A Characteristic Map for the Symmetric Space of Symplectic Forms Over A

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ch : C[Sp2n(Fq)\ GL2n(Fq)/ Sp2n(Fq)] → Λ n from the space ofM bi-invariant functions to a ring of symmetrOic functions and establish some basic properties. In particular, the spherical functions are shown to map to Macdonald polynomials with parameters (q,q2). Unfortu- nately, the multiplication on the ring of symmetric functions does not seem to have an easy representation-theoretic interpretation, and in particular is not given by parabolic induction in the sense defined by Grojnowski [10] (a previous version of this paper claimed to show this). Nevertheless, a "mixed product" can be defined using parabolic induction which takes a bi-invariant function on GL2n(Fq) and a class function on GLm(Fq) and produces a bi-invariant function on GL2(n+m)(Fq) and this (almost) corresponds to the product of symmetric functions under the characteristic map. It would be interesting to find a representation-theoretic interpretation of symmetric function multiplication under the characteristic map. This result should be seen as an application of the results in [3] and [13] to develop a framework to translate between the representation theory of GL2n(Fq)/ Sp2n(Fq) and the theory of symmetric functions. As examples of this, two applications are given. The first is an application to Schur-positivity of certain (unmodified) Mac- donald polynomials. If Pλ/µ(x; q,t) denotes the skew Macdonald polynomi- ν als, let Cλ/µ(q,t) be defined by ν Cλ/µ(q,t)= hPλ/µ(q,t),sν i where the inner product is the standard Hall inner product. These are the coefficients of the expansion of Pλ/µ in terms of Schur functions. They can be considered a deformed Littlewood-Richardson coefficient, which can be defined as hsλ/µ,sνi. The following theorem is proven using the characteristic map. Theorem 1.1. Let q = pe denote an odd prime power. Then for any partitions µ, ν, λ ν 2 Cλ/µ(q,q ) ≥ 0. This theorem is related to a conjecture of Haglund [26], which states that hJ (q,qk),s i λ µ ∈ N[q] (1 − q)|λ| and provides further evidence of this conjecture. In particular, taking µ = 0 in Theorem 1.1 shows hJ (q,q2),s i λ µ ≥ 0 (1 − q)|λ| A CHARACTERISTIC MAP FOR F F 3 GL2n( q )/ Sp2n( q ) when q is an odd prime power. The Schur expansion of integral Macdonald polynomials has been previously studied by Yoo who gave combinatorial formulas for the expansion coefficients in some special cases showing that they were polynomials in q with positive integer coefficients [26, 27]. Some related coefficients in the Jack case when q → 1 case was studied in [2]. The Schur-positivity proven is distinct from the well-known positivity result proven by Haiman [11], which deals with the modified Macdonald polynomials. The second application is to combinatorial formulas for the values of spherical functions on certain double cosets. Although there are formulas in [3] already for all values of the spherical functions, they are alternating and so are unsuitable for asymptotic analysis. Asymptotic analysis of spherical functions appears in the study of Markov chains on Gelfand pairs, see [6, 7] for examples.

1.3. Related work. In addition to the examples already mentioned for Sn, GLn(Fq) and S2n/Bn, there are further examples of characteristic maps appearing in the literature. In particular, Thiem and Vinroot constructed such a map for Un(Fq2 ) [24] and in [1] a map is constructed between the supercharacters of the unipotent upper triangular matrices over a finite field and symmetric functions in non-commuting variables. The connections between the representation theory of GL2n(Fq)/ Sp2n(Fq) and Macdonald polynomials are not new and were already noticed in [3]. A further connection was made by Shoji and Sorlin between a related space and the modified Kostka polynomials which are the change of basis from the Hall-Littlewood polynomials to the Schur functions [20]. As there is a rational parabolic for the Levi subgroup used in the parabolic induction, elementary proofs have been given for any necessary results. The statements and proofs were all inspired by a more general construction of Grojnowski [10] for parabolic induction of l-adic sheaves and subsequently studied by Henderson [13]. Some of this work was later extended in the work of Shoji and Sorlin [19–21] which can also be found in the survey [22]. The more general construction has the benefit of working even without a rational parabolic subgroup, and in particular the construction of a characteristic map for U2n(Fq2 )/ Sp2n(Fq) should be relatively straightforward although the combinatorial application in Section 5 would not extend. The q → 1 case was previously studied by Bergeron and Garsia [4] (see also [17]), where the Gelfand pair S2n/Bn played a similar role. In this case the multiplication on symmetric functions has a representation-theoretic interpretation giving results on the structure coefficients of Jack polynomials for α = 2. The lack of representation-theoretic interpretation in the q- deformed case means that their ideas cannot be directly applied here. Macdonald polynomials with parameters (q,q2) also appeared previously in the study of quantum symmetric spaces [18], and it would be interesting to see if there is a connection. 4 JIMMY HE

1.4. Outline. The paper is organized as follows. In Section 2, notation and preliminary background is reviewed. Section 3 develops the theory of parabolic induction for bi-invariant functions in an elementary way. In Section 4, the characteristic map for GL2n(Fq)/ Sp2n(Fq) is constructed. Section 5 gives an application to positivity and vanishing of the Schur expansion of skew Macdonald polynomials with parameters (q,q2) and Section 6 gives an application to computing spherical function values. Section 7 explains how parabolic induction for functions is a special case of parabolic induction of sheaves on symmetric spaces. Section 7 is the only section which requires any familiarity with character sheaves and the rest of the paper is independent of it.

2. Preliminaries In this section, some needed background is reviewed and the notation and conventions that are used are explained. The notation and background on character sheaves needed is left for Section 7. 2 2.1. Notation. If f(q) is a rational function in q, define f(q)q7→q2 := f(q ). For example, n−1 2n 2i | GLn(Fq)|q7→q2 = (q − q ). Yi=0 For a symmetric function f with rational coefficients in q when written in terms of pµ, write fq7→q2 to denote the symmetric function obtained by replacing q with q2 in each coefficient. Let G be a finite group. If S ⊆ G is some subset, IS will denote the indicator function for that set. If H ⊆ G is a subgroup, let C[G]G denote the set of class functions on G and let C[H\G/H] denote the set of H bi-invariant functions on G. These carry the usual inner product hf, gi = x∈G f(x)g(x). If S,H are subgroups of any group G, let HS = H ∩ S. Given an element P x of some finite field extension of Fq, let fx denote its minimal polynomial. 2.2. Macdonald polynomials. For details on Macdonald polynomials including a construction and proofs, see [17]. Let Λ denote the ring of symmetric functions. Consider the inner product on the ring of symmetric functions defined by qλi − 1 hpλ,pµiq,t = δλµzλ , tλi − 1 mi(λ) where zλ = mi(λ)!i and mi(λ) denotesY the number of parts of size i in λ. When the context is clear, the dependence on (q,t) may be dropped. This specializesQ to the Hall inner product when q = t and setting q = tα and taking a limit gives the Jack polynomial inner product. The Macdon- ald polynomials Pλ(x; q,t) (indexed by partitions λ) are defined by the fact that they are orthogonal with respect to this inner product, and the change A CHARACTERISTIC MAP FOR F F 5 GL2n( q )/ Sp2n( q ) of basis to the monomial basis is upper triangular with 1 along the diagonal. When q = 0, the Macdonald polynomials are known as Hall-Littlewood polynomials, and are written Pλ(x; t)= Pλ(x; 0,t). Define a(s) l(s)+1 cλ(q,t) := (1 − q t ), sY∈λ where a(s) and l(s) denote the arm and leg lengths respectively (so a(s)+ l(s)+1= h(s)). Similarly define ′ a(s)+1 l(s) cλ(q,t) := (1 − q t ). sY∈λ The dual basis to the Pλ(x; q,t) under h, iq,t are denoted Qλ(x; q,t) and satisfy cλ(q,t) Qλ(x; q,t)= ′ Pλ(x; q,t). cλ(q,t) The Macdonald polynomials have an integral form ′ Jλ(x; q,t)= cλ(q,t)Pλ(x; q,t)= cλ(q,t)Qλ(x; q,t). ′ Then hJλ, Jλi = cλ(q,t)cλ(q,t). The symmetric functions Jλ can be thought of as a deformation of the Jack polynomials, which are given by taking q = tα and sending t → 1 after dividing by (1 − t)n. There are also various homomorphisms defined on Λ, defined through their action on pn. These are also commonly written using plethystic notation but to match the notation in Macdonald this is avoided. n−1 Let ω : Λ → Λ denote the involution taking pn to (−1) pn (and extending to make it an algebra homomorphism). Let ωq,t : Λ → Λ denote the involution defined by qn − 1 ω p = (−1)n−1 p . q,t n tn − 1 n The involution ω interacts well with Schur functions, with

ωsλ = sλ′ while the involution ωq,t satisﬁes

ωq,tPλ(x; q,t)= Qλ′ (x; t,q)

ωq,tQλ(x; q,t)= Pλ′ (x; t,q). 2.3. Linear Algebraic groups. The point of view taken is to view the finite groups of interest as rational points of an algebraic group over Fq. Thus, the conventions and notation may differ slightly from more classical sources such as Carter [5] which work directly over the finite field. The definitions and conventions taken mostly follow Digne and Michel [8]. In general, linear algebraic groups G will be defined over Fq. Let F denote the Frobenius endomorphism, which will always be the one taking the matrix q F (xij) to (xij). Then G(Fq) or G will denote the Fq points. 6 JIMMY HE

∗ n A torus is a group which is isomorphic to (Fq ) for some n. Note that the maximal tori contained in GLn are precisely the subgroups which are conjugate to the standard maximal torus, which consists of diagonal matrices. A Levi subgroup L ⊆ G is a subgroup that is the centralizer of some torus

T . The Levi subgroups of GLn are all of the form GLni for ni = n. A Borel subgroup in GLn is some subgroup conjugate to the subgroup of upper triangular matrices which is the standard Borel subgroup.Q P A parabolic subgroup is a subgroup containing a Borel subgroup. In GLn these subgroups are conjugate to some subgroup of block upper-triangular matrices which are the standard parabolics. All parabolic subgroups have a decomposition P = LU where U is the unipotent radical of P and L is a choice of some Levi subgroup, called the Levi factor of P . In GLn, for a standard parabolic consisting of block upper-triangular matrices, the unipotent radical consists of matrices which are the identity in each block and arbitrary above the diagonal blocks. As an example, in GL4 ∗∗∗∗ 1 0 ∗ ∗ ∗ ∗ 0 0 ∗∗∗∗ 0 1 ∗ ∗ ∗ ∗ 0 0 P =   ,U =   , L =    0 0 ∗ ∗   0010   0 0 ∗ ∗         0 0 ∗ ∗   0001   0 0 ∗ ∗        are an example of a parabolic subgroup and the unipotent radi cal and Levi       subgroup respectively (the ∗’s denote arbitrary entries although the matrices must be invertible). As the unipotent radical is normal, every parabolic subgroup has a canonical projection P → L which for a standard parabolic in GLn amounts to replacing the blocks on the diagonal of a matrix in P with identity matrices. In general the projection of x ∈ P will be denoted by x. As Borel subgroups are parabolic, the same applies with L being a maximal torus. Say that a subgroup H ⊆ G is rational (or Fq-rational) if it is stable under F . Any rational subgroup H deﬁnes a subgroup HF ⊆ GF . Conversely, a subgroup of GF is always algebraic and so deﬁnes a rational subgroup H ⊆ G by base change. The standard Levi and parabolic subgroups are rational. In GLn, the rational points of rational maximal tori are of the form Fqi and two rational maximal tori are conjugate under GLF if and only if their n Q rational points are isomorphic.

