Arithmetic and representation theory of wild character varieties

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.


Introduction 1.A conjecture
Let C be a complex smooth projective curve of genus g ∈ Z ≥0 , with divisor 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 ) ∈ P k n 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; End(E) ⊗ 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 → A µ,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=1 H 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 H(M µ,r Dol ; q, t) := dim Gr P i (H k (M µ,r Dol )) q i t k .
The recent paper [CDDP] by Chuang-Diaconescu-Donagi-Pantev gives a string theoretical derivation of the following mathematical conjecture.
Here, H µ,r (z, w) ∈ Q(z, w) is defined by the generating function formula Hλ (z 2 , w 2 ; x i ) .(1.1.2) The notation is explained as follows.For a partition λ ∈ P we denote 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 Q(q, t).Finally, Log is the plethystic logarithm, see e.g.[HLV1,§2.3.3.] for a definition.
The paper [CDDP] gives several pieces of evidence for Conjecture 1.1.1.On physical grounds it argues that the left hand side should be the generating function for certain refined BPS 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 1.1.1.Another approach is to use another duality conjecture-the so-called "geometric engineering"-which conjecturally relates the left hand side of Conjecture 1.1.1 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 1.1.1.
Purely mathematical evidence for Conjecture 1.1.1comes 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 B , which we will define more carefully below.The cohomology of M µ B carries a weight filtration, and we denote by the mixed Hodge polynomial of M µ B .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 B ) via non-abelian Hodge theory.In particular, P = W would imply P H(M µ Dol ; q, t) = W H(M µ B ; q, t), and Conjecture 1.1.1 for r = 0; P H(M µ Dol ; q, t) replaced with W H(M µ B ; q, t) was the main conjecture in [HLV1].
It is interesting to recall what inspired Conjecture 1.1.1 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 H(M (2) B ; q, t).In fact, it was observed in [dCHM] that the computation in [HV,Remark 2.5.3]amounted to a formula for P H(M µ Dol ; q, t), which is the first non-trivial instance of Conjecture 1.1.1.This twist by r was first extended for the conjectured P H(M (n),r Dol ) in [Mo] to match the recursion relation in [CDP]; it was then generalized in [CDDP] to Conjecture 1.1.1.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 1.1.1does 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 H(M µ,r Dol ; q, t) above.Problem 1.1.4.Is there a character variety whose mixed Hodge polynomial agrees with P H(M 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.By the wild non-abelian Hodge theorem [BB] those will be diffeomorphic with wild character varieties.

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.Boalch in [B3] gives the following construction.
Let G := GL n (C) and let T ≤ G be the maximal torus of diagonal matrices.Let B + ≤ G (resp.B − ≤ G) be the Borel subgroup of upper (resp.lower) triangular matrices.Let U = U + ≤ B + (resp.U − ≤ B − ) 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 in other words, C i has eigenvalues with multiplicities µ i j .Finally we fix an m-tuple of regular diagonal matrices, such that the k + m tuple of semisimple conjugacy classes is generic in the sense of Definition 2.2.9.Then define where the affine quotient is by the conjugation action of G on the matrices A i , B i , X i , C i and the trivial action on S j i .Under the genericity condition as above, M µ,r B is a smooth affine variety of dimension d µ,r of (2.2.14).In particular, when m = 0, we have the character varieties M µ B = M µ,∅ B of [HLV1].
The main result of this paper is the following: Theorem 1.2.1.Let µ ∈ P k n be a k-tuple of partitions of n and r be an m-tuple of positive integers and M µ,r B be the generic wild character variety as defined above.Then we have W H(M µ,r B ; q, −1) = q dµ,r H μ,r (q −1/2 , q 1/2 ), where μ := (µ 1 , . . ., µ k , (1 n ), . . ., (1 n )) ∈ P k+m n is the type of (C 1 , . . ., C k , Gξ 1 , . . ., Gξ m ) and 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 H(M µ,r B ; q, −1) to counting M µ,r B (F q ), i.e., the F q points of M µ,r B .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 where U is as above.The main computational result, Theorem 4.3.4,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 3.7 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 3.9.5 clarifies this relationship, establishing an analogue of a result proved by Halverson and Ram [HR,Theorem 4.9(b)], though by different techniques.
Our main result Theorem 1.2.1 then leads to the following conjecture.
Conjecture 1.2.2.We have This gives a conjectural partial answer to our Problem 1.1.4originally 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 2 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 [B3].In Section 3 we recall the abstract approach to Hecke algebras; the explicit character theory of the Iwahori-Hecke and Yokonuma-Hecke algebras is also reviewed and clarified.In Section 4 we recall the arithmetic Fourier transform approach of [HLV1] and perform the count on the wild character varieties.In Section 5 we prove our main Theorem 1.2.1 and discuss our main Conjecture 1.2.2.In Section 6 we compute some specific examples of Theorem 1.2.1 and Conjecture 1.2.2, when n = 2, with particular attention paid to the cases when M µ,r B is a surface.Acknowledgements.We thank Philip Boalch, Daniel Bump, Maria Chlouveraki, Alexander Dimca, Mario García-Fernández, Eugene Gorsky, Emmanuel Letellier, András Némethi, Loïc Poulain d'Andecy, Vivek Shende, Szilárd Szab ó and Fernando R. Villegas for discussions and/or correspondence.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.320593 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 SFB/TR 45 "Periods, moduli and arithmetic of algebraic varieties", subproject M08-10 "Moduli of vector bundles on higher-dimensional varieties".

