Arithmetic and representation theory of wild character varieties

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1 Arithmetic and representation theory of wild character varieties Tamás Hausel IST Austria Michael Lennox Wong Universität Duisburg-Essen August 14, 2017 Martin Mereb IST Austria U de uenos Aires mmereb@gmail.com Abstract We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P = W conjecture for a suitable wild Hitchin system. 0.1 A conjecture Let C be a complex smooth projective curve of genus g Z 0, with divisor D = p p k + rp where p 1,..., p k, p C are distinct points, p having multiplicity r Z 0 with k 0 and k+r 1. For n Z 0, let P n denote the set of partitions of n and set P := n P n. Let µ = µ 1,..., µ k Pn k denote a k-tuple of partitions of n, and we write µ := n. We denote by M µ,r Dol the moduli space of stable parabolic Higgs bundles E, φ with quasi-parabolic structure of type µ i at the pole p i, with generic parabolic weights and fixed parabolic degree, and a twisted meromorphic Higgs field φ H 0 C; EndE K C D with nilpotent residues compatible with the quasi-parabolic structure at the poles p i but no restriction on the residue at p. Then M µ,r Dol is a smooth quasi-projective variety of dimension d µ,r with a proper Hitchin map χ µ,r : M µ,r Dol Aµ,r defined by taking the characteristic polynomial of the Higgs field φ and thus taking values in the Hitchin base A µ,r := n i=1h 0 C; K C D i. As χ µ,r is proper it induces as in [dcm] a perverse filtration P on the rational cohomology H M µ,r Dol of the total space. We define the perverse Hodge polynomial as P HM µ,r Dol ; q, t := dim Gr P i H k M µ,r Dol q i t k. 1

2 2 The recent paper [CDDP] by Chuang Diaconescu Donagi Pantev gives a string theoretical derivation of the following mathematical conjecture. Conjecture We expect P HM µ,r Dol ; q, t = qt2 dµ,r H µ,r q 1/2, q 1/2 t 1. Here, H µ,r z, w Qz, w is defined by the generating function formula 1r µ w dµ,r H µ,rz, w z 2 11 w 2 µ P k k m µi x i = Log i=1 λ P The notation is explained as follows. For a partition λ P we denote k H g,r λ z, w H λ z 2, w 2 ; x i i= H g,r λ z, w = z 2a w 2l r z 2a+1 w 2l+1 2g z 2a+2 w 2l z 2a w 2l+2, where the product is over the boxes in the Young diagram of λ and a and l are the arm length and the leg length of the given box. We denote by m λ x i the monomial symmetric functions in the infinitely many variables x i := x i1, x i2,... attached to the puncture p i. Hλ q, t; x i denotes the twisted Macdonald polynomials of Garsia Haiman [GH], which is a symmetric function in the variables x i with coefficients from Qq, t. Finally, Log is the plethystic logarithm, see e.g. [HLV1, ] for a definition. The paper [CDDP] gives several pieces of evidence for Conjecture On physical grounds it argues that the left hand side should be the generating function for certain refined PS invariants of some associated Calabi Yau 3-orbifold Y, which they then relate by a refined Gopakumar Vafa conjecture to the generating function of the refined Pandharipande Thomas invariants of Y. In turn they can compute the latter in some cases using the recent approach of Nekrasov Okounkov [NO], finding agreement with Conjecture Another approach is to use another duality conjecture the so-called geometric engineering which conjecturally relates the left hand side of Conjecture to generating functions for equivariant indices of some bundles on certain nested Hilbert schemes of points on the affine plane C 2. They compute this using work of Haiman [Hai] and find agreement with the right hand side of Conjecture Purely mathematical evidence for Conjecture comes through a parabolic version of the P = W conjecture of [dchm], in the case when r = 0. In this case, by non-abelian Hodge theory we expect the parabolic Higgs moduli space M µ Dol := M µ,0 Dol to be diffeomorphic with a certain character variety M µ, which we will define more carefully below. The cohomology of Mµ carries a weight filtration, and we denote by W HM µ ; q, t := i,k dim Gr W 2i H k M µ q i t k, the mixed Hodge polynomial of M µ. The P = W conjecture predicts that the perverse filtration P on H M µ Dol is identified with the weight filtration W on H M µ via non-abelian Hodge theory. In particular, P = W would imply P HM µ Dol ; q, t = W HMµ ; q, t, and Conjecture for r = 0; P HM µ Dol ; q, t replaced with W HMµ ; q, t was the main conjecture in [HLV1]. It is interesting to recall what inspired Conjecture for r > 0. Already in [HV, Section 5], detailed knowledge of the cohomology ring H M 2,r Dol from [HT] was needed for the computation of W HM 2 ; q, t. In fact, it was observed in [dchm] that the computation in [HV, Remark 2.5.3] amounted to a formula for P HM µ Dol ; q, t, which is the first non-trivial instance of Conjecture This twist by r was first extended for the conjectured P HM n,r Dol in [Mo] to

3 3 match the recursion relation in [CDP]; it was then generalized in [CDDP] to Conjecture We notice that the twisting by r only slightly changes the definition of H g,r z, w above and the rest of the right hand side of Conjecture does not depend on r. It was also speculated in [HV, Remark 2.5.3] that there is a character variety whose mixed Hodge polynomial would agree with the one conjectured for P HM µ,r Dol ; q, t above. Problem Is there a character variety whose mixed Hodge polynomial agrees with P HM µ,r Dol ; q, t? A natural idea to answer this question is to look at the symplectic leaves of the natural Poisson structure on M µ,r Dol. The symplectic leaves should correspond to moduli spaces of irregular or wild Higgs moduli spaces. y the wild non-abelian Hodge theorem [] those will be diffeomorphic with wild character varieties. 0.2 Main result In this paper we will study a class of wild character varieties which will conjecturally provide a partial answer to the problem above. Namely, we will look at wild character varieties allowing irregular singularities with polar part having a diagonal regular leading term. oalch in [3] gives the following construction. Let G := GL n C and let T G be the maximal torus of diagonal matrices. Let + G resp. G be the orel subgroup of upper resp. lower triangular matrices. Let U = U + + resp. U be the respective unipotent radicals, i.e., the group of upper resp. lower triangular matrices with 1 s on the main diagonal. We fix m Z 0 and r := r 1,..., r m Z m >0 an m-tuple of positive integers. For a µ P k n we also fix a k-tuple C 1,..., C k of semisimple conjugacy classes, such that the semisimple conjugacy class C i G is of type µ i = µ i 1, µ i 2,... P n ; in other words, C i has eigenvalues with multiplicities µ i j. Finally we fix ξ 1,..., ξ m T reg m an m-tuple of regular diagonal matrices, such that the k + m tuple C 1,..., C k, Gξ 1,..., Gξ m of semisimple conjugacy classes is generic in the sense of Definition Then define M µ,r := {A i, i i=1..n G G g, X j C j, C j G, S j 2i 1, Sj 2i i=1,...,r j U U + rj A 1, 1 A g, g X 1 X k C1 1 ξ 1S2r 1 1 S1C 1 1 Cm 1 ξ m S2r m m S1 m C m = I n }//G, where the affine quotient is by the conjugation action of G on the matrices A i, i, X i, C i and the trivial action on S j i. Under the genericity condition as above, Mµ,r is a smooth affine variety of dimension d µ,r of In particular, when m = 0, we have the character varieties M µ = Mµ, of [HLV1, Conjecture 0.2.1]. The main result of this paper is the following: Theorem Let µ Pn k be a k-tuple of partitions of n and r be an m-tuple of positive integers and be the generic wild character variety as defined above. Then we have M µ,r W HM µ,r ; q, 1 = qdµ,r H µ,r q 1/2, q 1/2,