2.4. Representation theory of GLn(Fq). To fix notation, the representation theory of GLn(Fq) is briefly reviewed. The representation theory of GLn(Fq) was originally developed by Green in [9] but this section follows the conventions in [17]. Let M denote the group of units of Fq and let Mn denote the fixed points of F n with F the Frobenius endomorphism F (x)= xq. ∗ Note that Mn may be identified with Fqn . Let L be the character group of the inverse limit of the Mn with norm maps between them. The Frobenius n endomorphism F acts on L in a natural manner so let Ln denote the F A CHARACTERISTIC MAP FOR F F 7 GL2n( q )/ Sp2n( q )

ﬁxed points in L, and note there is a natural pairing of Ln with Mn for each n (but these pairings are not consistent). The F -orbits of M can be viewed as irreducible polynomials over Fq under O 7→ α∈O(x − α). Denote by O(M) and O(L) the F -orbits in M and L respectively. Use P to denote the set of partitions. Then the conjugacy Q classes of GLn(Fq), denoted Cµ, are indexed by partition-valued functions µ : O(M) → P such that kµk := d(f)|µ(f)| = n, M f∈XO( ) where d(f) denotes the degree of f. This is because µ contains the infor- mation necessary to construct the Jordan canonical form. That is, given µ, construct a matrix in GLn(Fq) in Jordan form by taking for each orbit f ∈ O(M), l(µ(f)) blocks, of sizes µ(f)i, for each root of f. The resulting matrix has d(f) blocks of size µ(f)i for each f and i, and adding this all up gives kµk = n. For example, the partition-valued function corresponding to the set of transvections, which have Jordan form 1 1 0 . . . 0 0 1 0 . . . 0   0 0 1 . . . 0 ,  . . . . .   ......     0 0 0 . . . 1      n−2 correspond to the partition-valued function µ with µ(f1) = (21 ) and d(f) µ(f) = 0 for f 6= f1. Use qf to denote q . There is a formula for the sizes of conjugacy classes given by

| GLn(Fq)| |Cµ| = aµ(q) where mi(µ(f)) n 2n(µ(f)) −j aµ(q)= q qf (1 − qf ), M f∈YO( ) Yi≥1 jY=1 with n(λ)= (i − 1)λi. Similarly, the irreducible characters of GLn(Fq), denoted χλ, are indexed by functions Pλ : O(L) → P such that kλk := d(ϕ)|λ(ϕ)| = n, L ϕ∈XO( ) where d(ϕ) denotes the size of the orbit ϕ. The dimension of the irreducible representation corresponding to λ is given by n(λ(ϕ)′) −1 dλ = ψn(q) qϕ Hλ(ϕ)(qϕ) , L ϕ∈YO( ) 8 JIMMY HE

n i d(ϕ) h(x) where ψn(q)= i=1(q − 1), qϕ = q and Hλ(t) = x∈λ(t − 1), h(x) denoting the hook length. Note that with this convention, the trivial rep- Q Q n resentation corresponds to the partition-valued function λ(χ1) = (1 ) (χ1 being the trivial character) and 0 otherwise (this diﬀers from the usual convention for Sn, where the trivial representation corresponds to the partition (n)). As a matter of convention, µ will always be used to denote partition- valued functions O(M) → P while λ will be used to denote partition-valued functions O(L) → P. In general, if there is some expression involving a partition µ, F (µ), with F (0) = 1, then the same expression with µ : O(M) → P will be deﬁned as the product f∈O(M) F (µ(f)). Thus, write

zµ =Q zµ(f). M f∈YO( ) If the expression contains q, in each factor it should be replaced by qf . A similar convention is used for λ : O(L) → P. The Deligne-Lusztig characters (also called basic functions for GLn(Fq)) give another basis for the space of class functions on GLn(Fq). For a more detailed overview of Deligne-Lusztig characters, including their construction and properties, see the book of Carter [5]. GLn Given any rational maximal torus T ⊆ GLn, a (virtual) character ζT (·|θ) F of GLn(Fq) associated to some irreducible character θ of T can be constructed. The character constructed depends only on the GLn(Fq)-conjugacy class of the pair (T,θ). Deligne-Lusztig characters are rational functions in q in the following sense. Given an element g ∈ GLn(Fq) with Jordan decomposition g = su (s semisimple and u unipotent), the Deligne-Lusztig characters can be computed as GLn −1 Z(s) ζT (g|θ)= θ(xsx )Qx−1Tx(u), F x∈(GLXn /Z(s)) xsx−1∈T F Z(s) where Qx−1Tx(u) is a rational function of q, known as the Green function. The Green functions can be computed as Z(s) µ(f) Qx−1Tx(u)= Qγ(f)(qf ), M f∈YO( ) µ where Qρ (q) denotes the Green polynomials, µ is a partition valued function indexing the conjugacy class of g and γ(f) is the partition given by taking −1 F ∼ s ∈ x T x = Mki and including as parts the ki/d(f) for which f kills s restricted to Mki . Since isomorphic maximal tori in GLn(Fq) are always Q −1 F F conjugates, the tori x T x for x ∈ (GLn /Z(s)) are exactly those contain- F ing s up to conjugation by Z(s) , which in turn are of the form Mγi(f)d(f) for γ a partition valued function such that |γ(f)| = |µ(f)| for all f ∈ O(M). Q A CHARACTERISTIC MAP FOR F F 9 GL2n( q )/ Sp2n( q )

This means that the Deligne-Lusztig character can be written (2.1) ζGLn (g|θ)= θ(t) Qµ(f) (q ), T γt(f) f M t∈T,|γtX(f)|=|µ(f)| f∈YO( ) where γt denotes the partition valued function corresponding to the torus x−1T F x for t = xsx−1 and s is the semisimple element as described above. The Green polynomials give the change of basis from power sum to Hall- Littlewood polynomials and so satisfy µ −n(µ) −1 pρ(x)= Qρ (t)t Pµ(x; t ) µ X for a formal parameter t. The Deligne-Lusztig characters are invariant under conjugation of (T,θ) by GF . There is a correspondence between GF orbits of pairs (T,θ) and functions λ : O(L) → P with kλk = n. Call the function λ the combinatorial data associated to (T,θ). The correspondence is as follows. Given a torus T with rational points F isomorphic to Mki , and a character θ of T , consider the partition-valued function sending ϕ to the partition with parts ki/d(ϕ) for all i such that θ Q restricted to Mki lies in the orbit ϕ. Conversely, given λ, T can be constructed as a torus with rational points isomorphic to ϕ,i Mλ(ϕ)id(ϕ) and θ is given by picking for each factor M an element of the orbit ϕ λ(ϕ)id(ϕ) Q (which determines T and θ up to conjugacy by GF ). Given some λ : O(L) → P, consider the Levi subgroup Lλ whose rational points are given by

GL|λ(ϕ)|(Fqd(ϕ) ) ⊆ GLn(Fq), L ϕ∈YO( ) with Weyl group W (λ) = ϕ∈O(L) S|λ(ϕ)|. To each Weyl group element w there is an associated partition-valued function sending ϕ to the cycle Q type of w(ϕ). Let (Tw,θw) denote the torus and character associated to the combinatorial data deﬁned by w. Then there is a formula relating the Deligne-Lusztig characters to the irreducible characters of GLn(Fq) given by (−1)n−|λ| χ (g)= χSym (w(ϕ))ζGLn (g|θ ), λ λ′(ϕ) Tw w |W (λ)| L w∈XW (λ) ϕ∈YO( ) Sym where the χλ are the irreducible characters of the symmetric group. Here |λ| = ϕ |λ(ϕ)|. P 10 JIMMY HE

2.5. The symmetric space GL2n(Fq)/ Sp2n(Fq). Let −1 .  ..  −1 J =    1     .   ..     1    define the standard symplectic form (where any empty space in the matrix is 0) and define the involution ι(X)= −J(XT )−1J of GL2n. This choice is made for convenience so that the upper triangular matrices are stable under ι. This clearly commutes with the Frobenius map F . Then Sp2n denotes the subgroup of GL2n fixed by ι. Now for any ι-stable subgroup S, let Sι denote the subgroup of ι fixed points and S−ι denote the set of ι-split elements, which are elements s ∈ S such that ι(s)= s−1. A Gelfand pair is a (finite) group G, with a subgroup H ⊆ G such that inducing the trivial representation from H to G gives a multiplicity-free representation. For a Gelfand pair G/H, any representation ρ of G has either no non-zero H-fixed vectors, or a 1-dimensional subspace fixed pointwise by H. Say that ρ is a spherical representation if it has an H-fixed vector, and define the corresponding spherical function to be φ(g)= hvρ, ρ(g)vρi, where vρ is a unit H-fixed vector. The spherical functions can also be computed by averaging characters over −1 H. That is, φ(g) = |H| h∈H χ(hg) for χ the character of ρ (if ρ is not a spherical representation, then this average is 0). The spherical functions are the replacement for charactersP of a group and in particular, they form a basis for the space of bi-invariant functions on G. Now GLn(Fq)/ Spn(Fq) is a Gelfand pair, and the spherical functions, denoted φλ, are indexed by partition-valued function λ : O(L) → P with kλk = n (see [3], although note a different convention is used in this paper so all partitions labeling representations are transposed). For a partition λ, let λ ∪ λ denote the partition which contains every part of λ twice. Then φλ is the spherical function corresponding to the representation with character χλ∪λ of GL2n(Fq). Let Mµ ∈ GLn(Fq) denote a conjugacy class representative of Cµ. Let gµ ∈ GL2n(Fq) denote the matrix acting only on the first n coordinates by Mµ. The Sp2n(Fq)-double cosets of are indexed by µ : O(M) → P, with kµk = n, with the gµ being double coset representatives. Two key results from [3] are reproduced below. The first relates the sizes of double cosets in GL2n(Fq)/ Sp2n(Fq) to the sizes of conjugacy classes in GLn(Fq). A CHARACTERISTIC MAP FOR F F 11 GL2n( q )/ Sp2n( q )

Proposition 2.1 ( [3, Proposition 2.3.6]). Let µ : O(M) → P with kµk = n. Then F F F |H gµH | = |H ||Cµ|q7→q2 , F F where H gµH denotes the double coset indexed by µ in GL2n(Fq) and Cµ denotes the conjugacy class indexed by µ in GLn(Fq). The second result gives a formula for the values of spherical functions in terms of Deligne-Lusztig characters on GLn(Fq). GLn The function ζT (·|θ) (or any other class function on GLn(Fq)) may be F turned into an H -bi-invariant function on GL2n(Fq) by deﬁning