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 c (X, Q) and a decreasing Hodge filtration F • on H j c (X, C).The compactly supported mixed Hodge numbers of X are defined as the compactly supported mixed Hodge polynomial of X by and the E-polynomial of X by E(X; x, y) := H c (X; x, y, −1).
One observes that the compactly supported Poincaré polynomial P c (X; t) is given by Suppose that there exists a separated scheme X over a finitely generated Z-algebra R, such that for some embedding R ֒→ C we have 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 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 2.1.1.(N.Katz, [HV,Theorem 6.1.2])Suppose that the complex algebraic variety X is of polynomial count with counting polynomial P X .Then E(X; x, y) = P X (xy).
Remark 2.1.2.Thus, in the polynomial count case we find that the count polynomial P X (q) = E(X; q) agrees with the weight polynomial.We also expect our varieties to be Hodge-Tate, i.e., h p,q;j c (X) = 0 unless p = q, in which case H c (X; x, y, t) = W H(X; xy, t).Thus, in these cases we are not losing information by considering W H(X; xy, t) (resp.E(X; q)) instead of the usual H c (X; x, y, t) (resp.E(X; x, y)).

Wild character varieties
The wild character varieties we study in this paper were first mentioned in [B2, §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 [B3, §8].We give a direct definition here for which knowledge of quasi-Hamiltonian geometry is not required; however, as we appeal to results of [B3, §9] on smoothness and the dimension of the varieties in question, we use some of the notation of [B3, §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 := Lie(T) be the corresponding Lie algebras.Let B + ≤ G (resp., B − ≤ G) be the Borel subgroup of upper (resp., lower) triangular matrices.Let U = U + ≤ B + (resp., U − ≤ B − ) 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 2.2.1.We will use the following notation.For r ∈ Z >0 , we set An element of A r will typically be written (C, S, t) with 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 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 µ := (µ 1 , . . ., µ k ) ∈ P k n which we call the type of C. Fix also We will write r := m α=1 r α .Now consider the product The affine variety R g,C,r admits an action of G × T m given by where the indices run Now, fixing an element ξ = (ξ 1 , . . ., ξ m ) ∈ T m , we define a closed subvariety of R g,C,r by U g,C,r,ξ := (A i , B i , X j , C α , S α , t α ) ∈ R g,C,r : where the product means we write the elements in the order of their indices: 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 (2.2.4) 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 B and U µ,r .
Remark 2.2.5.The space A r defined at the beginning of Definition 2.2.1 is a "higher fission space" in the terminology of [B3, §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 [BB], 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 (2.2.3).The interested reader is referred to [B1, §2] for details about Stokes data.
Remark 2.2.6.As mentioned above, these wild character varieties were constructed in [B3, §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 (2.2.2) and the corresponding moment map Φ : R µ,r → G × T m would be Then one sees that U µ,r = Φ −1 ((I n , ξ −1 )) and so M µ,r Remark 2.2.7.By taking determinants in (2.2.3) we observe that a necessary condition for U µ,r , and hence M µ,r B , to be non-empty is that
Definition 2.2.9.The k-tuple is, where Gξ i is the conjugacy class of ξ i in G.
Remark 2.2.11.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 (2.2.12) hold.Write [A] := α∈A a for any multiset A ⊆ A i .The non-equalities are Theorem 2.2.13.For a generic choice of C × ξ (in the sense of Definition 2.2.9), the wild character variety M µ,r B is smooth.Furthermore, the G × T m action on U µ,r is scheme-theoretically free.Finally, one has where r := m α=1 r α and The first statement is a special case of [B3,Corollary 9.9], the second statement follows from the observations following [B3,Lemma 9.10], and the dimension formula comes from [B3,§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 [B3,Example 2.3]) is a special case of a higher fission variety, as noted at [B3, §3, Example (1)], and that D/ / C −1 G ∼ = C for a conjugacy class C ⊂ G. Then one may form the space in the notation of [B3, § §2,3] and see that M µ,r B 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 [B3, p.342].
To see that the genericity condition given at [B3,§9,Equations (38),(39)] specializes to ours (Definition 2.2.9), 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 [B3,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 .But this is the condition in Definition 2.2.9.

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 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 ] will be denoted by H(u).Note that this abbreviates the notation C(u)H 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 H(z).

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, Let M be the vector space of functions f : Clearly, M can be identified with the space of complex-valued functions on H\G and so has dimension Such a choice gives a basis The Hecke algebra associated to G and H, which we denote by H (G, H), is the vector space of functions ϕ : 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 = w∈W HwH. (3.1.6) Then for w ∈ W, we will set these form a basis of H (G, H).
Proposition 3.1.7.[I,Proposition 1.4 Remark 3.1.8.(Relation with the group algebra) The group algebra C[G] may be realized as the space of functions σ : given by It is clear that we have an embedding of vector spaces and it is easy to see that if 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 (3.1.9),we easily see that for (3.1.10) Proof.It is easy to check that this map is well-defined, i.e., that if To see that it preserves the convolution product, one uses the fact that E is a central idempotent and (3.1.10)to see that By the remark preceding Lemma 3.1.11,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 We write * K and * H for the convolution products in H (L, K) and H (G, H), respectively.
Proof.We write x = h 1 yh 2 for some h 1 , h 2 ∈ H. Since H = K ⋉U , we may write and hence x = ky(vh 2 ) ∈ KyK.
Proposition 3.1.14.One has an inclusion of Hecke algebras Proof.It is clear that this is a map of vector spaces.To show that it preserves multiplication, let Making the substitution x = k −1 z, this becomes It is easy to see that ½ 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 {T } 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 This says that the T x∈V is linearly independent, and hence so is the image (T x∈V of the basis of H (L, K).