4 4 where is the type of and µ := µ 1,..., µ k, 1 n,..., 1 n P k+m n C 1,..., C k, Gξ 1,..., Gξ m r := r r m. The proof of this result follows the route introduced in [HV, HLV1, HLV2]. Using a theorem of Katz [HV, Appendix], it reduces the problem of the computation of W HM µ,r ; q, 1 to counting M µ,r F q, i.e., the F q points of M µ,r. We count it by a non-abelian Fourier transform. The novelty here is the determination of the contribution of the wild singularities to the character sum. The latter problem is solved via the character theory of the Yokonuma Hecke algebra, which is the convolution algebra on C[UF q \GL n F q /UF q ], where U is as above. The main computational result, Theorem 3.3.4, is an analogue of a theorem of Springer cf. [GP, Theorem 9.2.2] which finds an explicit value for the trace of a certain central element of the Hecke algebra in a given representation. This theorem, in turn, rests on a somewhat technical result relating the classification of the irreducible characters of the group N = F q n S n to that of certain irreducible characters of GL n F q. To explain briefly, if Q n denotes the set of maps from Γ 1 = F q the character group of F q to the set of partitions of total size n see Section 2.7 for definitions and details, then Q n parametrizes both Irr N and a certain subset of Irr GL n F q. Furthermore, both of these sets are in bijection with the irreducible characters of the Yokonuma Hecke algebra. Theorem clarifies this relationship, establishing an analogue of a result proved by Halverson and Ram [HR, Theorem 4.9b], though by different techniques. Our main result Theorem then leads to the following conjecture. Conjecture We have W HM µ,r ; q, t = qt2 dµ,r H µ,r q 1/2, q 1/2 t 1. This gives a conjectural partial answer to our Problem originally raised in [HV, Remark 2.5.3]. Namely, in the cases when at least one of the partitions µ i = 1 n, we can conjecturally find a character variety whose mixed Hodge polynomial agrees with the mixed Hodge polynomial of a twisted parabolic Higgs moduli space. This class does not yet include the example studied in [HV, Remark 2.5.3], where there is a single trivial partition µ = n. We expect that those cases could be covered with more complicated, possibly twisted, wild character varieties. Finally, we note that a recent conjecture [STZ, Conjecture 1.12] predicts that in the case when g = 0, k = 0, m = 1 and r = r 1 Z >0, the mixed Hodge polynomial of our and more general wild character varieties, are intimately related to refined invariants of links arising from Stokes data. Our formulas in this case should be related to refined invariants of the n, rn torus links. We hope that the natural emergence of Hecke algebras in the arithmetic of wild character varieties will shed new light on Jones s approach [Jo] to the HOMFLY polynomials via Markov traces on the usual Iwahori Hecke algebra and the analogous Markov traces on the Yokonuma Hecke algebra, c.f. [J1, CL, JP]. The structure of the paper is as follows. Section 1 reviews mixed Hodge structures on the cohomology of algebraic varieties, the theorem of Katz mentioned above, and gives the precise definition of a wild character variety from [3]. In Section 2 we recall the abstract approach to Hecke algebras; the explicit character theory of the Iwahori Hecke and Yokonuma Hecke

5 5 algebras is also reviewed and clarified. In Section 3 we recall the arithmetic Fourier transform approach of [HLV1] and perform the count on the wild character varieties. In Section 4 we prove our main Theorem and discuss our main Conjecture In Section 5 we compute some specific examples of Theorem and Conjecture 0.2.2, when n = 2, with particular attention paid to the cases when M µ,r is a surface. Acknowledgements. We thank Philip oalch, Daniel ump, Maria Chlouveraki, Alexander Dimca, Mario García-Fernández, Eugene Gorsky, Emmanuel Letellier, András Némethi, Loïc Poulain d Andecy, Vivek Shende, Szilárd Szabó and Fernando R. Villegas for discussions and/or correspondence and a referee for pointing out some necessary corrections. We are also indebted to Lusztig s observation [L, 1.3.a] which led us to study representations of Yokonuma Hecke algebras. This research was supported by École Polytechnique Fédérale de Lausanne, an Advanced Grant Arithmetic and physics of Higgs moduli spaces no of the European Research Council and the NCCR SwissMAP of the Swiss National Foundation. Additionally, in the final stages of this project, MLW was supported by SF/TR 45 Periods, moduli and arithmetic of algebraic varieties, subproject M08-10 Moduli of vector bundles on higher-dimensional varieties. 1 Generalities 1.1 Mixed Hodge polynomials and counting points To motivate the problem of counting points on an algebraic variety, we remind the reader of some facts concerning mixed Hodge polynomials and varieties with polynomial count, more details of which can be found in [HV, 2.1]. Let X be a complex algebraic variety. The general theory of [D1, D2] provides for a mixed Hodge structure on the compactly supported cohomology of X: that is, there is an increasing weight filtration W on H j X, Q and a decreasing Hodge filtration F on H j X, C. The mixed Hodge numbers of X are defined as the mixed Hodge polynomial of X by h p,q;j X := dim C Gr p F GrW p+qh j X, C, HX; x, y, t := h p,q;j Xx p y q t j, and, when X is smooth, the E-polynomial of X by EX; x, y := xy dimx HX; 1/x, 1/y, 1. We could also define the mixed weight polynomial W HX; q, t = dim C Gr W k H j X, Cq k/2 t j = HX; q 1/2, q 1/2, t which, for a smooth X, specializes to the weight polynomial Eq := q dimx W HX; 1/q, 1 = EX; q 1/2, q 1/2. One observes that the Poincaré polynomial P X; t is given by P X; t = HX; 1, 1, t. Suppose that there exists a separated scheme X over a finitely generated Z-algebra R, such that for some embedding R C we have X R C = X;