GLn GLn ζT (gµ|θ)= ζT (Mµ|θ). and extending to the double-coset. It turns out that the correct analogue for Deligne-Lusztig characters are given by the basic functions

G/H GLn ζT (·|θ)= ζT (·|θ)q7→q2 . Here the Deligne-Lusztig characters are viewed as rational functions in q as described above. Remark 2.2. It may seem a little strange to index the basic functions by maximal tori of GLn. It is more natural to view (T,θ) as a maximal torus of G stable under ι along with a (T ∩ H)F bi-invariant function θ of T F . The basic functions are then indexed by such pairs up to HF -conjugacy. The basic functions may be seen as parabolically induced from the torus T although instead of working with representations, l-adic sheaves must be used. Since there is a correspondence between these two indexing sets, there is no harm in choosing to index with maximal tori of GLn. The following theorem relates the spherical functions to the basic functions. Theorem 2.3 ( [3, Theorem 6.6.1]). Let λ : O(L) → P with kλk = n. Then

|λ| (−1) −n(λ(ϕ)′) 2 φλ = qϕ cλ(ϕ)′ (qϕ,qϕ)dλ(ϕ)′ (w)(qϕ) |W (λ)|  L  w∈XW (λ) ϕ∈YO( )  G/H  ζ (·|θw) × sgn(w) Tw , |T |ζG/H (1|θ ) w Tw w where cλ(q,t) denotes the scaling from the two parameter Macdonald polynomials to their integral forms, dλ(w)(q) denotes the change of basis from power 2 sum to two parameter Macdonald polynomials Pλ(q ,q) and sgn denotes the sign character of W (λ). Note that the convention used to label spherical representations in [3] is opposite the one used in this paper so all partitions are transposed. 12 JIMMY HE

3. Induction and restriction This section defines a bi-invariant version of the usual parabolic induction and restriction on reductive groups. Only the case when G = GL2n will be needed, but all results except those concerning basic functions hold for any connected reductive group G and involution ι (of algebraic groups). Only induction through rational parabolic subgroups will be considered. Everything in this section follows from (and was inspired by) more general results on a bi-invariant parabolic induction defined for sheaves on symmetric spaces originally due to Grojnowski. These results can be found in [10, 13, 19–21]. The connection between the two induction procedures and how the results below follow from results for sheaves is explained in Section 7. To keep the paper accessible and self-contained as much as possible, more elementary proofs of the needed results are given below. 3.1. Tori, Borel subgroups and parabolic subgroups. Let G be a connected reductive group defined over Fq (q odd) with Frobenius endomorphism F and an involution ι of algebraic groups commuting with F . Let H ⊆ G be the subgroup of fixed points of ι. Fix a rational ι-stable pair (T,B) of a maximal torus T and Borel subgroup B containing T . Say that a parabolic subgroup is standard if it contains B. For convenience in later sections, if G = GLn, T can be taken to be the set of diagonal matrices in GLn. All pairs of rational ι-stable (T,B) with T ⊆ B are conjugate under HF (see [19] for example). Furthermore, all rational ι-stable Levi subgroups in an ι-stable parabolic P are conjugate under HF ∩U F since they’re conjugate by a unique element of the unipotent radical. Since ι acts on G and preserves T and B, it preserves the simple roots and so determines an automorphism of the Dynkin diagram of G. Standard parabolic subgroups correspond to subsets of the Dynkin diagram and it’s not hard to see that ι-stable parabolics correspond to ι-stable subsets. This gives an easy way to classify the Levi subgroups with ι-stable rational parabolics. In the case of G = GL2n and ι fixing the symplectic group, the Dynkin diagram is A2n−1 and the action of ι is given by reflecting along the middle. Then the ι-stable parabolic subgroups correspond to subsets of the simple roots which are symmetric about the middle, consisting of pairs of the type Ani and one of type A2n0−1 with ni = n. The corresponding Levi subgroups are then of the form GL2n × GLn × GLn where GL2n 0 P i i 0 corresponds to the middle component of the diagram (and n0 = 0 if the Q middle vertex is not included) and the GLni × GLni factors correspond to the symmetric pairs of components. Take T ⊆ B to be the diagonal matrices and upper triangular matrices. Both are stable under ι. Then the standard parabolic subgroups are exactly the block upper triangular matrices, and the ι-stable ones are those whose block structure is symmetric about the anti-diagonal. A CHARACTERISTIC MAP FOR F F 13 GL2n( q )/ Sp2n( q )

3.2. Bi-invariant parabolic induction. Let P be a rational ι-stable parabolic subgroup with rational ι-stable Levi factor L and unipotent radical U. Then define a function G/H F F F F F F IndL⊆P : C[HL \L /HL ] → C[H \G /H ] with the formula G F F −2 ′ IndL⊆P (f)(x)= |H ∩ P | f(hxh ). ′ F h,hX∈H hxh′∈P F It is clear that this defines an HF bi-invariant function. This definition reduces to the standard definition of parabolic (or Harish-Chandra) induction when considering the group G × G and the involution switching the two factors. Remark 3.1. The choice is made to work with bi-invariant functions on GF rather than invariant functions on GF /HF , which is the convention taken in [3] and [13]. Of course these two points of view are completely equivalent and indeed the induction implicitly used in [3] can be viewed as G/H −1 F ΦG ◦ IndL⊆P ◦ΦL where ΦG(f)(xH )= f(x). When studying induction, many times it is convenient to work with standard parabolic subgroups. The next proposition shows that induction is invariant under conjugation by HF and so it always suffices to study induction through standard parabolics. F F F F Lemma 3.2. For any h0 ∈ H and f ∈ C[HL \L /HL ],

G/H h0 G/H Ind −1 −1 ( f) = IndL⊆P (f) h0Lh0 ⊆h0Ph0

h0 h0 −1 where f is the function f(x)= f(h0 xh0). −1 −1 Proof. Note that the projection map h0P h0 → h0Lh0 can be written as −1 −1 x 7→ h0h0 xh0h0 where x denotes the projection for P . Then compute

G/H h0 Ind −1 −1 ( f)(x) h0Lh0 ⊆h0Ph0 F F −2 −1 −1 ′ −1 =|H ∩ P | f(h0 h0h0 hxh h0h0 h0) h,h′∈HF ′ X F −1 hxh ∈h0P h0 =|HF ∩ P F |−2 f(hxh′) ′ F h,hX∈H hxh′∈P F G/H =IndL⊆P (f)(x) −1 −1 where H ∩ h0P h0 = h0(H ∩ P )h0 . 14 JIMMY HE

Proposition 3.3. If M ⊆ L are rational ι-stable Levi subgroups with Q ⊆ P rational ι-stable parabolics with Levi factors M,L respectively, then

G/H G/H L/HL IndM⊆Q = IndL⊆P ◦ IndM⊆Q∩L . Proof. This follows from computing both sides. In particular, note that the quotient map P → L is HL bi-equivariant and so F F −2 F F −2 ′ ′ |H ∩ P | |HL ∩ (Q ∩ L) | f(h2h1xh1h2) h ,h′ ∈HF h ,h′ ∈HF 1 X1 2 X2 L h xh′ ∈P F ′ ′ F 1 1 h2h1xh1h2∈(Q∩L) =|HF ∩ QF |−2 f(hxh′) ′ F h,hX∈H hxh′∈P F ′ ′ ′ ′ F ′ ′ since h2h1xh1h2 = h2h1xh1h2 lies in (Q ∩ L) if and only if h2h1xh1h2 lies F ′ F F 2 in Q and the sum over h2, h2 ∈ HL contributes a factor of |HL | leaving F F −2 F F −2 F F −2 F F −2 |H ∩ UP | |H ∩ M | |H ∩ (UQ ∩ L) | = |H ∩ Q | where UP and UQ denote the unipotent radicals of P and Q respectively. 3.3. Bi-invariant parabolic restriction. Similar to the definition of induction, define a function G/H F F F F F F ResL⊆P : C[H \G /H ] → C[HL \L /HL ] with the formula G/H ResL⊆P (f)(x) := f(p). F p∈PX,p=x F This function is HL bi-invariant. Note that this definition coincides with the usual definition of parabolic restriction of class functions. The first ob- servation is that this operation is the adjoint of parabolic induction. F F F Proposition 3.4. If f is H bi-invariant on G and g is HL bi-invariant on LF , then G/H F 2 F F −2 G/H hf, IndL⊆P gi = |H | |H ∩ P | hResL⊆P f, gi. Proof. Compute G/H G/H hf, IndL⊆P (g)i = f(x)IndL⊆P (g)(x) F xX∈G = |HF ∩ P F |−2 f(hxh′)g(hxh′) F ′ F xX∈G h,hX∈H hxh′∈P F = |HF |2|HF ∩ P F |−2 f(p)g(p) F pX∈P A CHARACTERISTIC MAP FOR F F 15 GL2n( q )/ Sp2n( q ) where the fact that f is HF bi-invariant is used. On the other hand, G/H hResL⊆P (f), gi = f(p)g(x) F F xX∈L p∈PX,p=x = f(p)g(p). F pX∈P

3.4. Induction of basic functions. The following results concern only −1 T G = GL2n with ι(x)= −J(x ) J. Let L0 denote a rational Levi subgroup of the form GLn × GLn in GL2n such that L0 ∩ H =∼ GLn and L0/(L0 ∩ H) =∼ GLn. Let P0 denote a rational ι-stable parabolic for L0. With the conventions taken, an example is given by having L0 act on the first and last n coordinates separately. Then bi-invariant functions on L0 can be identified with class functions on GLn, ι-stable Levi and parabolic subgroups can be identified with Levi and parabolic subgroups of GLn and bi-invariant parabolic induction corresponds to the usual parabolic induction. −ι −ι To see this, note that the map G → G restricts to L0 → L0 and −ι ∼ F L0 = GLn. Then if f is a bi-invariant function on L0 , there is a unique HF -invariant function f on L−ι such that f(x) = f(xι(x)−1). Also, H - L0 0 L0 −ι conjugacy in L0 corresponds to GLn-conjugacy and ι-stable Levi and parabolic subgroups are mappede to Levi and parabolice subgroups of GLn by L 7→ L−ι and P 7→ P −ι. Under this correspondence, L /H 0 L0 F −1 −1 −1 IndL⊆P (f)(x)= | GLn | f(hxι(x) h ) F −1 −ι h∈GLn ,hxhX ∈P G−ι −1 e = IndL−ι⊆P −ι (f)(xι(x) ) −ι −ι which is just the value of the usual parabolic induction of f from L to L0 −1 e at the point xι(x) . Thus, for all intents and purposes working in L0 is equivalent to the group situation for GLn. For any ι-stable rational Levi subgroup L ⊆ G, with ι-stable rational parabolic, F ∼ L = GL2n0 (Fq) × GLni (Fq) × GLni (Fq) ∼ where ni = n and L/HL = GLY2n0 / Sp2n0 × GLni . Let T = Ti be a rational maximal torus of GLni . Then if θ = θi is an irreducible characterP of T F , let ζL(·|θ) be the function on LQF defined by Q T Q Q GL / Sp L/HL 2n0 2n0 GLni ζ (l|θ)= ζ (l |θ ) 2 ζ (l |θ ) T T0 0 0 q7→q Ti i i GL where l = (l , l , · · · ) and ζ ni (·|θ ) is viewed asY a function on GL (F ) × 0 1 Ti i ni q

GLni (Fq) under the correspondence described above. These are called basic functions and form a basis for the bi-invariant functions on LF . This is a simple extension of the deﬁnitions already given for basic functions on 16 JIMMY HE symmetric spaces of the form GL2n / Sp2n and GLn to products of these spaces. The following proposition is proved in [3, Theorem 5.3.2] for a particular choice of L0 and P0 but this suﬃces as all such choices are conjugate under HF .