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 B ≤ G be the Borel subgroup of upper triangular matrices.Let the finite dimensional algebra H (G, B) be as defined above.The Bruhat decomposition for G, with respect to B, allows us to think of S n as a set of double B-coset representatives, and hence {T w } w∈Sn gives a basis of H (G, B).Furthermore, the choice of B 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, B) in terms of generators and relations.

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) 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,(68.11)Proposition])

Yokonuma-Hecke Algebras
Let T ≤ B ≤ G be as in the previous example, let U ≤ B be the unipotent radical of B, 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, B, T. Let N (T) be the normalizer of T in G; N (T) 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 N (T) = T ⋊ S n , where the Weyl group S n acts by permuting the entries of a diagonal matrix.We will often write N for N (T).By the Bruhat decomposition, one may take N ≤ G as a set of double U-coset representatives.Section 3.1 describes how H (G, U) has a basis {T v , v ∈ N (T)}.

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 T i , i = 1, . . ., n − 1 and h j for j = 1, . . ., n subject to the following relations: (a) (c) h i h j = h j h i for all i, j = 1, . . ., n; (d) h j T i = T i h si(j) 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; where for i = 1, . . ., n − 1, and In Theorem 3.4.3below, we will see that H (G, U) arises as the specialization Y q−1,n (q).
Remark 3.3.3.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 3.4.3and its proof.

Some computations in
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 p(v); we will denote this by ℓ(v).
Proof.This follows readily from [I, Lemma 1.2] using the fact that, in the notation there, for v ∈ N, ind(v) = q ℓ(v) .
Our first task is to describe the relationship of Y d,n to H (G, U), as alluded to at the beginning of Section 3.3.1.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 the group algebra of the normalizer in G of the torus T (F q ).
Theorem 3.4.3.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 Then one has an isomorphism of C-algebras Y q−1,n (q) ∼ = H (G, U) under which Note that ω i is the matrix obtained by replacing the (2 × 2)-submatrix of the identity matrix formed by the ith and (i + 1)st rows and columns by Proof.It is sufficient to show that T ωi , T hj(tg ) satisfy the relations prescribed for T i , h j in Section 3.3.1.Lemma 3.4.1 makes most of these straightforward.The computation in [Y1, Théorème 2.4 • ] gives The longest element w 0 ∈ W = S n is the permutation 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 (3.4.4), we define 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 (3.4.4).Now, Lemma 3.4.1 shows that and Theorem 3.4.3shows that this corresponds to T 0 ∈ Y q−1,n (q).Lemma 3.4.6.The element Proof.Proceeding as in [GP,§ 4.1], we define a monoid B + generated by Observe that these are simply relations (a), (b) and (d) of those given for Y d,n in Section 3.3.1;one can define the monoid algebra C[u ±1 ][B + ] of which Y d,n will be a quotient via the mapping 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 (4.2.5) give The following will be useful when we look at representations.
Lemma 3.4.7.The element e i defined in (3.3.1)commutes with T i in Y d,n (u).
Proof.By relation (d), we see that The Lemma follows by averaging over j = 1, . . ., d and observing that both h i and h i+1 have order d.Proof.From relation (f) in Section 3.3.1,we see that each T i is invertible, with inverse and the statement follows by transitivity of the conjugacy relation.

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 3.5.1.Let W a finite-dimensional vector space over K, let A ⊆ End K M be a semisimple subalgebra and let be its centralizer subalgebra.Then A ′ is also semisimple and there is a direct sum decomposition 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 Remark 3.5.2.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 3.1, W = Ind G H ½ 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 3.1.7,we know A ′ = H (G, H).We can then conclude the following.
Furthermore, one observes that the kernel of the induced representation Ind G H ½ H is given by g∈G gHg −1 .Thus, applying Theorem 3.5.1 with A = H (G, H), since its commuting algebra is the image of the G-action, we may state the following.

Representations of Hecke algebras
We return to the abstract situation of Section 3.1.By 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 be the subspace fixed by H. Then V H is a representation of the Hecke algebra H (G, H) via the action Note that upon choosing a basis vector for the trivial representation ½ H of H, we may identify by taking an H-morphism to the image of the basis vector.Thus, we have defined a map

Let us now set
Irr(G : where ( , ) is the pairing on characters; the condition is equivalent to Hom H (½ 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 [BZ,Proposition 2.10].
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 which the decomposition in (3.6.4) confirms.

Traces
Let M ∼ = Ind G H ½ H be as in Section 3.1.Observe that if X ∈ End C M , then using the basis (3.1.2),its trace is computed as Then where χ V is the character of the G-module V , and Proof.In the notation of (3.1.2), where we use (3.1.1)at the last line.On the other hand, if g, ϕ are as in the Lemma, then applying gϕ = ϕg to the decomposition (3.6.4),we get tr(gϕ) = V ∈Irr(G:H) χ V (g)χ DH (V ) (ϕ).

Induced representations
Let G, H, L, K and U be as in Section 3.1.2.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 3.1.14allows us to induce representations from We also have a "parabolic induction" functor: for V ∈ Rep L, we define We know that V ∈ Irr(L : K) if and only if Hom K (½ K , Res L K V ) = 0.The latter implies that 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 3.6.7.Assume that ℓ∈L ℓKℓ −1 is trivial.Then the following diagram commutes: Proof.By the assumption, Lemma 3.5.4gives us Therefore, given V ∈ Irr(L : K), we may rewrite this as where now we are taking the L-action on the first factor Ind L K ½ K .Using (3.6.8)gives Now, applying R G L to both sides, one gets (3.6.9) as we had said that the L-action is on the first factor.Now, using the natural isomorphisms of functors and so (3.6.9)becomes Applying now the functor to both sides, and again noting that the G-action in the right hand side is on the first factor, we get

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 A := Hom(A, C × ).