6 6 in such a case we say that X is a spreading out of X. If, further, there exists a polynomial P X w Z[w] such that for any homomorphism R F q where F q is the finite field of q elements, one has X F q = P X q, then we say that X has polynomial count and P X is the counting polynomial of X. The motivating result is then the following. Theorem N. Katz, [HV, Theorem 6.1.2] Suppose that the complex algebraic variety X is of polynomial count with counting polynomial P X. Then EX; x, y = P X xy. Remark Thus, in the polynomial count case we find that the count polynomial P X q = EX; q agrees with the weight polynomial. We also expect our varieties to be Hodge Tate, i.e., h p,q;j X = 0 unless p = q, in which case HX; x, y, t = W HX; xy, t. Thus, in these cases we are not losing information by considering W HX; xy, t resp. EX; q instead of the usual HX; x, y, t resp. EX; x, y. 1.2 Wild character varieties The wild character varieties we study in this paper were first mentioned in [2, 3 Remark 5], as a then new example in quasi-hamiltonian geometry a multiplicative variant of the theory of Hamiltonian group actions on symplectic manifolds with a more thorough and more general construction given in [3, 8]. We give a direct definition here for which knowledge of quasi- Hamiltonian geometry is not required; however, as we appeal to results of [3, 9] on smoothness and the dimension of the varieties in question, we use some of the notation of [3, 9] to justify the applicability of those results Definition We now set some notation which will be used throughout the rest of the paper. Let G := GL n C and fix the maximal torus T G consisting of diagonal matrices; let g := gl n C, t := LieT be the corresponding Lie algebras. Let + G resp., G be the orel subgroup of upper resp., lower triangular matrices. Let U = U + + resp., U be the unipotent radical, i.e., the group of upper resp., lower triangular matrices with 1s on the main diagonal; one will note that each of these subgroups is normalized by T. Definition We will use the following notation. For r Z >0, we set A r := G U + U r T. An element of A r will typically be written C, S, t with S = S 1,..., S 2r U + U r, where S i U + if i is odd and S i U if i is even. The group T acts on U + U r by x S = xs 1 x 1,..., xs 2r x 1 ; the latter tuple will often be written simply as xsx 1. We fix g, k, m Z 0 with k + m 1. Fix also a k-tuple C := C 1,..., C k

7 7 of semisimple conjugacy classes C j G; the multiset of multiplicities of the eigenvalues of each C j determines a partition µ j P n. Hence we obtain a k-tuple which we call the type of C. Fix also µ := µ 1,..., µ k P k n r := r 1,..., r m Z m >0. We will write r := m α=1 r α. Now consider the product R g,c,r := G G g C 1 C k A r1 A rm. The affine variety R g,c,r admits an action of G T m given by y, x 1,..., x m A i, i, X j, C α, S α, t α = ya i y 1, y i y 1, yx j y 1, x α C α y 1, x α S α x 1 α, t α, where the indices run 1 i g, 1 j k, 1 α m. Now, fixing an element ξ = ξ 1,..., ξ m T m, we define a closed subvariety of R g,c,r by { g k m U g,c,r,ξ := A i, i, X j, C α, S α, t α R g,c,r : A i, i X j Cα 1 ξ α S2r α α S1 α C α = I n, i=1 j=1 α=1 t α = ξ α, 1 α m }, where the product means we write the elements in the order of their indices: d y i = y 1 y d. i=1 It is easy to see that U g,c,r,ξ is invariant under the action of G T m. Finally, we define the generic genus g wild character variety with parameters C, r, ξ as the affine geometric invariant theory quotient M g,c,r,ξ := U g,c,r,ξ /G T m = Spec C[U g,c,r,ξ ] G Tm Since g will generally be fixed and understood, we will typically omit it from the notation. Furthermore, since the invariants we compute depend only on the tuples µ and r, rather than the actual conjugacy classes C and ξ, we will usually abbreviate our notation to M µ,r and U µ,r. Remark The space A r defined at the beginning of Definition is a higher fission space in the terminology of [3, 3]. These are spaces of local monodromy data for a connection with a higher order pole. To specify a de Rham space which are constructed in [], along with their Dolbeault counterparts at each higher order pole, one specifies a formal type which is the polar part of an irregular connection which will have diagonal entries under some trivialization; this serves as a model connection. The de Rham moduli space then parametrizes holomorphic isomorphism classes of connections which are all formally isomorphic to the specified formal type. Locally these holomorphic isomorphism classes are distinguished by their Stokes data, which live in the factor U + U r appearing in A r. The factor of T appearing is the formal monodromy which differs from the actual monodromy by the product of the Stokes matrices, as appearing in the last set of factors in the expression The interested reader is referred to [1, 2] for details about Stokes data.

8 8 Remark As mentioned above, these wild character varieties were constructed in [3, 8] as quasi-hamiltonian quotients. In quasi-hamiltonian geometry, one speaks of a space with a group action and a moment map into the group. In this case, we had an action on R µ,r given in and the corresponding moment map Φ : R µ,r G T m would be g k A i, i, X j, C α, S α, t α [A i, i ] i=1 m X j j=1 α=1 Then one sees that U µ,r = Φ 1 I n, ξ 1 and so M µ,r Hamiltonian quotient. Cα 1 t α S2r α α S1 α C α, t 1 1,..., t 1 m. = Φ 1 I n, ξ 1 /G T m is a quasi- Remark y taking determinants in we observe that a necessary condition for U µ,r, and hence M µ,r, to be non-empty is that k m det C j det ξ α = 1, j=1 α=1 noting that det S α p = 1 for 1 α m, 1 p 2r α Smoothness and dimension computation We recall [HLV1, Definition 2.1.1]. Definition The k-tuple C = C 1,..., C k is generic if the following holds. If V C n is a subspace stable by some X i C i for each i such that k det X i V = i=1 then either V = 0 or V = C n. When, additionally, we say that is generic if is, where Gξ i is the conjugacy class of ξ i in G. ξ = ξ 1,..., ξ m T m C ξ = C 1,..., C k, ξ 1,..., ξ m C 1,..., C k, Gξ 1,..., Gξ m Remark It is straightforward to see that the genericity of C 1,..., C k for a k-tuple of semisimple conjugacy classes can be formulated in terms of the spectra of the matrices in C i as follows. Let A i := {α i 1,..., α i n} be the multiset of eigenvalues of a matrix in C i for i = 1... k. Then C 1,..., C k is generic if and only if the following non-equalities hold. Write [A] := α A a for any multiset A A i. The non-equalities are for A i A i of the same cardinality n with 0 < n < n. [A 1] [A k]

9 9 Theorem For a generic choice of C ξ in the sense of Definition 1.2.9, the wild character variety is smooth. Furthermore, the G Tm action on U µ,r is scheme-theoretically free. Finally, one has M µ,r dim M µ,r = 2g + k 2n2 µ 2 + nn 1 m + r + 2 =: d µ,r, where r := m α=1 r α and for k µ 2 = l j j=1 p=1 µ j p 2 µ j = µ j 1,..., µj l j. The first statement is a special case of [3, Corollary 9.9], the second statement follows from the observations following [3, Lemma 9.10], and the dimension formula comes from [3, 9, Equation 41]. To see that our wild character varieties are indeed special cases of those constructed there, one needs to see that the double D = G G see [3, Example 2.3] is a special case of a higher fission variety, as noted at [3, 3, Example 1], and that D / C 1G = C for a conjugacy class C G. Then one may form the space S g,k,r := D Gg G D Gk G A r1 G G A rm /G, in the notation of [3, 2,3] and see that M µ,r is a quasi-hamiltonian quotient of the above space by the group G k T m at the conjugacy class C ξ, and is hence a wild character variety as defined at [3, p.342]. To see that the genericity condition given at [3, 9, Equations 38, 39] specializes to ours Definition 1.2.9, we observe that for G = GL n C the Levi subgroup L of a maximal standard parabolic subgroup P corresponds to a subgroup of matrices consisting of two diagonal blocks, and as indicated earlier in the proof of [3, Corollary 9.7], the map denoted pr L takes the determinant of each factor. In particular, it takes the determinant of the relevant matrices restricted to the subspace preserved by P. ut this is the condition in Definition Finally, using [3, 9, Equation 41] and the fact that n dim U + = dim U =, 2 it is straightforward to compute the dimension as Hecke Algebras In the following, we describe the theory of Hecke algebras that we will need for our main results. Let us first explain some notation that will be used. Typically, the object under discussion will be a C-algebra A which is finite-dimensional over C. We will denote its set of isomorphism classes of representations by Rep A and the subset of irreducible representations by Irr A; since it will often be inconsequential, we will often also freely confuse an irreducible representation with its character. Of course, if A = C[G] is the group algebra of a group G, then we often shorten Irr C[G] to Irr G. We will also sometimes need to consider deformations or generalizations of these algebras. If H is an algebra free of finite rank over C[u ±1 ] the extension Cu C[u ±1 ] H of such an algebra to the quotient field Cu of C[u ±1 ] will be denoted by Hu. Note that this abbreviates the notation CuH in [CPA] and [GP, 7.3]. Now, if z C and θ z : C[u ±1 ] C is the C-algebra homomorphism which takes u z, then we may consider the specialization C θz H of H to u = z which we will denote by Hz.

10 Definitions and Conventions Let G be a finite group and H G a subgroup. Given a subset S G, we will denote its indicator function by I S : G Z 0, that is to say, { 1 x S I S x = 0 otherwise. Let M be the vector space of functions f : G C such that fhg = fg for all h H, g G. Clearly, M can be identified with the space of complex-valued functions on H\G and so has dimension [G : H]. We may choose a set V of right H-coset representatives, so that G = v V Hv Such a choice gives a basis of M. Furthermore, we have a G-action on M via {f v := I Hv } v V g fx := fxg With this action, M is identified with the induced representation Ind G H 1 H of the trivial representation 1 H on H. The Hecke algebra associated to G and H, which we denote by H G, H, is the vector space of functions ϕ : G C such that ϕh 1 gh 2 = ϕg, for h 1, h 2 H, g G. It has the following convolution product ϕ 1 ϕ 2 g := 1 H a G ϕ 1 ga 1 ϕ 2 a = 1 H ϕ 1 bϕ 2 b 1 g Furthermore, there is an action of H G, H on M, where, for ϕ H G, H, f M, g G, one has ϕ.fg := 1 ϕxfx 1 g H x G One easily checks that this is well-defined by which we mean that ϕ.f M. It is clear that H G, H may be identified with C-valued functions on H\G/H, and hence it has a basis indexed by the double H-cosets in G. Let W G be a set of double coset representatives which contains e the identity element of G, so that G = HwH Then for w W, we will set these form a basis of H G, H. w W T w := I HwH ; b G

11 11 Proposition [I, Proposition 1.4] Under the convolution product 2.1.4, H G, H is an associative algebra with identity T e = I H. The action yields a unital embedding of algebras H G, H End C M whose image is End G M. Thus we may identify H G, H = End G Ind G H 1 H. Remark Relation with the group algebra The group algebra C[G] may be realized as the space of functions σ : G C with the multiplication given by G : C[G] C C[G] C[G] σ 1 G σ 2 x = a G σ 1 aσ 2 a 1 x. It is clear that we have an embedding of vector spaces ι : H G, H C[G] and it is easy to see that if ϕ 1, ϕ 2 H G, H, we have ιϕ 1 G ιϕ 2 = H ϕ 1 ϕ 2, where the right hand side is the convolution product in H G, H. Furthermore, the inclusion takes the identity element T e H G, H to I H, which is not the identity element in C[G]. Thus, while the relationship between the multiplication in H G, H and that in C[G] will be important for us, we should be careful to note that ι is not an algebra homomorphism. When we are dealing with indicator functions for double H-cosets, we will write T v, v W when we consider it as an element of H G, H, and in contrast, we will write I HvH when we think of it as an element of C[G]. We will also be careful to indicate the subscript in G when we mean multiplication in the group algebra as opposed to the Hecke algebra. We will need some refinements regarding Hecke algebras taken with respect to different subgroups Quotients Let G be a group, H G a subgroup and suppose that H = K L for some subgroups K, L H. Our goal is to show that there is a natural surjection H G, K H G, H. We note that since K H, there is an obvious inclusion of vector spaces H G, H H G, K, when thought of as bi-invariant G-valued functions. We will write K and H to denote the convolution product in the respective Hecke algebras. From 2.1.9, we easily see that for ϕ 1, ϕ 2 H G, H ϕ 1 K ϕ 2 = L ϕ 1 H ϕ Let W = W K G be a set of double K-coset representatives such that L W ; note that for l L, KlK = lk = Kl. If l L and w W, then it follows from [I, Lemma 1.2] that T l K T w = T lw. From this it is easy to see that E = E L := 1 L is an idempotent in H G, K. In fact, one notes that E = L 1 I H = L 1 1 H G,H, thought of as bi-invariant functions on G. l L T l