Proposition 3.5. Let L0 ⊆ P0 be deﬁned as above. Then if T is a rational F maximal torus of GLn and θ an irreducible character on T , F |T | 2 IndG/H (ζGLn (·|θ)) = q7→q ζG/H (·|θ). L0⊆P0 T |T F | T The next proposition extends the previous one to any maximal ι-stable Levi and parabolic subgroup. Proposition 3.6. Let L be a rational ι-stable Levi subgroup of the form ∼ GL2n × GLm × GLm with HL = Sp2n × GLm and P a rational ι-stable parabolic with L as its Levi factor. Let T = T1 × T2 be a rational maximal torus of GLn+m with T1 and T2 maximal tori in GLn and GLm respectively, and θ an irreducible character of T F . Then F 2 G/H L/HL |T2 |q7→q G/H IndL⊆P (ζT (·|θ)) = F ζT (·|θ). |T2 | ′ Proof. Let L0 and P0 be chosen so that there is a parabolic subgroup P with Levi factor L0 ∩ L contained in both P and P0. This can be done by taking a rational ι-stable Borel in P , and choosing P0 to be the parabolic corresponding to the subset of all simple roots except the middle one. Using Proposition 3.5 on the GL2n factor write F G/H L/HL |T1 | G/H L/HL L∩L0/HL∩L0 Ind (ζ (·|θ)) = Ind (Ind ′ (ζ (·|θ))). L⊆P T F L⊆P L∩L0⊆P ∩L T |T1 |q7→q2 ′ Then by Proposition 3.3, as P is contained in both P and P0,

G/H L G/H G/H L0/HL0 Ind ◦ Ind ′ = Ind ′ = Ind ◦ Ind ′ . L⊆P L∩L0⊆P ∩L L∩L0⊆P L0⊆P0 L∩L0⊆P ∩L0

L0/HL0 Now Ind ′ is the usual parabolic induction on GL and so L∩L0⊆P ∩L0 n

L0/HL0 L0∩L/HL0∩L L0/HL0 Ind ′ (ζ (·|θ)) = ζ (·|θ) L∩L0⊆P ∩L0 T T and a ﬁnal application of Proposition 3.5 establishes the result.

4. The Characteristic Map In this section, a characteristic map

ch : C[Sp2n(Fq)\ GL2n(Fq)/ Sp2n(Fq)] → Λ M f∈OO( ) is constructed from the Sp2n(Fq) bi-invariant functions on GL2n(Fq). Here the tensor product ⊗Λ is taken over irreducible f and each factor is isomorphic to a copy of the symmetric functions. The notational convention will A CHARACTERISTIC MAP FOR F F 17 GL2n( q )/ Sp2n( q ) be to drop the tensor and if p ∈ Λ corresponds to the factor indexed by f, then write p(f) = p(x1,f ,... ). Before proceeding, the theory for GLn(Fq) is reviewed because the construction for GL2n(Fq)/ Sp2n(Fq) is both very similar and some facts about the characteristic map for GLn(Fq) are used in the proofs. The exposition follows [17].

4.1. The GLn(Fq) Theory. Deﬁne

GLn(Fq ) chGL : C[GLn(Fq)] → Λ n M M f∈OO( ) by −n(µ(f)) −1 chGLn (ICµ )= qf Pµ(f)(f; qf ), M f∈YO( ) where ICµ denotes the indicator function for Cµ with µ : O(M) → P and Pµ(x; t) is the Hall-Littlewood symmetric function. GLn(Fq ) Define a multiplication on ⊕C[GLn(Fq)] given by parabolic induction. That is, given two class functions, f on GLn(Fq) and g on GLm(Fq), define a class function on GLn+m(Fq) by embedding GLn(Fq)×GLm(Fq) as a standard Levi subgroup, viewing f ×g as a function on this Levi subgroup, extending to a rational parabolic subgroup by the canonical quotient, and then inducing to the full group. With this multiplication, ch is an isomor- GLn(Fq ) phism of graded algebras on ⊕nC[GLn(Fq)] . In addition, this map GLn(Fq ) is an isometry when C[GLn(Fq)] is equipped with the usual inner product and Λ with the inner product defined by 1 hpµ,pµiGLn = zµ(f) µ(f)i M q − 1 f∈YO( ) Yi f for µ : O(M) → P. Then define

pn/d(fx)(fx) if d(fx)|n pn(x) := , (0 else n−1 L e (−1) x∈Mn ξ(x)pn(x) if ξ ∈ n pn(ξ) := , 0 else ( P e pen(ϕ) := pnd(ϕ)(ξ), where ξ ∈ ϕ for ϕ ∈ O(L). Since they are algebraically independent, the pn(ϕ) may be viewed ase power sum symmetric functions in "dual variables xi,ϕ" (that is, they may be formally viewed as symmetric functions in these variables, and the deﬁnition of pn(ϕ) = pn(xi,ϕ) already available can be used to deﬁne other symmetric functions, e.g. sλ(ϕ), eλ(ϕ)). As a matter of convention, µ will denote functions O(M) → P and λ will denote functions O(L) → P, and symmetric functions indexed by µ will always be in variables xi,f for f ∈ O(M) and symmetric functions indexed 18 JIMMY HE by λ will always be in the variables xi,ϕ. Symmetric functions labeled by a partition valued function are interpreted as a product. Thus,

pµ = pµ(f)(f). M f∈YO( ) Now the characteristic map on the characters can be computed as

chGLn (χλ)= sλ where the sλ denote the Schur functions. Note by orthonormality of the characters that the inner product on the dual variables is just the standard Hall inner product. That is,

hpλ,pλiGLn = zλ. F Finally, note that if T is a rational maximal torus in GLn with T =∼ F Mk1 ×···× Mkr and θ an irreducible character of T , then

GLn n−r n−l(λ) chGLn (ζT (·|θ)) = (−1) pki/d(ϕi)(ϕi) = (−1) pλ, Yi where ϕ is the orbit of θ|M and λ is the combinatorial data associated i ki to (T,θ). In fact, this could be used as a definition of the Deligne-Lusztig characters for GLn(Fq). 4.2. Definition of the Characteristic Map. Now return to the case of interest. Define the inner product hf, gi = f(x)g(x) on x∈GL2n(Fq ) C[Sp (F )\ GL (F )/ Sp (F )] and the inner product on ⊗Λ by 2n q 2n q 2n q P 1 hpµ,pµi = zµ . 2µ(f)i M q − 1 f∈YO( ) Yi f Define the characteristic map ch by −2n(µ(f)) −2 F F ch(IH gµH )= qf Pµ(f)(f; qf ) M f∈YO( ) and extending linearly.

F F Note that the coeﬃcients of pµ in ch(IH gµH ) (which are rational in 2 q), are the same as those in chGLn (ICµ ) but with q replaced with q . It is natural to expect in light of Theorem 2.3 that the same is true for the spherical functions. To establish this, the following lemma is needed.

GLn Lemma 4.1. Let t denote a formal variable and use ζT (Cµ|θ,t) to denote GLn the Deligne-Lusztig character ζT (·|θ), which is a rational function in q, in F ∼ r terms of t. Here, T = i=1 Mki and θ = θi with θi ∈ ϕi. Then

GLn −n(µ(f)) −1 n−r ζT (Mµ|θ,t) Q tf Pµ(f)(f; Qtf ) = (−1) pki/d(ϕi)(ϕi) µ M X f∈YO( ) Yi n−l(λ) = (−1) pλ A CHARACTERISTIC MAP FOR F F 19 GL2n( q )/ Sp2n( q ) as polynomials with coefficients in C(t), where λ is the combinatorial data associated to (T,θ) and Mµ lies in the conjugacy class indexed by µ. Proof. This proof follows that of Theorem 4.2 in [24], where it is proven for the case of t = −q, but the proof given works for a formal parameter. −1 It suffices to show that that the coefficient of Pµ(f)(f; qf ) in pki/d(ϕi)(ϕi) GLn is ζT (Cµ|θ,t). Write Q (−1)n−r p (ϕ )= θ (x )p (f ) ki/d(ϕi) i i i ki/d(fxi ) xi i i x ∈M Y Y iXki = θ(x) p (f ). ki/d(fxi ) xi xX∈T Yi Now rewrite p (f )= M p (f) and use the fact that the i ki/d(fxi ) xi f∈O( ) γx(f) Green polynomials are the change of basis from power sum to Hall-Littlewood polynomialsQ to obtain Q n−r (−1) pki/d(ϕi)(ϕi) Y µ(f) −n(µ(f)) −1 = θ(x) Q (tf )t Pµ(f)(f,t )  γx(f) f f  F M xX∈T f∈YO( ) |µ(f)X|=|γx(f)|   = θ(t)Qµ (t) t−n(µ(f))P (f,t−1)  γx  f µ(f) f µ M X x∈T,|γxX(f)|=|µ(f)| f∈YO( ) GLn −n(µ(f)) −1 = ζT (Cµ|θ,t) tf Pµ(f)(f,tf ) µ M X f∈YO( ) as required. F Proposition 4.2. Let T be a rational maximal torus of GLn with T =∼ F Mk1 ×···× Mkr and θ a character of T . Then

GL2n / Sp2n n−r n−l(λ) ch(ζT (·|θ)) = (−1) pki/d(ϕi)(ϕi) = (−1) pλ, Yi where ϕ is the F -orbit of θ|M ∈ L and λ is the combinatorial data associ- i ki ated to (T,θ). Proof. Write

GL2n / Sp2n GLn 2 F F ζT (·|θ)= ζT (Mµ|θ)q7→q IH gµH µ X and apply the characteristic function, giving

GLn −2n(µ(f)) −2 n−r 2 ζT (Mµ|θ)q7→q qf Pµ(f)(f; qf ) = (−1) pki/d(ϕi)(ϕi) µ M X f∈YO( ) Yi by the lemma with t = q2.