Character table of GL n (F q )
We follow the presentation of [Ma, Chapter IV].Fix a prime power q.Let Γ n := F × q n be the dual group of F × q n .For n|m, the norm maps Nm n,m : F × q m → F × q n yield an inverse system, and hence the Γ n form a direct system whose colimit we denote by There is a natural action of the Frobenius Frob q : F × q → F × q , given by γ → γ q , restricting to each F × q n and hence inducing an on action each Γ n and hence on Γ; we identify Γ n with Γ Frob n 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 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 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 By [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 where (3.7.4) Remark 3.7.5.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.
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., 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 Now, given Λ ∈ Q n , we will often write ψ 1 , . .., ψ r ∈ Γ 1 for the distinct elements for which For each 1 ≤ i ≤ r, we have the unipotent representation χ G λi of GL ni (F q ), described above, as well as the character 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 (1dimensional) 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 ∈ Irr(Stab ψ), we get ψ ⊗ χ ∈ Irr(T ⋊ 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 , . . ., , where the n i are the multiplicities with which the ψ j appear.Further, χ ∈ Irr(Stab ψ) = Irr S n1 × • • • × Irr S nr so is an exterior tensor product χ = χ S λ1 ⊗ • • • ⊗ χ S λr with λ i ∈ P ni .Thus, from Ind ψ ⊗ χ, we may define a map Λ : Γ 1 → P by setting Λ(ψ j ) = λ i , where j is among the indices permuted by S ni and Λ(ϕ) 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 3.7.5.
Conversely, given Λ ∈ Q n , let ψ i ∈ Γ 1 , n i and λ i ∈ P ni be as in the paragraph preceding (3.7.7).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 with multiplicity n i , then r i=1 N i = T ⋊ Stab ψ.Now, ψ i defines a character of N i by and χ S λi defines an irreducible representation of S ni and hence of N i .Hence we get Taking their exterior tensor product and then inducing to N gives the irreducible representation It is in this way that we will realize the bijection Q n ∼ − → Irr N. It will be convenient to define for Λ ∈ Q n the function Λ ∈ Q n as Λ (ψ) := Λ (ψ) ′ for ψ odd, Λ (ψ) for ψ even (3.7.9) where ψ is said to be even if ψ(−1 Fq ) = 1 C and odd otherwise.

Parameter sets for Irr H n
Here, we take up again the notation introduced at the beginning of Section 3.2.Given a partition λ ∈ P n , we will be able to associate to it three different characters: the unipotent character χ G λ of G described in Remark 3.7.5, which we will see below is, in fact, an element of Irr(G : B); a character χ H λ ∈ Irr H (G, B); and the irreducible character χ S λ of the symmetric group S n .The discussions in Sections 3.2 and 3.6 suggest that there are relationships amongst these and the purpose of this section is indeed to clarify this.
To a partition ν ∈ P n , say ν = (ν 1 , . . ., ν ℓ ), one can associate a subgroup S ν := S ν1 × • • • × S ν ℓ ≤ S n and then consider the character τ S ν of the induced representation Ind Sn Sν ½ Sν .Then these characters are related to those of the irreducible representations χ S λ (see Section 3.7.1)by the Kostka numbers K λν [FH,Corollary 4.39], via the relationship (3.8.1) Also to ν ∈ P n one can associate a standard parabolic subgroup Pν ½ Pν of the parabolic induction of the trivial representation.2Then if λ ∈ P n and χ G λ denotes the corresponding unipotent characters (as described in Remark 3.7.5), the following remarkable parallel with representations of the symmetric group was already observed at [Ste,Corollary 1]: In particular, as K λλ = 1 for all λ ∈ P n , we see that Let us now consider the specializations (3.2.1) of H n corresponding to θ q , θ 1 : [CR,(68.12)Corollary], Tits's deformation theorem ( [CR,(68.20)Corollary], [GP,7.4.6 Theorem]) applies to give bijections where a character X : ] by [GP,Proposition 7.3.8]) is taken to its specialization X z : We can thus define the composition Now, we have bijections (using Proposition 3.6.3for the one on the left) x x q q q q q q q q q q Irr H (G, B) (3.8.5) and since | Irr S n | = |P n | all of the sets are of this size, so it follows that the inclusion in (3.8.3) is in fact an equality Furthermore, the following holds.
Proposition 3.8.6.[HR,Theorem 4.9(b)] For λ ∈ P n , one has Proof.[CR,(68.24)Theorem] states that the bijection for all ν ∈ P n .But the right hand side is, from (3.8.1),K λν , but then from (3.8.2), we must have This allows us to unambiguously define, for each λ ∈ P n , an irreducible representation χ H λ ∈ Irr H (G, B) by

Parameter sets for Irr Y d,n
In Section 3.7, we saw that the set Q n was used to parametrize both the irreducible representations of Irr N (Section 3.7.2) as well as a subset of those of G = GL n (F q ) (Remark 3.7.5).We will see (in Remark 3.9.20)that this latter subset is in fact Irr(G : U), which by Proposition 3.6.3yields the irreducible representations of H (G, U), and furthermore, that Tits's deformation theorem again applies to the generic Yokonuma-Hecke algebra, which gives a bijection of these with Irr N. The purpose of this section is to establish, as in Section 3.8, the precise correspondence between the relevant irreducible representations in terms of the elements of the parameter set Let us proceed with the argument involving Tits's deformation theorem.Recall from Theorem 3.4.3that we have isomorphisms by specialising Y q−1,n at u = q and u = 1, respectively.Thus, both H (G, U) ∼ = Y q−1,n (q) and C[N] ∼ = Y q−1,n (1) are split semisimple by [CPA,Proposition 9] and Y d,n (u) is also split by [CPA,5.2].3Thus, the deformation theorem ( [CR,(68.20)Corollary], [GP,7.4.6 Theorem]) again applies to yield bijections where we denote by θ q : C[u ±1 ] → C the C-algebra homomorphism sending u to q.Again, [GP,Proposition 7.3.8]applies to say that if and the bijections in (3.9.1) are in fact the specializations of X .We may put these together to obtain a bijection (3.9.2) Furthermore, [GP,Remark 7.4.4]tells us that d θq (X ) = θ q (X ) and d θ1 (X ) = θ 1 (X ). (3.9.3) In particular, the dimensions of the irreducible representations of H (G, U) and C[N] agree.Thus, we may conclude from Wedderburn's theorem that the Hecke algebra H (G, U) and the group algebra C[N] are isomorphic as abstract C-algebras.
Defining D U : Irr(G : U) → Irr H (G, U) as in Proposition 3.6.3,we get a diagram like that at (3.8.5): Irr N TU x x r r r r r r r r r r r Irr H (G, U).
(3.9.4)For every Λ ∈ Q n we will define a character X Y Λ ∈ Irr Y d,n (u) (see (3.9.24) in Section 3.9.4below) such that is the one described in Section 3.7.2for the modified Λ.This, together with the u = q specialization will satisfy As in Section 3.6, for a character χ G Λ ∈ Irr(G : U) one defines namely, the subspace of U-invariants, as in Proposition 3.6.3.
Our definition of X Y Λ will be in such a way that the χ G Λ from Section 3.7.1 is mapped by D U to the same χ H Λ , proving thus the main result of this section.Theorem 3.9.5.Let the characters χ G Λ and χ N Λ be those described in Sections 3.7.1 and 3.7.2,respectively.Then, the set Q n parametrizes the pairs of irreducible representations (V, V U ) from Proposition 3.6.3 in such a way that the characters Inspired by the construction of Irr N in Section 3.7.2,we establish a parallel with the technique of parabolic induction to build the character table of G using the unipotent characters as building blocks.
The rest of this section is devoted to studying more carefully the bijections D U , T U , D B and T B .In Section 3.9.1 we prove the compatibility between them.In Section 3.9.2we analyze their behaviour with a twist by a degree one character.In Section 3.9.3we check their interplay with parabolic induction and exterior tensor products.Finally, in Section 3.9.4we construct the character table of the generic Yokonuma-Hecke algebra, as a common lift of both Irr N and Irr(G : U).

From B to U
Let us start by checking the correspondence for the unipotent characters χ G λ in the H (G, U)-case agrees with the one in the H (G, B)-case (cf.Remark 3.7.5).Taking G = G, H = B, K = U and L = T in Proposition 3.1.12,noting that Lemma 3.1.11applies as the set N of double U-coset representatives normalizes T , we are provided with a surjective homomorphism H (G, U) ։ H (G, B), and hence we obtain an inflation map Infl : Rep H (G, B) → Rep H (G, U), taking Irr H (G, B) to Irr H (G, U).We can then make the following precise statement.Proposition 3.9.6.The following diagram is commutative where the top horizontal arrows are the bijections in (3.8.5), the bottom horizontal arrows those in (3.9.4), the leftmost vertical arrow is the natural inclusion and the other two are the inflation maps.
Proof.We know from Proposition 3.6.3that D B maps χ G λ to the character χ H λ ∈ Irr H (G, B) given by We want to replace Ind G B ½ B by Ind G U ½ U .Let us take a closer look at the latter.It is canonically where C[T] is the regular representation of T. Since C[T] is the direct sum of the degree one characters φ : T → C × , the group of which we denote by T, one gets (3.9.8) For any φ ∈ T one has which by the Mackey decomposition becomes where σ runs over a full set of B-double coset representatives.Since T ⊆ B ∩ σ −1 Bσ acts nontrivially on Res B B∩σ −1 Bσ (φ) for φ nontrivial, the only non-vanishing term in this last sum is the one Since χ G λ is a constituent of Ind G B ½ B , only one summand in the right hand side of (3.9.8) does not vanish and we end up with The irreducible H (G, U)-representation associated to χ G λ is with the H (G, U)-module structure induced by the surjection H (G, U) → H (G, B) coming from Proposition 3.1.12.This proves the commutativity of the left square in (3.9.7).
The right one is also commutative since all the non-horizontal arrows in arise from taking tensor products.

Twisting by characters of F × q
In order to deal with character twists, we extend the Iwahori-Hecke algebra as follows.For d, n ≥ 1 we introduce the C[u ±1 ]-algebra H d,n defined by where C d is the cyclic group of order d and h ∈ C d a generator.As usual, denote by is a tensor product of semisimple algebras it is also semisimple and its set of characters is the cartesian product of those of its factors.Concretely, for λ ∈ P n and ψ λ,ψ to be the exterior tensor product: (3.9.9) These are all the irreducible representations of H d,n (u), and they all come from localizing certain finitely generated representations of H d,n that we also denote by X (3.9.10) and the u = q specialization gives which in the d = q − 1 case gives where B 1 = B ∩ SL n (F q ).Here one has and the u = q specialization of the natural map (3.9.10) becomes where, for σ ∈ S n and t ∈ T, the corresponding basis elements are mapped as follows: Remark 3.9.12.The T i from Y q−1,n corresponds to ω i ∈ T ⋊ S n , whereas the T i from H q−1,n corresponds to σ i ∈ S n ⊆ F × q × S n .Therefore, the u = 1 specialization of (3.9.10) gives the surjection (3.9.13)where sgn σ = (−1) ℓ(σ) ∈ F × q .For this reason we define as composition with the map (3.9.13).Thus, for χ (3.9.14) Remark 3.9.15.In general, we reserve the notation Infl for inflation by the natural quotient.In this case it is . Therefore, by (3.9.14) Since ψ • sgn ∈ Irr S n is the sign representation when ψ is odd (i.e., a non-square) character, and is trivial when ψ is even (i.e., the square of a character), and tensoring with the sign representation amounts to taking the transpose partition λ ′ we see that where the superscripts F × q S were omitted.Remark 3.9.16.In any case we have Infl χ for t ∈ T ⊆ N, and where ω i , s i ∈ N as in Theorem 3.4.3.Thus, by definition of the characters χ N Λ ∈ Irr Z ≥0 for Λ ∈ Q n in Section 3.7.2 and (3.7.9) Thus, taking tensor products gives a map Irr(G : B) × F × q → Irr(G : U).
Proposition 3.9.18.Let ψ ∈ Γ 1 be as above.Then the following diagram is commutative Proof.The commutativity of the squares on the left largely comes from the formula (3.6.1) for the action of the Hecke algebra, as well as noting that the H (G, U)-action factors through the surjection onto H (G, B)[F × q ] = H (G, B 1 ), which comes from either (3.9.11) or Proposition 3.1.12.The squares on the right are also commutative since all the non-horizontal arrows in arise from tensor products.

Parabolic induction
We now check that the parabolic induction of a product of irreducible representations is compatible with the operations D U (coming from Section 3.6) and T U (defined at (3.9.2)).
Suppose n 1 , n 2 are such that n = n 1 +n 2 .For i = 1, 2, let G i := GL ni (F q ), and let B i , U i , and N i be the respective subgroups of G i as described at the beginning of Section 3.3.Let L = G 1 × G 2 and we will view it as the subgroup of G = GL n (F q ) of block diagonal matrices; let U 12 ≤ G be the subgroup of upper block unipotent matrices and P := LU 12 the corresponding parabolic subgroup.If B L := L ∩ B, then its unipotent radical is U L = L ∩ U = U 1 × U 2 .One may also identify N 1 × N 2 as a (proper) subgroup of N.
Remark 3.9.19.Observe that U = U L ⋉ U 12 and that L normalizes U 12 , so that we, with G = G, H = U, L = L, K = U L and U = U 12 , we are in the situation of Section 3.1.2,so that Proposition 3.1.14gives an inclusion H (L, U L ) ֒→ H (G, U).Furthermore, the functor R G L : Rep L → Rep G defined at (3.6.6) is the usual parabolic induction functor.
Remark 3.9.20.For i = 1, 2, let ψ 1 , ψ 2 ∈ F × q be distinct and let λ i ∈ P ni .Then by the discussion preceding Proposition 3.9.18,χ G λi ⊗ ψ Gi i ∈ Irr(G i : U i ).Thus, Remark 3.9.19makes the discussion preceding Proposition 3.6.7 relevant: it says that χ However, this •-product is irreducible, as mentioned in Remark 3.7.5, so in fact it lies in Irr(G : U).By a straightforward inductive argument, we see that for every Λ ∈ Q n , one has χ G Λ ∈ Irr(G : U) (as defined in Remark 3.7.5).We repeat the argument of Section 3.8 to show that Irr(B : G) consists of precisely the unipotent representations: one has an inclusion Remark 3.9.21.Since as modules over the algebra one has (3.9.22) Proposition 3.9.23.The following diagram is commutative: Proof.Commutativity of the square on the left comes from Proposition 3.6.7 which applies by Remark 3.9.19 and observing that ℓ∈L ℓU L ℓ −1 is trivial.
Let us take representations X Y 1 , X Y 2 of Y q−1,n1 (u) and Y q−1,n2 (u) and denote their corresponding u = 1 specializations by χ N1 1 and χ N2 2 , and their u = q specializations by χ H1 1 and χ H2 2 , respectively, so that For i = 1, 2 we write χ Gi i for the corresponding G i representation, in such a way that χ Hi i agrees with . By (3.9.22), the commutativity of the rightmost square follows since the vertical and diagonal arrows in come from taking tensor products.Namely happens to be the u = q specialization of Taking the u = 1 specialization gives 3.9.4Proof of Proposition 3.9.5 We define a family of representations of Y q−1,n parametrized by Q n whose u = 1 specialization matches that of Section 3.7.2,and whose u = q specialization gives the D U image of the characters described in Remark 3.7.5.Given a partition λ ⊢ n and ψ : of Y q−1,n by inflating the H q−1,n -representation X Hn λ ⊗ ψ from (3.9.9) via the natural quotient (3.9.10).Now, for Λ ∈ Q n , let ψ i , λ i , n i be as in the paragraph preceding (3.7.7) and consider the which may be embedded in Y q−1,n by a choice of ordering of the indices i.This has the representation , where the tensor is over C[u ±1 ], and we may define X Y Λ as the induced character . (3.9.24) We are now in position to prove Proposition 3.9.5.
Proof of Proposition 3.9.5.Take a Λ ∈ Q n and the corresponding pairs (λ i , ψ i ) with Λ(ψ i ) = λ i , and define (F q ), L = i G i viewed as the subgroup of block diagonal matrices of G, and U L the product of the upper triangular unipotent subgroups.
Consider the Y q−1,n -representation X Y Λ defined in (3.9.24).Its u = 1 and u = q specializations give, by the commutativity of the squares on the right in Proposition 3.9.18, (3.9.25) The character χ G Λ ∈ Irr(G : U) was defined at (3.7.7).Applying D U , one gets, by Proposition 3.9.23 and an inductive argument, where we have used Proposition 3.9.18 and Proposition 3.9.6 for the following two equalities.But this is exactly χ H Λ .So we may conclude by taking a second look at (3.9.25).
Remark 3.9.26.One can use this construction to give another proof of the splitting of Y d,n (u), together with a list of their irreducible representations parametrized by the set where C d is the cyclic group with d elements.
Given a function Λ ∈ Q d,n we consider its set of pairs , where the X λ,ψ stands for the inflation of the X H d,n λ,ψ from (3.9.9).
The induced characters are defined over C(u) since the X Hn λ are (cf.[BC,Theorem 2.9]).We extend scalars to some finite Galois extension K/C(u) so that Y d,n becomes split.One can also extend the u = 1 specialization as in [GP,8.1.6]

Counting on quasi-Hamiltonian fusion products
Here we describe the technique we use to count the points on the wild character varieties, which was already implicitly used in [HV,HLV1].The idea is to use the construction of the wild character variety as a quotient of a fusion product and reduce the point-counting problem to one on each of the factors.Then the counting function on the entire variety will be the convolution product of those on each of the factors.This can be handled by a type of Fourier transform as in the references above.

Arithmetic harmonic analysis
In carrying out our computations, we will employ the technique of "arithmetic harmonic analysis," which is an of analogue of the Fourier transform for non-abelian finite groups such as GL n (F q ).This is described in [HLV1,§3], a part of which we reproduce here for the convenience of the reader.
Let G be a finite group, Irr G the set of irreducible character of G, C(G • ) the vector space of class functions (i.e., functions which are constant on conjugacy classes) on G and C(G • ) the space of functions on the set Irr G.We define isomorphisms Note that these are not quite mutually inverse, but will be up to a scalar; precisely, ; it is not difficult to verify that it is in fact a subalgebra for the convolution product * G .We can define a product on C(G • ) simply by pointwise multiplication: Then F • and F • have the important properties that

Set-theoretic fusion
Let G be a finite group, M a (left) G-set and µ : M → G an equivariant map of sets (where G acts on itself by conjugation).We may define a function N : G → Z ≥0 by The equivariance condition implies that m → a • m gives a bijection µ −1 (x) ↔ µ −1 (axa −1 ) for a, x ∈ G, and hence it is easy to see that N ∈ C(G • ).
Suppose M 1 and M 2 are two G-sets and µ 1 : M 1 → G, µ 2 : M 2 → G equivariant maps, and let M := M 1 × M 2 and define µ : a straightforward computation gives

Counting via Hecke algebras
Recall the notation of Section 3.1.Let V ⊆ G be a set of double H-coset representatives as in For h ∈ H we set Often we will abbreviate N (x, v) := N 1 (x, v).We are interested in the function Proof.For h ∈ H, we have bijections From this, we get On the other hand, one sees that Therefore, if we set then Lemma 3.6.5 applied to (4.2.3) gives us Proof.One finds in [I, §1] a bijection (H ∩ v −1 Hv)\H↔H\HvH given by This quickly yields (a).Now we let x ∈ G and evaluate If x ∈ HvH, then h −1 x ∈ vH for any h ∈ H, and so the above is zero.On the other hand, if x = h 0 vh for some h 0 ∈ H, h ∈ H, then the above evaluates to using (a) for the last step.The second equation in (b) is proved similarly.For (c), we have Finally, for (d) we compute We can conclude the proof of Proposition 4.2.2 by noting that This and (4.2.4) imply Proposition 4.2.2.
Remark 4.2.7.When k = 1 and v 1 = 1, Proposition 4.2.2 gives a character formula for the cardinality of the intersection of conjugacy classes and Bruhat strata and appears at [L, 1.3.(a)].In fact, the computation there is what led us to Proposition 4.2.2.

Character values at the longest element
From now on we will let G, T, B, U and N be as in Section 3.3.We need to compute certain values of the characters of H (G, U).The element e i is the idempotent projector to the subspace V i of V where h i h −1 i+1 acts trivially and there is a direct sum decomposition V = V i W i , where W i := ker e i .Lemma 3.4.7 shows that this decomposition is preserved by T i .Over V i , the endomorphism T i satisfies the following quadratic relation Remark 4.3.11.For X = X Λ , Λ ∈ Q n , we have by Remark 3.9.16 and Theorem 3.9.5 with the notation of (3.7.4) from Section 3.7.1.
Proof.If λ ∈ P n and χ S λ ∈ Irr S n is the corresponding irreducible character of S n and s ∈ S n is a simple transposition then by [FH,Exercise 4.17 (c)] or [F, §7 (16.) where the a i and b i are the number of boxes below and to the right of the ith box of the diagonal in the Young diagram of λ.By writing j − i in the box (i, j) and computing the sum in two ways we see at once that From this and (4.3.16),we get that It remains to prove the analogous formula for χ N Λ ∈ Irr N with Λ ∈ Q n .Since we are working in N, we will omit the subscript and simply write χ Λ .The description of the character χ Λ was given in Section 3.7.2 and in particular by the induction formula (3.7.8).We will use the notation established there.We will make the further abbreviations N(ψ) :  Proof.Recall that the multiplication map T × U − × U + → G is an open immersion and that U − = ω 0 U + ω 0 , so that every element of ω 0 Uω 0 U has a unique factorization in to a pair from U − × U + and similarly for vω 0 Uω 0 U and vU − × U + .Thus, it follows that where v = (v 1 , w 1 , . . ., v r , w r ) with v 1 = vω 0 , v i = ω 0 for i > 1 and w i = ω 0 for all i.So by Proposition 4.2.2 we have Theorem 4.3.4states that T 2 ω0 acts by the scalar q fΛ in the irreducible representation corresponding to Λ, and the result follows.

Values at generic regular semisimple F q -rational elements
Here we compute our count function in the case when v ∈ T reg , i.e., when v has distinct eigenvalues.
where the first sum is over the characters χ G Λ ∈ Irr G defined in 3.7.1 for functions Λ ∈ Q n , while the second sum is over all irreducible characters parametrised by Λ : Γ → P of size n.Let I n be the n × n identity matrix, let ξ i be the diagonal matrix with diagonal elements a i+k 1 , . . ., a i+k n for i = 1, . . ., m.Finally, for elements A, B of a group, put (A, B) := ABA −1 B −1 .Define I 0 ⊆ A 0 to be the radical of the ideal generated by the entries of and the coefficients of the polynomial det(tI n − X i ) − ri j=1 (t − a i j ) µ i j in an auxiliary variable t.Finally, let A := A 0 /I 0 and U µ,r := Spec(A) an affine R-scheme.
Let φ : R → K be a map to a field K and let U φ µ,r be the corresponding base change of U µ,r to K. A K-point of U φ µ,r is a solution in GL n (K) to (5.1.1)where X i ∈ C φ i and C φ i is the semisimple conjugacy class in GL n (K) with eigenvalues φ(a i 1 ), . . ., φ(a i ri ) of multiplicities µ i 1 , . . ., µ i ri and ξ φ i ∈ T reg (K) is a diagonal matrix with diagonal entries φ(a k+i 1 ), . . ., φ(a k+i n ).
By construction (C φ 1 , . . ., C φ k , ξ φ 1 , . . ., ξ φ m ) is generic.Finally G = GL n × T k acts on U µ,r via the formulae (2.2.2).We take M µ,r = Spec(A G(R) ) the affine quotient of U µ,r by G(R).Then for φ : R → C the complex variety M φ µ,r agrees with our M µ,r B thus M µ,r is its spreading out.We need the following Proposition 5.1.2.Let φ : R → K a homomorphism to a field K. Then if A i , B i , X j , C α ∈ GL n (K), a solution to (5.1.1)representing a K-point in U φ µ,r is stabilized by is a scalar matrix.Equivalently, if D = {λI n , . . ., λI n } ≤ G φ is the corresponding subgroup then G φ := G φ /D acts set-theoretically freely on U φ µ,r .
Proof.By assumption x α C α y −1 = C α (5.1.3)thus the matrices y, x 1 , . . ., x m are all conjugate and split semisimple.Let λ ∈ K be one of their eigenvalues and V λ < K n be the λ-eigenspace of y then by (5.1.3)C α (V α ) ⊆ K n is the λ-eigenspace of x α .As y commutes with all of A i , B i , X j we see that they leave V λ invariant.While x α commutes with S α i and ξ α thus they leave C α (V λ ) invariant or equivalently C −1 α S α i C α and C −1 α ξ α C α leave V λ invariant.As S α i is unipotent and the determinant of the equation (5.1.1)restricted to V α gives By assumption (C φ 1 , . . ., C φ k , ξ φ 1 , . . ., ξ φ m ) is generic.Thus, we get from (2.2.10) that V λ = K n .
Let now K = F q a finite field and assume we have φ : R → F q .Because G is connected and G(K) acts freely on U φ µ,r we have by similar arguments as in [HLV1,Theorem 2.1.5],[HV,Corollaries 2.2.7,2.2.8] and by Theorem 4.6.4 that #G(F q ) = q d μ,r H μ,r (q −1/2 , q 1/2 ).
Proof.This is a consequence of Theorem 1.2.1 and the combinatorial Lemma 5.2.4 proved below.

Mixed Hodge polynomial of wild character varieties
In this section we discuss Conjecture 1.2.2.First we recall the the combinatorics of various symmetric functions from [HLV1,§2.3].Let Λ(x) := Λ(x 1 , . . ., x k ) be the ring of functions separately symmetric in each of the set of variables x i = (x i,1 , x i,2 , . . .).
For a partition, let λ ∈ P n s λ (x i ), m λ (x i ), h λ (x i ) ∈ Λ(x i ) be the Schur, monomial and complete symmetric functions, respectively.By declaring {s λ (x i )} λ∈P to be an orthonormal basis, we get the Hall pairing , , with respect to which {m λ (x i )} λ∈P and {h λ (x i )} λ∈P are dual bases.We also have the Macdonald polynomials of [GH] Hλ (q, t) = µ∈Pn Kλµ s µ (x) ∈ Λ(x) ