ≥ X λ ∈P δ U ( λ, d ) u | λ | d ! N ∗ ( q ;2 d ) Y d ≥ X λ ∈P δ GL ( λ, d ) u | λ | d ! M ∗ ( q ; d ) W ( u ) . 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