Now ch(φλ) may be computed. 20 JIMMY HE

Proposition 4.3. Let λ : O(L) → P with kλk = n. Then |λ| (−1) −n(λ′(ϕ)) 2 ch(φλ)= 2 qϕ Jλ(ϕ)(qϕ,qϕ). ψn(q ) L ϕ∈YO( ) Proof. First, note that

|λ| (−1) −n(λ(ϕ)′) 2 φλ = qϕ cλ(ϕ)′ (qϕ,qϕ)dλ(ϕ)′ (w)(qϕ) |W (λ)|  L  w∈XW (λ) ϕ∈YO( )  GLn  ζ (·|θw)q7→q2 × sgn(w) Tw F GLn |T |ζ (1|θ ) 2 w Tw w q7→q from Theorem 2.3. Recall that cλ denotes the scaling from Pλ to Jλ, dλ denotes the change of basis from power sum symmetric functions to Macdonald G symmetric functions and ζT denotes a Deligne-Lusztig character. GLn n−l(w) F F Now ζ (1) = (−1) | GL /T | ′ (see [5, Thoerem 7.5.1]), where for Tw n w q an integer n, nq′ denotes the q-free part. Here l(w) denotes the length of the GLn partition corresponding to w and not the Coxeter length. Thus, ζT (1) = n−l(w) 2 F (−1) ψn(q )/|Tw |q7→q2 . Then apply ch, obtaining

|λ| (−1) −n(λ(ϕ)′) 2 qϕ cλ(ϕ)′ (qϕ,qϕ)dλ(ϕ)′ (w)(qϕ) |W (λ)|  L  w∈XW (λ) ϕ∈YO( ) F  p (ϕ)|T | 2 |λ|−l(w) ϕ∈O(L) w(ϕ) w q7→q × (−1) F 2 Q |Tw |ψn(q ) as applying Proposition 4.2 gives

GLn n−l(w) ch(ζ (·|θ ) 2 ) = (−1) p (ϕ). Tw w q7→q w(ϕ) L ϕ∈YO( ) Next, notice that if ωq,t denotes the automorphism of ⊗Λ which sends n−1 n n pn(ϕ) to (−1) (qϕ − 1)/(tϕ − 1)pn(ϕ) then F p (ϕ)|T | 2 |λ|−l(w) ϕ∈O(L) w(ϕ) w q7→q ω 2 p = (−1) q ,q w(ϕ) |T F | L Q w ϕ∈YO( ) and so |λ| (−1) −n(λ(ϕ)′) 2 ch(φλ)= 2 qϕ cλ(ϕ)′ (qϕ,qϕ) dλ(ϕ)′ (w)(qϕ)ωq2,qpw(ϕ) ψn(q ) L   ϕ∈YO( ) w∈XS|λ(ϕ)| |λ|   (−1) −n(λ(ϕ)′) 2 = ωq2,q 2 qϕ Jλ(ϕ)′ (ϕ; qϕ,qϕ) ψn(q ) L ϕ∈YO( ) |λ| (−1) −n(λ(ϕ)′) 2 = 2 qϕ Jλ(ϕ)(ϕ; qϕ,qϕ) ψn(q ) L ϕ∈YO( ) A CHARACTERISTIC MAP FOR F F 21 GL2n( q )/ Sp2n( q )

4.3. The Isometry Property. Next, it will be shown that the map ch is an isometry, up to a constant. Lemma 4.4. The map ch satisﬁes hφ, ψi = q−n|HF |2hch(φ), ch(ψ)i. Proof. It suﬃces to compute the norms of indicator functions. It is clear F F F F F F that hIH gµH ,IH gµH i = |H gµH |. Now

F F F F 2 hch(IH gµH ), ch(IH gµH )i = (hch(ICµ ), ch(ICµ )iGLn )q7→q , where Cµ denotes the conjugacy class of GLn(Fq) associated to µ. This can F F 2 be shown by noting that ch(IH gµH ) = ch(ICµ )q7→q , and expanding into power sum symmetric functions, and then using the fact that hpµ,pµi =

Remark 4.5. The inner product on C[Sp2n(Fq)\ GL2n(Fq)/ Sp2n(Fq)] could be renormalized so that ch is actually an isometry. This would also remove some of the constants appearing in Proposition 3.4 and so in some sense this inner product would be more natural although to avoid confusion with the usual one this is not done. These constants also appear in the characteristic map for the Gelfand pair S2n/Bn, see [17, VII, §2]. Remark 4.6. Lemma 4.4 could also be proven by computing the norms of spherical functions. Strictly speaking, this is not necessary, but the computation helps illustrate the use of the dual variables so it is included. This computation naturally leads to a proof of Theorem 2.3, with the triangu- larity between the Pλ and mµ the remaining condition to check. This is basically how Theorem 2.3 is proven in [3] and so for the sake of brevity is not reproduced here. 22 JIMMY HE

To compute the analogue for the spherical functions, ﬁrst h, i must be computed on the dual variables xi,ϕ (since the indicator functions are in terms of f ∈ O(M) and the spherical functions are in terms of ϕ ∈ O(L)). Proposition 4.7. For λ : O(L) → P,

λ(ϕ)i qϕ − 1 hpλ,pλi = zλ . 2λ(ϕ)i L qϕ − 1 ϕ∈YO( ) Yi

Proof. Let pλ(ϕ) denote the symmetric function obtained by expanding pλ(ϕ) in terms of pλ(f) and taking the complex conjugate of all coeﬃcients, extended multiplicatively. First note that

pn(ϕ) ⊗ pn(ϕ)= ϕ(x)ϕ(y)pn(x) ⊗ pn(y) L L M ϕX∈ n ϕX∈ n x,yX∈ n n e e = (q − 1) pn(x) ⊗epn(y). e M xX∈ n Then e e

pn(ϕ) ⊗ pn(ϕ)= d(ϕ)pn/d(ϕ)(ϕ) ⊗ pn/d(ϕ)(ϕ) L L ϕX∈ n ϕ∈O(X),d(ϕ)|n and e e

pn(x) ⊗ pn(x)= d(fx)pn/d(fx)(fx) ⊗ pn/d(fx)(fx) M L xX∈ n f∈O(X),d(f)|n e e q2n−1 and so multiplying by n(qn−1) and summing over all n gives 2n 1 2n 1 qϕ − 1 (qf − 1)pn(f) ⊗ pn(f)= n pn(ϕ) ⊗ pn(ϕ). n M n L qϕ − 1 nX≥1 f∈XO( ) nX≥1 ϕ∈XO( ) Finally, exponentiate both sides to obtain

2λ(ϕ)i 1 qϕ − 1 1 2µ(f)i pλ⊗pλ = (qf − 1) pµ⊗pµ z  λ(ϕ)i  z   λ L qϕ − 1 µ µ M Xλ ϕ∈YO( ) Yi X f∈YO( ) Yi and this power series identity implies that  

λ(ϕ)i qϕ − 1 hpλ,pλi = zλ . 2λ(ϕ)i L qϕ − 1 ϕ∈YO( ) Yi Now the norms of spherical functions may be computed as follows. Alternative proof of Lemma 4.4. Note that for any spherical function, it is always the case that | GL2n(Fq)| hφλ, φλi = dλ∪λ A CHARACTERISTIC MAP FOR F F 23 GL2n( q )/ Sp2n( q )

(see e.g. [17, VII, §1]). Now compute F 2 F 2 |H | |H | −2n(λ(ϕ)′) 2 2 n hch(φλ), ch(φλ)i = n 2 2 qϕ hJλ(q,q ), Jλ(q,q )i. q q ψn(q ) L ϕ∈YO( ) 2 2 2 2 Because hJλ(q,q ), Jλ(q,q )i = cλ(q,q )cλ′ (q,q )= Hλ∪λ(q), this is equal to −1 −n((λ(ϕ)∪λ(ϕ))′ ) | GL2n(Fq)|ψ2n(q) qϕ Hλ∪λ(q) L ϕ∈YO( ) and ﬁnally the dimension formula n((λ(ϕ)∪λ(ϕ))′ ) −1 dλ∪λ = ψ2n(q) qϕ Hλ∪λ(q) L ϕ∈YO( ) gives F 2 |H | | GL2n(Fq)| n hch(φλ), ch(φλ)i = q dλ∪λ as desired.

4.4. Induction. Let G = GL2(n+m) and let H = Sp2(n+m). Let L be an F ι-stable Levi subgroup with L = GL2n(Fq) × GLm(Fq) × GLm(Fq), such F F that L ∩H = Sp2n(Fq)×GLm(Fq). Let P be a rational ι-stable parabolic with L as its Levi factor. With the conventions taken, an explicit realization of these subgroups are given by ∗ 0 0 0 ∗∗∗∗ 0 ∗ ∗ 0 0 ∗ ∗ ∗ L =   , P =    0 ∗ ∗ 0   0 ∗ ∗ ∗       0 0 0 ∗   0 0 0 ∗      where the sizes of the rows and columns are m,n,n,m and it is easy to check     that these are both stable under ι. Then the function G/H F F F F F F IndL⊆P : C[HL \L /HL ] → C[H \G /H ] may be viewed as taking f a Sp2n(Fq) bi-invariant function on GL2n(Fq) and ∼ g a class function on GLm(Fq) (since there is an isomorphism GLm × GLm / GLm = GLm) and producing a Sp2(n+m)(Fq) bi-invariant function on GL2(n+m)(Fq), denoted by f ∗ g. This can be done for any n,m and so deﬁnes a graded bilinear map

GLn(Fq ) C[Sp2n(Fq)\ GL2n(Fq)/ Sp2n(Fq)] × C[GLn(Fq)] n n M M → C[Sp2n(Fq)\ GL2n(Fq)/ Sp2n(Fq)]. n M 24 JIMMY HE

If f1,f2 are functions on G1 and G2 respectively, let f1 × f2 denote the function on G1 × G2 given by (f1 × f2)(x1,x2)= f1(x1)f2(x2). Then G/H f ∗ g(x):=IndL⊆P (f × g) 1 = (f × g)(hxh′). | SpF |2| GLF |4qm(m+1)+4nm 2n m h,h′∈SpF X2(n+m) hxh′∈P F

GLn(Fq ) Remark 4.8. Zelevinsky showed that R = ⊕C[GLn(Fq)] can be given the structure of a positive self-adjoint Hopf algebra [28]. Then the operation just deﬁned turns

C[Sp2n(Fq)\ GL2n(Fq)/ Sp2n(Fq)] n M into an R-module. Similarly, the restriction operation deﬁned in Section 3 turns it into an R-comodule and the two operations are adjoint. This gives the structure of a positive self-adjoint module over R as deﬁned in [25]. The key result of this section is the following compatibility with the characteristic map.

Theorem 4.9. Let f be an Sp2n(Fq) bi-invariant function on GL2n(Fq) and let g be a GLm(Fq) bi-invariant function on GLm(Fq) × GLm(Fq) (or equivalently a class function on GLm(Fq)). Then

ch(f ∗ g) = ch(f)ωωq2,q chGLm (g).

Proof. It suffices to show that if T1 and T2 are rational maximal tori in GLn F F and GLm respectively, and if θ1 and θ2 are characters of T1 and T2 , then F |T | 2 ζGL2n / Sp2n (·|θ ) ∗ ζGLm (·|θ )= 2 q7→q ζG/H (·|θ × θ ). T1 1 T2 2 F T1×T2 1 2 |T2 | This is because applying the characteristic map implies F F |T | 2 |T | 2 ch ζGL2n / Sp2n (·|θ ) ∗ 2 q7→q ζGLm (·|θ ) = (−1)n+m−l(λ) 2 q7→q p T1 1 F T2 2 F λ |T2 | ! |T2 | which is exactly the desired result, and as the basic functions form a basis and the equation is linear in f, g, the result follows. Now let L ⊆ P be the Levi and parabolic defining the ∗ operation defined above. Then ζGLn (·|θ ) ∗ ζGLm (·|θ ) = IndG/H (ζL/HL (·|θ × θ )) T1 1 T2 2 L⊆P T1×T2 1 2 and the desired equality follows directly from Proposition 3.6. Remark 4.10. The above induction operation can be defined for any Levi subgroup with a rational ι-stable parabolic. However, as every such Levi subgroup is of the form GL2n × GLni × GLni , there is no real gain to considering more general subgroups because the induction can be broken Q A CHARACTERISTIC MAP FOR F F 25 GL2n( q )/ Sp2n( q ) ∼ into two steps, first all the GLni × GLni / GLni = GLni factors, and then the remaining GL2n0 / Sp2n0 factor, and the first step simply corresponds to the classical theory for GLn. It would be very interesting if an analogous operation could be defined for the Levi subgroups GL2n × GL2m with ι-fixed points Sp2n × Sp2m. Unfortu- nately, it is not hard to see that this cannot be given by the same formula (and so does not correspond to the bi-invariant parabolic induction defined by Grojnowski [10]). For example, the parabolic induction of the indicator function for a double coset generated by a unipotent element may be non-zero on double cosets generated by a non-unipotent element. Such an operation would define a multiplication operation on

C[Sp2n(Fq)\ GL2n(Fq)/ Sp2n(Fq)] n M analogous to the ones defined for the more classical examples of a characteristic map. A nice formula for this operation would have consequences for the structure constants of the Macdonald polynomials (the Macdonald analogue of the Littlewood-Richardson coefficients) similar to the applications in [17] in the S2n/Bn case (or Theorems 1.1 and 5.9) on positivity and vanishing. 5. Schur expansion of skew Macdonald polynomials In this section, Theorem 1.1 on positivity of the coefficients of the Schur expansion of skew Macdonald polynomials with parameters (q,q2) is proven. The argument for Theorem 1.1 is in the same spirit as that of Macdonald [17, VII, §2] for a similar result in the Jack case with parameter 2. A condition for the vanishing of these coefficients is also given and in the special case of a non-skew Macdonald polynomial, relates to the classical Littlewood- Richardson coefficients. The strategy will be to use the characteristic map and the mixed product ν 2 to reinterpret Cλ/µ(q,q ) in terms of bi-invariant functions on GL2n(Fq) and then utilize general facts about Gelfand pairs and the formula for parabolic induction to establish the results. 5.1. Skew Macdonald polynomials. If λ/µ is a skew shape, the skew Macdonald polynomial Pλ/µ(x; q,t) is defined by requiring

hPλ/µ(q,t),fi = hPλ(q,t),Qµ(q,t)fi ν for all symmetric functions f. Define the coefficients Cλ/µ(q,t) by ν Pλ/µ(x; q,t)= Cλ/µ(q,t)sν(x). ν X These are the coefficients of the Schur expansion of Pλ/µ. Remark 5.1. Since q |λ|−|µ| P (q,t)= P (q−1,t−1), λ/µ t λ/µ 26 JIMMY HE

Theorem 1.1 immediately extends to parameters (q−1,q−2) for q an odd prime power. 5.2. Positive-definite functions. For any finite group G, a positive-definite function on G is a function f : G → C such that the matrix indexed by G whose x,y entry is f(x−1y) is a positive-definite matrix. A key fact is that any bi-invariant positive-definite function is a positive linear combination of spherical functions. Proposition 5.2 ( [17, VII, §1]). Let G/H be a Gelfand pair. Any spherical function is positive-definite, and moreover if f is an H bi-invariant function on G that is positive-definite, then for any spherical function φ on G, hf, φi≥ 0. If α : G → H is a group homomorphism, then given functions f : G → C and g : H → C, define the functions α∗g : G → C, or the pullback, and α∗f : H → C, or the pushforward, by α∗g(x)= g(α(x))

α∗f(x)= f(y). α(Xy)=x ∗ ∗ ∗ From the definition, it’s clear that (α ◦ β) = β ◦ α and (α ◦ β)∗ = α∗ ◦ β∗. There are two basic properties that are needed, namely that pullback and pushforward are adjoint and that pushforward and pullback preserve positive-definite functions. Lemma 5.3. If α : G → H, and f : H → C and g : G → C, then ∗ hα f, gi = hf, α∗gi. Proof. Note that hα∗f, gi = f(α(x))g(x) xX∈G and hf, α∗gi = f(y) g(x) yX∈H α(Xx)=y which are equal. Lemma 5.4. If f : G → C is positive-definite, and α : G → H and β : H → ∗ G are group homomorphisms, then α∗f and β f are also positive-definite. Proof. Note positive definiteness is equivalent to having f(x−1y)h(x)h(y) ≥ 0 x,yX∈G for all functions h : G → C. Then ∗ −1 −1 β f(x y)h(x)h(y)= f(x y)β∗h(x)β∗h(y) ≥ 0 x,yX∈H x,yX∈G so β∗f is positive-definite. A CHARACTERISTIC MAP FOR F F 27 GL2n( q )/ Sp2n( q )

For α∗f, note that α can always be factored as a surjection and an injec- tion. If α is surjective, then −1 −1 −1 ∗ ∗ α∗f(x y)h(x)h(y)= | ker α| f(x y)α h(x)α h(y) ≥ 0 x,yX∈H x,yX∈G and if α is injective, then −1 −1 α∗f(x y)h(x)h(y)= f(x y)h(x)h(y) −1 x,yX∈H x,y∈H,xXy∈α(H) = f(x−1y)h(zx)h(zy) z∈G/αX(H) x,yX∈H is non-negative as each summand is non-negative. Next, it is shown that the parabolic restriction of bi-invariant functions takes positive-deﬁnite functions to positive-deﬁnite functions.

Lemma 5.5. Let f be a positive-definite Sp2(n+m)(Fq) bi-invariant function on GL2(n+m)(Fq), and let L ⊆ P be any rational ι-stable parabolic subgroup GL2(n+m) and its Levi factor. Then ResL⊆P (f) is a positive-definite function on LF . F F F F Proof. Let i : P → GL2(n+m) denote the inclusion map and pr : P → L denote the canonical projection. Then G ∗ ResL⊆P (f)= pr∗i (f), G and so if f is a positive-definite function, then by Lemma 5.4 so is ResL⊆P (f).

F G Corollary 5.6. If f is positive-definite bi-invariant on L , then IndL⊆P (f) is a positive-definite function. Proof. Since parabolic induction and restriction are adjoints, if f is a positive- definite bi-invariant function on LF , then for any spherical function φ on GF G F 2 F −2 F −2 G hIndL⊆P (f), φi = | Sp2(n+m) | | Sp2n | | Sp2m | hf, ResL⊆P φi is non-negative by expanding both arguments in terms of spherical functions. G Thus, IndL⊆P (f) is positive-definite. 5.3. Proof of Theorems 1.1. Proof of Theorem 1.1. Fix µ, ν, λ and let |µ| = m and |ν| = n. Assume ν that |λ| = m + n as otherwise Cλ/µ(q,t) = 0. When necessary, view these partitions as partition-valued functions O(L) → P by taking λ(ϕ) = λ if ϕ is the trivial character and 0 otherwise. Note that

ν 2 hJλ, Jµωωq2,qsνiq,q2 Cλ/µ(q,q )= hPλ/µ,sνi = 2 ′ 2 cλ(q,q )cµ(q,q ) 28 JIMMY HE is a positive scalar multiple of hφλ, φµ ∗ χν i by Proposition 4.3 and Theorem 4.9. G Now by Proposition 5.2, it is enough to show that φµ ∗ χν = IndL⊆P (φµ × χν) is a positive-definite bi-invariant function on GL2(m+n)(Fq). But since G φµ × φν is positive-definite, then by the corollary so is IndL⊆P (φµ × χν). Remark 5.7. This proof gives a representation-theoretic interpretation of the coefficients in the Schur expansion of skew Macdonald polynomials (with q an odd prime power) in terms of coefficients of spherical functions in the bi-invariant parabolic induction, similar to how the classical Littlewood- Richardson coefficients can be viewed as the multiplicities of irreducible representations in the Young induction. ⊥ 2 2 Remark 5.8. Let Jµ (q,q ) denote the adjoint of multiplication by Jµ(q,q ) with respect to the h, iq,q2 inner product. Some computations for small partitions in Sage suggest that ⊥ 2 2 hJ (q,q )Jλ(q,q ),sν i µ ∈ N[q] (1 − q)|λ|+|µ| extending the conjecture of Haglund. It is unclear if this is the correct ⊥ 2 normalization although replacing Jµ (q,q ) by the normalized version (that 2 takes Jµ(q,q ) to 1) leads to rational and not polynomial expressions. If this conjecture holds, it would be interesting to see what combinatorial interpretation the coefficients might have. 5.4. A vanishing theorem. The characteristic map also gives a condition ν 2 for vanishing of Cλ/µ(q,q ). In the special case that µ = 0, the theorem 2 states that sν appears in Jλ(q,q ) only if sλ∪λ appears in the expansion of sνsν. ν 2 Theorem 5.9. If hsλ∪λ,sµ∪µsνsνi = 0, then Cλ/µ(q,q ) vanishes as a function of q.

Proof. Suppose that hsλ∪λ,sµ∪µsνsνi = 0. Take |µ| = m, |ν| = n and assume ν 2 |λ| = m + n as otherwise Cλ/µ(q,q ) = 0. Then as in the proof of Theorem 1.1, it is equivalent to show that hφλ, φµ ∗ χνi = 0 since it’s a positive scalar ν 2 multiple of Cλ/µ(q,q ), again interpreting partitions as partition-valued functions in the same way as above. Let Avk denote the averaging operation sending a function f on GL2k(Fq) F −2 ′ F to x 7→ |H | h,h′∈HF f(hxh ) where the group H should be deter- mined from context (either Sp (F ) or GL (F )). Let G = GL , P 2k q k q 2(n+m) H = Sp2(n+m) and L ⊆ P denote the Levi and parabolic deﬁning ∗. Then A CHARACTERISTIC MAP FOR F F 29 GL2n( q )/ Sp2n( q ) note hφλ, φµ ∗ χνi is a non-zero multiple of G hResL⊆P (Avn+m(χλ∪λ)), Avn × Avm(χµ∪µ × χν × χν)i ∗ =hpr∗i Avn+m(χλ∪λ),χµ∪µ × χν × χνi ∗ =hAvn+m(χλ∪λ), i∗pr (χµ∪µ × χν × χν)i ∗ = Avn+m(χλ∪λ · i∗pr (χµ∪µ × χν × χν))(1) where in the last line the multiplication on functions is convolution. Now ∗ χλ∪λ · i∗pr (χµ∪µ × χν × χν)(x)

−1 ∗ = Tr ρλ∪λ(x) ρλ∪λ(y )pr (χµ∪µ × χν × χν)(y)  F  yX∈P where ρλ∪λ denotes the corresponding representation and this is0 unless ∗ hχλ∪λ, pr (χµ∪µ × χν × χν)iP F 6= 0, ∗ because pr (χµ∪µ × χν × χν) is an irreducible character. By Frobenius reci- procity G hχλ∪λ, IndL⊆P (χµ∪µ × χν × χν)i 6= 0, G where IndL⊆P is the standard parabolic induction for GLn. This happens exactly when hsλ∪λ,sµ∪µsνsνi is non-zero. ν 2 Here, the theorem can be extended to hold for all q because Cλ/µ(q,q ) is a rational function in q.

6. Computation of Spherical Function Values In this section, the values of spherical functions on the double coset of non-symplectic transvections is computed. Similar computations could be ∗ done for the double cosets generated by diag(a, 1,..., 1) for a ∈ Fq. This section should be seen as an application of the characteristic map to use the Pieri rule for Macdonald polynomials to compute spherical function values. Proposition 6.1 ( [17, VI, §6]). Let

′ bλ(s; q,t) ψλ/µ := , bµ(s; q,t) s∈Cλ/µY\Rλ/µ where 1 − qa(s)tl(s)+1 b (s; q,t) := , λ 1 − qa(s)+1tl(s) and where Cλ/µ denotes the columns of λ intersecting λ/µ and similarly Rλ/µ but for rows. Then ′ Pµ(x; q,t)er(x)= ψλ/µPλ(x; q,t), Xλ where the sum is over partitions λ such that λ \ µ is a vertical strip with r boxes. 30 JIMMY HE

First the value at the identity will be computed as a similar computation shows up in the transvection computation, even though the value is already known to be 1. Deﬁne δ as the specialization homomorphism on ⊗Λ given by 1 δ(pn(ϕ)) = n . qϕ − 1 The following lemma is essentially proven in [17, IV, §6].

Lemma 6.2. For any F ∈⊗Λ, hF, en(f1)i = δ(ωωq,q2 F ).

Proof. Since hF, en(f1)i = hωωq,q2 F, en(f1)iGLn and ωωq,q2 is invertible it suﬃces to check that hF, en(f1)iGLn = δ(F ). Since both sides are linear in F , it is enough to check on a basis, which is done in [17, IV, §6]. −2 The value of φλ(1) can be computed as follows. First note that Pλ(x; q )= n en when λ = (1 ) and so F −1 φλ(1) = |H | hφλ,IHF i can be computed by using the characteristic map and Lemma 4.4, giving |λ| −n 2 −1 −(n2−n) −n(λ(ϕ)′) 2 (−1) q | Sp2n(Fq)|ψn(q ) q qϕ hJλ(q,q ), en(f1)i L ϕ∈YO( ) |λ| −n(λ(ϕ)′) 2 =(−1) qϕ hJλ(q,q ), en(f1)i. L ϕ∈YO( ) 2 2 Using Lemma 6.2 and the fact that ωq,q2 Jλ(q,q ) = Jλ′ (q ,q) along with ′ 2 |λ| a (s) δ(ωJλ′ (q ,q)) = (−1) s∈λ(ϕ) qϕ [17, VI, §8], the inner product can be computed giving Q −n(λ(ϕ)′) 2 qϕ δ(Jλ′ (q ,q)) L ϕ∈YO( ) −n(λ(ϕ)′) a′(s) = qϕ qϕ L ϕ∈YO( ) s∈Yλ(ϕ) =1 n−2 F F The analogous computation of φλ(IH gµH ) where µ(f1) = (21 ) and 0 otherwise requires an additional lemma. Lemma 6.3. We have ′ −n(λ0) ′ 2 q cλ(q,q ) ′ 2 en−1(f1)e1(f1)= ψ Jλ(q,q ), c′ (q,q2)(1 − q) λ/λ0 λ0 kλXk=n Xλ0 b where the second sum is over all partition-valued functions with kλ0k = n−1 2 obtained by removing one box from some λ(ϕ) with d(ϕ) = 1, and Jλ(q,q ) 2 denotes the dual basis to Jλ(q,q ) under the inner product. b A CHARACTERISTIC MAP FOR F F 31 GL2n( q )/ Sp2n( q )

|λ| 2 2 Proof. First, note that ek(f1)= kλk=k(−1) δ(ωJλ′ (q ,q))Jλ(q,q ) which is an easy consequence of Lemma 6.2. Thus P en−1(f1)e1(f1) c

|λ1| 2 2 |λ2| 2 2 = (−1) δ(ωJλ′ (q ,q))Jλ (q,q ) (−1) δ(ωJλ′ (q ,q))Jλ (q,q ) .  1 1   2 2  kλ1Xk=n−1 kλX2k=1 There are exactly q − 1 partitionb valued functions  λ with kλk = 1, whichb  2 give e1(ϕ) for ϕ ∈ L1 as the polynomials Jλ2 (q,q ). Thus, apply Pieri’s rule for r = 1, and so Cλ/µ consists of the column that is added, and similarly for the row. The arm/leg lengths in λ are exactly one more than in µ because of the added box, and so after relabeling λ1 with λ0 ′ −n(λ0) ′ 2 q cλ(q,q ) ′ 2 en−1(f1)e1(f1)= ψ Jλ(q,q ), c′ (q,q2)(1 − q) λ/λ0 λ1 kλXk=n Xλ0 ′ 2 b where δ(e1(ϕ)) = 1, and the remaining factors cλ(q,q ) and (1 − q) come 2 ′ 2 −1 2 from the scaling Jλ(q,q )= cλ(q,q ) Pλ(q,q ). Now the spherical functions of interest may be computed. b n−2 Proposition 6.4. Let µ : O(M) → P given by µ(f1) = (21 ) and 0 otherwise. Then 2n−2 2 c′ (q,q2)ψ′ 2n q (q − 1) λ λ/λ0 ′ ′ q − 1 φ (g )= qn(λ0)−n(λ ) − λ µ (q2n − 1)(q2n−2 − 1)  c′ (q,q2)(1 − q) q2n−2(q2 − 1) λ0 Xλ0 for all spherical functions φλ where λ0 is obtained from λ by removing a  single box from some λ(ϕ) with d(ϕ) = 1. Proof. Note that 1 F F φλ(gµ)= F F hφλ,IH gµH i |H gµH | 2n−2 2 q (q − 1) |λ| −n(λ(ϕ)′) 2 −2 = 2n 2n−2 (−1) qϕ Jλ(q,q ), Pµ(f1)(f1; q ) (q − 1)(q − 1) * L + ϕ∈YO( ) by Lemma 4.4. From [14, Eq. 2.4], n−1 i Pµ(f1)(f1; q)= en−1(f1)e1(f1) − q en(f1). ! Xi=0 Lemma 6.2 can handle the en(f1) term, and Lemma 6.3 will give the other ′ term. Note that although the notation cλ, and other functions indexed by partitions, is used for partition-valued functions by taking a product over the domain, in almost all cases due to cancellation only one partition will be relevant. 32 JIMMY HE

Now evaluate the desired inner product, obtaining |λ| 2 −2 h(−1) Jλ(q,q ), Pµ(f1)(f1,q )i ′ −n(λ0) ′ 2 ′ 2n q cλ(q,q )ψλ/λ q − 1 ′ = 0 − q−n(λ ) c′ (q,q2)(1 − q) q2n−2(q2 − 1) λ0 Xλ0 which gives the desired result with the remaining factors since ′ n(λ0(ϕ) ) 2 q = δ(J ′ (q ,q)). ϕ λ0 L ϕ∈YO( )

7. Character sheaves on symmetric spaces Here, the bi-invariant parabolic induction deﬁned in Section 3 is connected to the theory developed by Grojnowski [10] and Henderson [13] regarding character sheaves on symmetric spaces. References are also given to the work of Shoji and Sorlin [19–21] where possible for any cited results as both [10] and [13] are unpublished. Some familiarity with the theory of l-adic sheaves and algebraic groups is assumed, although beyond the results of [13], only some basic formal properties will be needed to translate the results into the language of functions.

7.1. Motivation. For GLn(Fq), most properties of the characteristic map are more easily seen when viewing the domain not as the ring of class functions but rather the representation ring. In the symmetric space setting, the relationship between the spherical functions and representations is less well-behaved, so representations should be replaced by sheaves instead. All operations that can be done on representations have analogues for sheaves (for instance, tensor products, induction) but there are additional operations available. Moreover, the relationship between these sheaves and functions is well-understood and so anything that can be proven for sheaves has a function analogue. Working with sheaves, the usual notions of parabolic induction and restriction for groups also have a natural analogue. 7.2. Preliminaries on l-adic sheaves. This section mostly follows [13], see there for further details and references. Fix some prime l not equal to the characteristic of Fq (nothing will depend on the choice of l). Since only algebraic numbers appear, all statements in this section about Ql functions can be readily transferred to C. All varieties will be defined over Fq. An Fq-structure on a variety X is given by a Frobenius map F : X → X. A morphism of varieties with F Fq structures is defined over Fq if it is F -equivariant. Let X denote the corresponding variety over Fq. If X is a variety, let D(X) denote the bounded derived category of con- structible Ql-sheaves of finite rank on X. The objects of this category are called complexes. A CHARACTERISTIC MAP FOR F F 33 GL2n( q )/ Sp2n( q )

Say that a complex K is F -stable if F ∗K =∼ K. Now to any F -stable complex K, and a choice of isomorphism φ : F ∗K → K, there is an associated F function χK : X → Ql called the characteristic function. It is defined by i i χK(x) := (−1) Tr(φ, HxK), Xi i where HxK denotes the stalk at x of the cohomology sheaf of K. The dependence on φ will be dropped in the notation but a choice of such an isomorphism must always be made when taking the characteristic function. If f : X → Y is a morphism of varieties, there are two functors, f ∗ : D(Y ) → D(X), called the pullback and f! : D(X) → D(Y ), called the compactly supported pushforward. There is a shift functor [n] : D(X) → D(X) taking a complex K to the complex K[n] with K[n]m = Kn+m and this functor commutes with pushforward and pullback and the corresponding effect on the characteristic function is multiplication by (−1)n. Finally, direct sums of complexes in D(X) correspond to addition and the tensor product of complexes in D(X) corresponds to multiplication of the corresponding characteristic functions. If G is an algebraic group acting on X, let DG(X) denote the G-equivariant derived category. All the constructions above have analogues in the equivariant setting. In particular, if the action is defined over Fq then the characteristic function of an equivariant sheaf defined over Fq is invariant under the action of GF . If f : X → Y is a principal G-bundle, then there is an equivalence of categories f ∗ : D(Y ) → DG(X) (whose composition with the forgetful functor DG(X) → D(X) gives the usual pullback) and the inverse is denoted by G f♭ : D (X) → D(Y ). If f is defined over Fq then the characteristic functions satisfy F −1 χf♭K(y)= |G | χK (x). F f(x)=Xy,x∈X If H ⊆ G, then this equivalence also restricts to an equivalence DH×H (G) =∼ DH (G/H) between the H bi-equivariant sheaves on G and H equivariant sheaves on G/H and so when convenient bi-equivariant sheaves can be thought to live on the symmetric space G/H (this is like how there is no difference between working with bi-invariant functions on G and invariant functions on G/H). 7.3. Induction functors. Here, the induction functors analogous to Deligne- Lusztig induction for GLn are introduced. They were first introduced by Grojnowski [10] but the definition used follows that of Henderson [13] which was extended in the work of Shoji and Sorlin [21]. It is important to note that these functors do not preserve any sort of Fq structure unless the parabolic subgroup used is rational. Even if P is rational, to define induction for functions some choice of Fq structure must 34 JIMMY HE be placed on the sheaves themselves. For now, the definition does not involve any Fq structure whatsoever. For any ι-stable Levi subgroup L with P = LU an ι-stable parabolic, G HL×HL H×H define a parabolic induction functor IndL⊆P : D (L) → D (G) taking bi-equivariant sheaves on L to G as follows. Consider the diagram pr q i LH × P × HH ×HP P ×HP H G where pr(h, p, h′)= p, q is the quotient morphism and i(h, p, h′)= hph′ and ′ ′−1 the action of HP ×HP on L is given by (h, h )·l = hlh with H ×H acting ′ trivially, the action of HP × HP on H × P × H is by (h1, h2) · (h, p, h ) = −1 −1 ′ ′ (hh1 , hph2 , h2h ) and the action of H × H is by (h1, h2) · (h, p, h ) = ′ −1 (h1h, p, h h2 ), and H × H acts on the last two spaces on the left and right similarly. Then all the maps are equivariant, with pr being HP × HP equivariant G HL×HL as P is ι-stable, so define the induction functor IndL⊆P : D (L) → DH×H (G) by G ∗ IndL⊆P := i!q♭pr [dim U + 2dim H/HP ]. A key result is that the composition of two induction functors is again an induction functor. The following proposition is given in [13, Prop. 2.19] and also in [21, Prop. 4.3] and the proof is similar to the usual proof of transitivity of induction for groups. Proposition 7.1 ( [13, Prop. 2.19]). Let M ⊆ L be ι-stable Levi subgroups, and let Q ⊆ P be ι-stable parabolic subgroups with Levi factors M and L respectively. Then G L ∼ G IndL⊆P ◦ IndM⊆Q∩L = IndM⊆Q . The following proposition follows easily from the definition of induction. Proposition 7.2. Let L be a rational ι-stable Levi subgroup, with a rational parabolic P = LU. Then if K is a complex on L with an Fq structure, and G IndL⊆P (K) is given the induced Fq structure, F F −2 dim U ′ χ G = |H ∩ P | (−1) χK(hxh ). IndL⊆P (K) ′ F h,hX∈H hxh′∈P F 7.4. Characteristic functions of induced complexes. The induction functors define parabolic induction of complexes but give no rational structure on the resulting complex and so more is needed to define induction of functions. The idea is to restrict to a smaller set of sheaves which are closed under induction and have a canonical Fq structure. For more details on this section, see [13, Ch. 5, 6] or [22], noting that any sheaf on G/H × V can be restricted to G/H × {0}. First, consider the group case. For any maximal torus T and a tame rank one local system L on T , there is an associated complex K(T,L) defined A CHARACTERISTIC MAP FOR F F 35 GL2n( q )/ Sp2n( q ) using intersection cohomology. This complex depends only on the G orbit of (T, L). Moreover, if T is rational and L is F -stable, there is a unique Fq structure making the characteristic function an irreducible character of T F . Then K(T,L) has an induced Fq structure coming from the structures on T and L (through the construction in terms of intersection cohomology). The cor- F responding characteristic functions χK(T,L) depend only on the G orbit of (T, L). These complexes K(T,L) are exactly the ones obtained by inducing L from T with any choice of Borel subgroup. The benefit of the intersection cohomology definition is that it gives an Fq structure. In the symmetric space setting where G = GL2n and H = Sp2n, the above still holds with modifications, namely replacing a maximal torus with a maximal ι-stable torus and characters replaced with spherical functions. There is only one H-conjugacy class of maximal ι-stable tori, and so in particular every maximal ι-stable torus T contains a maximal ι-split torus −ι −ι T with T =∼ T/TH and is contained in an ι-stable Borel subgroup. The F F H conjugacy classes of maximal ι-split tori are in bijection with the GLn - conjugacy classes of F -stable maximal tori of GLn, as every maximal ι-split torus can be conjugated to lie in GLn × GLn (acting on the first and last n coordinates) where it acts by (t,t). There are complexes K(T −ι,L) associated to tame rank one local systems L −ι on maximal ι-split tori T . When both have an Fq structure, then K(T −ι,L) does as well and their characteristic functions are invariant under HF . See [13] or [22] for details on their construction and properties. The transitivity of induction implies that the collection of complexes G IndT ⊆B(L) is closed under bi-invariant parabolic induction. The method of defining parabolic induction for functions through the functor will be to give these complexes an Fq structure, and then show that their characteristic functions form a basis for the bi-invariant functions, uniquely defining induction for any bi-invariant function. The next theorem relates the induced complexes to the complexes K(T −ι,L) and is due to Grojnowski [10], although see also [20, Theorem 1.16] or [13, Theorem 5.5] where a more complete proof is given. Theorem 7.3 ( [10, Lemma 7.4.4]). Let T be a rational maximal ι-stable torus and L an F -stable tame rank one local system on T −ι. There are isomorphisms −ι G G L0 • ZH (T ) −ι ∼ ∼ −ι IndT ⊆B(L) = IndL0⊆P0 (K(T −ι,L)) = K(T ,L) ⊗ Hc (B )[dim T ]. Z (T −ι) −ι where B H denotes the flag variety of ZH (T ) and the second is defined over Fq. G This theorem means that IndT ⊆B(L) has a canonical Fq structure and so the characteristic function of induced complexes, at least from a maximal 36 JIMMY HE torus, may be taken. Also, since induction from a maximal torus does not G depend on the Borel chosen, the notation IndT ⊆B will be used even when a particular Borel is not specified. The characteristic functions of the complexes K(T,L) are related to the Deligne-Lusztig characters. These functions are essentially the basic functions already defined. Proposition 7.4. Let T be a rational maximal ι-stable torus and θ an ir- −ι F reducible character of (T ) . Let Lθ denote the corresponding local system on T −ι. Then −ι F n |(T ) | (−1) χ = χ G K(T −ι,L ) −ι F Ind (K −ι ) θ |(T ) |q7→q2 L0⊆P0 (T ,Lθ) GLn = ζT −ι (·|θ)q7→q2 . The first equality follows from the previous theorem by taking characteristic functions and the second equality follows from [13, Proposition 6.9] or Theorem 5.3.2 in [3]. As a corollary, this also shows that the χK −ι (T ,Lθ) form a basis for the space of bi-invariant functions on GF , since the basic functions do. This proposition give a way to define induction for any bi-invariant function. Define G IndL⊆P χ L = χ G IndT ⊆B (L) IndT ⊆B′ (L) where B′ is any Borel subgroup with Levi factor T in G and extend by F linearity. Since the χ L form a basis for the H bi-invariant functions IndT ⊆B (L) L on LF , this is well-defined. In the case that the parabolic subgroup P is G rational, the induction functor actually gives an Fq structure on IndL⊆P (K) if K has one. Finally, it is shown that the Fq-structure coming from induction through a rational parabolic subgroup agrees with the canonical one when applied to K(T −ι,L). Proposition 7.5. Let L be a rational ι-stable Levi subgroup and P a rational ι-stable parabolic with L as its Levi factor. Let T −ι be a rational maximal ι-split torus and an F -stable tame rank one local system, and θ the character −ι L G of T associated to L. Then if IndT ⊆B(L) and IndT ⊆B′ (L) are given their canonical rational structures, G L G IndL⊆P (IndT ⊆B(L)) = IndT ⊆B′ (L) where B and B′ are any ι-stable Borel subgroups and this is an isomorphism over Fq. Proof. First, note that it suffices to prove the proposition for Levi subgroups ∼ ∼ of the form L = GL2n × GLm × GLm where HL = Sp2n × GLm. This is because all ι-stable Levi subgroups are of the form GL2n × GLni × GLni and HL are of the form Sp × GLn (see [19, 5.6] for example) and all 2n i Q Q A CHARACTERISTIC MAP FOR F F 37 GL2n( q )/ Sp2n( q ) factors except the GL2n / Sp2n factor are isomorphic to the symmetric space GLn × GLn / GLn =∼ GLn, where the proposition is already known (the induction operation is equivalent to Deligne-Lusztig induction up to the twist and χK(T,L) is simply a Deligne-Lusztig character [16, Proposition 9.2]). Now let L0 ⊆ G be a subgroup of the form GLn+m × GLn+m with L0 ∩ ∼ L ⊆ L a subgroup of the form GLn × GLn × GLm × GLm with HL0 = GLn × GLm. Then the proposition for L0 is already given by Theorem 7.3. It’s clear from the proof of Proposition 7.1 that if all Levi and parabolic subgroups involved are rational, the isomorphism is defined over Fq (the isomorphism is constructed by chasing a diagram which will contain only varieties defined over Fq) and so this proposition can be reduced to the known case as G L G L0∩L ∼ ′ IndL⊆P (IndT ⊆B(L)) = IndL0∩L⊆P (IndT ⊆B′′ (L)) ∼ G L0 = IndL0⊆P0 (IndT ⊆B′′ (L)) ∼ G = IndT ⊆B′ (L) ′ where P is a rational ι-stable parabolic contained in P and P0 is a rational ι- ′ stable parabolic chosen to contain P (and all isomorphisms are over Fq). The key result of Section 3, Proposition 3.6, follows by taking characteristic functions of both sides. Acknowledgements

This research was supported in part by NSERC. The author would like to thank Anthony Henderson for some comments on an earlier draft and for pointing out some references, Arun Ram and Dan Bump for some insightful suggestions and Cheng-Chiang Tsai for clarifying some aspects of character sheaves, as well as Persi Diaconis, and Aaron Landesman for helpful discus- sions. References

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Department of Mathematics, Stanford University, Stanford, CA 94305 E-mail address: [email protected]