12 12 Lemma If W K N G L then E is central in H G, K. Proof. It is enough to show that for any w W, E K T w = T w K E. One has E K T w = 1 T l K T w = 1 T lw = 1 T L K H H ww 1 lw = 1 T wm = T w K E. H l L l L Proposition If E is central in H G, K then there exists a surjective algebra homomorphism H G, K H G, H which takes 1 H G,K to 1 H G,H, given by l L α L E K α. Proof. It is easy to check that this map is well-defined, i.e., that if α is K-bi-invariant, then L E K α is H-bi-invariant. To see that it preserves the convolution product, one uses the fact that E is a central idempotent and to see that L E K α K β = L E K α K E K β = L E K α H L E K β. y the remark preceding Lemma , this map preserves the identity. Finally, it is surjective, for given ϕ H G, H, as mentioned above, we may think of it as an element of H G, K and we find L 1 ϕ α Inclusions Suppose now that G is a group L, H G are subgroups and let K := H L. Assume H = K U for some subgroup U G and that L N G U. We write K and H for the convolution products in H L, K and H G, H, respectively. Lemma Suppose x, y L are such that x HyH. Then x KyK. Proof. We write x = h 1 yh 2 for some h 1, h 2 H. Since H = K U, we may write h 1 = ku for some k K, u U. Then x = kyy 1 uyh 2, but since y L N G U, v := y 1 uy U H and so vh 2 H. ut also vh 2 = ky 1 x L, so vh 2 K = H L, and hence x = kyvh 2 KyK. Proposition One has an inclusion of Hecke algebras H L, K H G, H, taking 1 H L,K to 1 H G,H, given by ϕ ϕ H, where ϕ H g := 1 K ϕxi H x 1 g. x L Proof. It is clear that this is a map of vector spaces. To show that it preserves multiplication, let ϕ 1, ϕ 2 H L, K. Then ϕ H 1 H ϕ H 2 g = 1 ϕ 1aϕ 2a 1 1 g = ϕ 1yI Hy 1 aϕ 2xI Hx 1 a 1 g H H K 2 = = a G 1 U K 3 1 U K 3 x,y L a yh x,y L k K,u U x,y L a G ϕ 1yϕ 2xI Hx 1 a 1 g = 1 U K 3 ϕ 1yϕ 2xI Hx 1 k 1 u 1 y 1 g. Making the substitution x = k 1 z, this becomes ϕ H 1 H ϕ H 1 2 g = ϕ 1yϕ 2k 1 zi Hz 1 u 1 y 1 g U K 3 = 1 U K 2 y,z L k K,u U y,z L u U x,y L h H ϕ 1yϕ 2zI H z 1 u 1 zz 1 y 1 g = 1 K 2 m L ϕ 1yϕ 2xI Hx 1 h 1 y 1 g ϕ 1yϕ 2zI H z 1 y 1 g y,z L

13 13 and now letting z = y 1 x, ϕ H 1 H ϕ H 2 g = 1 K 2 x,y L = ϕ 1 K ϕ 2 H g. ϕ 1 yϕ 2 y 1 xi H x 1 g = 1 K ϕ 1 K ϕ 2 xi H x 1 g It is easy to see that 1 H H L,K = IH K = I H = 1 H G,H, so we do indeed get a map of algebras. To see that it is injective, let V L be a set of double K-coset representatives, so that {Tx H L,K } x V is a basis of H L, K. Lemma says that if W is a set of double H-coset representatives in G, then we may take V W. We write {Tw H G,H } w W for the corresponding basis of H G, H, with the subscripts denoting which Hecke algebra the element lies in. Then we observe that T H L,K x H x = 1 K y L T H L,K x yi H y 1 x = 1 K x L y KxK I H y 1 x > 0. This says that the Tx H G,H -component of Tx H L,K H is non-zero, but since V W, the set { H G,H T x }x V is linearly independent, and hence so is the image { Tx H L,K H} of the basis x V of H L, K. 2.2 Iwahori Hecke Algebras of type A n 1 Let G be the algebraic group GL n defined over the finite field F q. Let T G be the maximal split torus of diagonal matrices. There will be a corresponding root system with Weyl group S n, the symmetric group on n letters, which we will identify with the group of permutation matrices. Let G be the orel subgroup of upper triangular matrices. Let the finite dimensional algebra H G, be as defined above. The ruhat decomposition for G, with respect to, allows us to think of S n as a set of double -coset representatives, and hence {T w } w Sn gives a basis of H G,. Furthermore, the choice of determines a set of simple reflections {s 1,..., s n 1 } S n ; we will write T i := T si. The main result of [I, Theorem 3.2] gave the following characterization of H G, in terms of generators and relations. a If w = s i1 s ik is a reduced expression for w S n, then T w = T i1 T ik. Hence H G, is generated as an algebra by T 1,..., T l. b For 1 i l, T 2 i = qt i + q A generic deformation If u is an indeterminate over C, we may consider the C[u ±1 ]-algebra H n generated by elements T 1,..., T n 1 subject to the relations a T i T j = T j T i for all i, j = 1,..., n 1 such that i j > 1; b T i T i+1 T i = T i+1 T i T i+1 for all i = 1,..., n 2; c T 2 i = u 1 + u 1T i; H n is called the generic Iwahori Hecke algebra of type A n 1 with parameter u. Setting u = 1, we see that the generators satisfy those of the generating transpositions for S n and [I, Theorem 3.2] shows that for u = q these relations give the Iwahori Hecke algebra above, so cf. [CR, Proposition] H n 1 = C[S n ] H n q = H G,

14 Yokonuma Hecke Algebras Let T G be as in the previous example, let U be the unipotent radical of, namely, the group of upper triangular unipotent matrices. Then the algebra H G, U, first studied in [Y1], is called the Yokonuma Hecke algebra associated to G,, T. Let NT be the normalizer of T in G; NT is the group of the monomial matrices i.e., those matrices for which each row and each column has exactly one non-zero entry and one has NT = T S n, where the Weyl group S n acts by permuting the entries of a diagonal matrix. We will often write N for NT. y the ruhat decomposition, one may take N G as a set of double U-coset representatives. Section 2.1 describes how H G, U has a basis {T v, v NT} A generic deformation The algebra H G, U has a presentation in terms of generators and relations due to [Y1, Y2], which we now describe and which we will make use of later on. For this consider the C[u ±1 ]- algebra Y d,n generated by the elements subject to the following relations: T i, i = 1,..., n 1 and h j for j = 1,..., n a T i T j = T j T i for all i, j = 1,..., n 1 such that i j > 1; b T i T i+1 T i = T i+1 T i T i+1 for all i = 1,..., n 2; c h i h j = h j h i for all i, j = 1,..., n; d h j T i = T i h sij for all i = 1,..., n 1, and j = 1,..., n, where s i := i, i + 1 S n ; e h d i = 1 for all i = 1,..., n; f T 2 i = uf if i+1 + u 1e i T i for i = 1,..., n 1, where for i = 1,..., n 1, and e i := 1 d d j=1 h j i h j i f i := { d/2 h i for d even, 1 for d odd for i = 1,..., n. In Theorem below, we will see that H G, U arises as the specialization Y q 1,n q. Remark The modern definition of Y d,n in terms of generators and relations takes f i = 1, regardless of the parity of d cf. [CPA] and [J2]. We decided to take that of [Y1] so that the meaning of the generators is more transparent. Again, this will be clearer from Theorem and its proof. 2.4 Some computations in H G, U and Y d,n u We continue with the context of the previous subsection. There is a canonical surjection p : N S n, which allows us to define the length of an element v N as that of pv; we will denote this by lv.

15 15 Lemma If v 1, v 2 N are such that lv 1 v 2 = lv 1 + lv 2, then T v1 T v2 = T v1v 2. In particular, if h T, v N then T h T v = T hv. Proof. This follows readily from [I, Lemma 1.2] using the fact that, in the notation there, for v N, indv = q lv. Our first task is to describe the relationship of Y d,n to H G, U, as alluded to at the beginning of Section To do this, we follow the approach in [GP, 7.4,8.1.6]. For example, the u = 1 specialization of Y q 1,n gives Y q 1,n 1 = C θ1 Y q 1,n = C[N] = C[F q n S n ] the group algebra of the normalizer in G of the torus T F q. Theorem Let q be a prime power and fix a multiplicative generator t g F q. For t F q, let h i t T be the diagonal matrix obtained by replacing the ith diagonal entry of the identity matrix by t. Finally, we let s i N denote the permutation matrix corresponding to i, i + 1 and ω i := s i h i 1 = h i+1 1s i N. Then one has an isomorphism of C-algebras Y q 1,n q = H G, U under which T i T ωi H G, U and h i T hit g H G, U. Note that ω i is the matrix obtained by replacing the 2 2-submatrix of the identity matrix formed by the ith and i + 1st rows and columns by [ ] Proof. It is sufficient to show that T ωi, T hjt g satisfy the relations prescribed for T i, h j in Section Lemma makes most of these straightforward. The computation in [Y1, Théorème 2.4 ] gives = qt hi 1h i T hith i+1t 1 T ωi, which is relation f. T 2 ω i The longest element w 0 W = S n is the permutation n/2 i=1 i, n + 1 i the order of the factors is immaterial since this is a product of disjoint transpositions and is of length n 2. We may choose a reduced expression With the same indices as in 2.4.4, we define t F q w 0 = s i1 s i n ω 0 := ω i1 ω i n 2 N, and T 0 := T i1 T i n 2 Y d,n Using the braid relations a and b and arguing as for Matsumoto s Theorem [GP, Theorem 1.2.2], one sees that T 0 is independent of the choice of reduced expression Now, Lemma shows that T ω0 = T ωi1 T ωi H G, U n 2 and Theorem shows that this corresponds to T 0 Y q 1,n q.

16 16 Lemma The element T 2 0 is central in Y d,n. It follows, by specialization, that Tω 2 0 H G, U. is central in Proof. Proceeding as in [GP, 4.1], we define a monoid + generated by and subject to the relations T 1,..., T n 1, h 1,..., h n a T i T j = T j T i for 1 i, j n 1 with i j > 1; b T i T i+1 T i = T i+1 T i T i+1 for 1 i n 2; c h j T i = T i h sij for 1 i n 1, 1 j n, where s i := i, i + 1 S n. Observe that these are simply relations a, b and d of those given for Y d,n in Section 2.3.1; one can define the monoid algebra C[u ±1 ][ + ] of which Y d,n will be a quotient via the mapping for 1 i n 1, 1 j n. Letting T i T i h j h j, T 0 := T i1 T i n 2 C[u±1 ][ + ], it is enough to show that T 0 is central in C[u ±1 ][ + ]. Arguing as in the proof of [GP, Lemma 4.1.9], one sees that T 2 0 commutes with each T i, 1 i n 1. Furthermore, relation c and give h j T 2 0 = T 2 0h w 2 0 j = T 2 0h j, noting that w 2 0 = 1 implies w 0 = w 1 0 = s i n 2 s i 1. The following will be useful when we look at representations. Lemma The element e i defined in commutes with T i in Y d,n u. Proof. y relation d, we see that T i h j i h j i+1 = h j i h j i+1 T i. The Lemma follows by averaging over j = 1,..., d and observing that both h i and h i+1 have order d. Lemma The elements T 1,..., T n 1 Y d,n u are all conjugate. Proof. From relation f in Section 2.3.1, we see that each T i is invertible, with inverse Then T 1 i = u 1 T i u 1e i f i f i+1. T i T i+1 T i T i+1 T i T i+1 T i 1 = T i T i+1 T i T i+1 T i+1 T i T i+1 1 = T i, and the statement follows by transitivity of the conjugacy relation.

17 The double centralizer theorem The following is taken from [KP, 3.2]. Let K be an arbitrary field, A a finite-dimensional algebra over K and let W be a finite-dimensional left A-module. Recall that W is said to be semisimple if it decomposes as a direct sum of irreducible submodules. If A is semisimple as a module over itself then it is called a semisimple algebra; the Artin Wedderburn theorem then states that any such A is a product of matrix algebras over finite-dimensional division K-algebras. If U is a finite-dimensional simple A-module, then the isotypic component of W of type U is the direct sum of all submodules of W isomorphic to U. The isotypic components are then direct summands of W and their sum gives a decomposition of W precisely when the latter is semisimple; in this case, it is called the isotypic decomposition of W. We recall the following, which is often called the double centralizer theorem. Theorem Let W a finite-dimensional vector space over K, let A End K M be a semisimple subalgebra and let A = End A W := {b End K W : ab = ba a A}, be its centralizer subalgebra. Then A is also semisimple and there is a direct sum decomposition W = r i=1 which is the isotypic decomposition of W as either an A-module or an A -module. In fact, for 1 i r, there is an irreducible A-module U i and an irreducible A -module U i such that if D i := End A U i this is a division K-algebra by Schur s lemma, then End A U i = D op and W i W i = Ui Di U i. Remark In the case where K is algebraically closed, then there are no non-trivial finitedimensional division algebras over K, and so in the statement above, the tensor product is over K. We are interested in the case where K = C so that we are within the scope of the Remark, H and G are as in Section 2.1, W = Ind G H 1 H is the induction of the trivial representation of a subgroup H G to G and A is the image of the group algebra C[G] in End C W. Then A is semisimple and via Proposition 2.1.7, we know A = H G, H. We can then conclude the following. Corollary The Hecke algebra H G, H is semisimple. Furthermore, one observes that the kernel of the induced representation Ind G H 1 H is given by g G ghg 1. Thus, applying Theorem with A = H G, H, since its commuting algebra is the image of the G-action, we may state the following. Corollary If g G ghg 1 is trivial, one has C[G] = End H G,H Ind G H 1 H. 2.6 Representations of Hecke algebras We return to the abstract situation of Section 2.1. y a representation of H G, H we will mean a pair W, ρ consisting of a finite-dimensional complex vector space V and an identity-preserving homomorphism ρ : H G, H End C W. Let V, π be a representation of G and let V H V

18 18 be the subspace fixed by H. Then V H is a representation of the Hecke algebra H G, H via the action ϕ.v := 1 ϕaπa v H a G for ϕ H G, H, v V H. It is easy to check that ϕ.v V H so that this is well-defined. Note that upon choosing a basis vector for the trivial representation 1 H of H, we may identify Hom H 1 H, Res G H V = V H by taking an H-morphism to the image of the basis vector. Thus, we have defined a map D H : Rep G Rep H G, H Let us now set V, π Hom H 1 H, Res G H V = V H. IrrG : H := { ζ Irr G : ζ, 1 G H > 0 }, where, is the pairing on characters; the condition is equivalent to Hom H 1 H, Res G H ζ 0. We can now give the following characterisation of irreducible representations of H G, H, which, in the more general case of locally compact groups, is [Z, Proposition 2.10]. Proposition If V, π is an irreducible representation of G, then V H is an irreducible representation of H G, H, and every irreducible representation of H G, H arises in this way, that is, D H restricts to a bijection D H : IrrG : H Irr H G, H. Since C[G] and H G, H are semisimple, we can apply Theorem to W = Ind G H 1 H. If we denote the set of irreducible representations of G by Irr G, we find that Ind G H 1 H = V V H, V Irr G V H {0} with elements of G acting on the left side of the tensor product and those of H G, H acting on the right. One has a consistency check here in that for an irreducible representation V of G, the multiplicity of V in Ind G H 1 H is given by dim Hom G Ind G H 1 H, V = dim Hom H 1H, Res G H V = dim V H, which the decomposition in confirms Traces Let M = Ind G H 1 H be as in Section 2.1. Observe that if X End C M, then using the basis 2.1.2, its trace is computed as tr M X = v V X.f v v. Lemma Let g G and ϕ H G, H and consider gϕ = ϕg End C Ind G H 1 H where g is thought of as an element of End C Ind G H 1 H via Then trgϕ = 1 ϕxgx 1 = H x G V IrrG:H χ V gχ DH V ϕ, where χ V is the character of the G-module V, and χ DH V is that of the H G, H-module D H V.

19 19 Proof. In the notation of 2.1.2, H trgϕ = H gϕ.fv v = H ϕ.f v vg = ϕyf v y 1 vg v V v V v V y G = ϕy = ϕy = ϕvgv 1 h 1 v V y 1 vg Hv v V y vgv 1 H v V h H = ϕ hvghv 1 = ϕxgx 1, v V h H x G where we use at the last line. On the other hand, if g, ϕ are as in the Lemma, then applying gϕ = ϕg to the decomposition 2.6.4, we get trgϕ = χ V gχ DH V ϕ Induced representations V IrrG:H Let G, H, L, K and U be as in Section Then one sees that L U is trivial and hence we may define P := L U. The inclusion H L, K H G, H given by Proposition allows us to induce representations from H L, K to H G, H: if V is an H L, K-representation, then Ind H G,H H L,K V := H G, H H L,K V, yielding a map Rep H L, K Rep H G, H. We also have a parabolic induction functor: for V Rep L, we define R G L V := Ind G P Infl P L V, which gives a map Rep L Rep G. Let RepG : H denote the set of isomorphism classes of finite-dimensional representations of G at least one of whose irreducible components lies in IrrG : H. Then we claim that if V IrrL : K, then R G L V RepG : H and hence RG L yields a map R G L : IrrL : K RepG : H. We know that V IrrL : K if and only if Hom K 1 K, Res L K V 0. The latter implies that Now, inducing to G in both factors 1 0 Hom H 1 H, Res P H Infl P L V = Hom P Ind P H 1 H, Infl P L V. 0 Hom G Ind G P Ind P H 1 H, Ind G P Infl P L V = Hom G Ind G H 1 H, R G L V = Hom H 1 H, Res G H R G L V, which proves the claim. Our goal here is to show that the D-operators are compatible with these induction operations. Here is the precise statement. Proposition Assume that l L lkl 1 is trivial. Then the following diagram commutes: IrrL : K D K Irr H L, K R G L RepG : H D H Ind Rep H G, H. 1 If Irr P, then 1 Ind End G Ind G P, IndG P, so Hom P, Res G P IndG P 0, and thus is an irreducible component of Res G P IndG P. It follows that if A and are any P -representations with Hom P A, 0 then Hom G Ind G P A, HomG P 0.

20 20 Proof. y the assumption, Lemma gives us C[L] = End H L,K Ind L K 1 K = Ind L K 1 K H L,K Ind L K 1 K. Therefore, given V IrrL : K, we may rewrite this as V = C[L] C[L] V = Ind L K 1 K H L,K Ind L K 1 K C[L] V, where now we are taking the L-action on the first factor Ind L K 1 K. Using gives Ind L K 1 K C[L] V = Hom L Ind L K 1 K, V = Hom K 1K, Res L K V = D K V, V = Ind L K 1 K H L,K D K V. Now, applying RL G to both sides, one gets RL G V = RL G Ind L K 1 K H L,K D K V = RL Ind G L K 1 K H L,K D K V, as we had said that the L-action is on the first factor. Now, using the natural isomorphisms of functors Infl P L Ind L K = Ind P H Infl H K Ind G P Ind P H = Ind G H and the fact that Infl H K 1 K = 1 H, we can simplify Ind L K 1 K = Ind G P Infl P L Ind L K 1 K = Ind G P Ind P H Infl H K 1 K = Ind G H 1 H R G L and so becomes RL G V = Ind G H 1 H H L,K D K V. Applying now the functor D H = Hom H 1H, Res G H to both sides, and again noting that the G-action in the right hand side is on the first factor, we get D H R G L V = Hom H 1H, Res G H Ind G H 1 H H L,K D K V = End G Ind G H 1 H, Ind G H 1 H H L,K D K V = H G, H H L,K D K V = Ind H G,H H L,K D KV. 2.7 Character tables We now review some facts about the character tables of some finite groups which will be used in our counting arguments later. As a matter of notation, in this section and later, if A is an abelian group, we often denote its group of characters by  := HomA, C.

21 Character table of GL n F q We follow the presentation of [Ma, Chapter IV]. Fix a prime power q. Let Γ n := F qn be the dual group of F q n. For n m, the norm maps Nm n,m : F q m F qn yield an inverse system, and hence the Γ n form a direct system whose colimit we denote by Γ := lim Γ n. There is a natural action of the Frobenius Frob q : F q F q, given by γ γ q, restricting to each F q and hence inducing an on action each Γ n n and hence on Γ; we identify Γ n with Γ Frobn q. Let Θ denote the set of Frob q -orbits in Γ. The weighted size of a partition λ = λ 1, λ 2,... P is nλ := i 1λ i = λ j 2 i 1 j 1 where, as usual, λ = λ 1, λ 2,... is the conjugate partition of λ, i.e., λ i is the number of λ j s not smaller than i. The hook polynomial of λ is defined as H λ q := λq h where the product is taken over the boxes in the Ferrers diagram cf. [Sta, 1.3] of λ, and h is the hook length of the box in position i, j defined as h := λ i + λ j i j + 1. y [Ma, IV 6.8] there is a bijection between the irreducible characters of GL n q and the set of functions Λ : Θ P which are stable under the Frobenius action and having total size Λ := γ Θ γ Λγ equal to n. Under this correspondence, the character χ G Λ corresponding to Λ has degree n i=1 qi 1 q nλ H Λ q where H Λ q := γ Θ H Λγ q γ and nλ := γ Θ γ nλγ Remark There is a class of irreducible characters of GL n F q, known as the unipotent characters, which are also parametrized by P n. Given λ P n, the associated unipotent character χ G λ is the one corresponding to, in the description above, the function Λ λ : Θ P, where Λ λ takes the singleton orbit of the trivial character in Γ 1 to λ P n and all other orbits to the empty partition.

22 22 In fact, any ψ Γ 1 = F q is a singleton orbit of Frob q and so we may view Γ 1 as a subset of Θ. Thus, if we let Q n denote the set of maps Λ : Γ 1 P of total size n, i.e., Λ = ψ Γ 1 Λψ = n, then Q n is a subset of the maps Θ P of size n. The set of characters of GL n F q corresponding to the maps in Q n will also be important for us later. The following description of the characters corresponding to Q n can be found in the work of Green [Gr]. Suppose n 1, n 2 are such that n = n 1 + n 2. Let L = GL n1 F q GL n2 F q and view it as the subgroup of GL n F q of block diagonal matrices, U 12 G the subgroup of upper block unipotent matrices, and P := L U 12 the parabolic subgroup of block upper triangular matrices. The -product : Irr GL n1 F q Irr GL n2 F q Rep GL n F q is defined as χ 1 χ 2 = R G L χ 1 χ 2 = Ind G P Infl P Lχ 1 χ Now, given Λ Q n, we will often write ψ 1,..., ψ r Γ 1 for the distinct elements for which n i := Λψ i > 0 note that i n i = n and λ i := Λψ i. For each 1 i r, we have the unipotent representation χ G λ i of GL ni F q, described above, as well as the character ψi G := ψ i det : GL ni F q F q C. Then the irreducible character χ G Λ of GL nf q associated to Λ Q n is χ G Λ := χ G Λψ 1 ψg 1 χ G Λψ r ψg r The fact that it is irreducible is attributable to [Gr]. One may also think of the tuple of characters of the GL ni F q as yielding one on the product, which may be viewed as the Levi of some parabolic subgroup of GL n F q. Then the above -product is the parabolic induction of the character on the Levi Character table of N Recall that we have isomorphisms N = T S n = F q n S n. Our aim is to describe Irr N, but let us begin with a description of the irreducible representations of each of its factors. One has Irr T = T, the dual group. Furthermore, it is well known that Irr S n is in natural bijection with the set P n of partitions of n: to λ P n one associates the left submodule of C[S n ] spanned by its Young symmetrizer [FH, 4.1]; we will denote the resulting character by χ S λ. To describe Irr N explicitly, we appeal to [Se, 8.2, Proposition 25], which treats the general situation of a semidirect product with abelian normal factor. If ψ T then ψ extends to a 1- dimensional character of T Stab ψ noting that if we identify T = F q n, then S n acts by permutations trivial on Stab ψ; so now, given χ IrrStab ψ, we get ψ χ IrrT Stab ψ. The result cited above says that Ind N T Stab ψ ψ χ is irreducible and in fact all irreducible representations of N arise this way with the proviso that we get isomorphic representations if we start with characters in the S n -orbit of ψ. If we write ψ = ψ 1,..., ψ n F q n, then Stab ψ = S n1 S nr, where the n i are the multiplicities with which the ψ j appear. Further, χ IrrStab ψ = Irr S n1 Irr S nr so is an exterior tensor product χ = χ S λ 1 χ S λ r with λ i P ni. Thus, from Ind ψ χ, we may define a map Λ : Γ 1 P by setting Λψ j = λ i, where j is among the indices permuted by S ni and Λϕ to be the empty partition if ϕ F q does not appear in ψ. In this way, we get a map Λ : Γ 1 P of total size n, i.e., an element of Q n defined in Remark Conversely, given Λ Q n, let ψ i Γ 1, n i and λ i P ni be as in the paragraph preceding Let T i := F q ni and set N i := T i S ni. Observe that if ψ T is defined by taking the ψ i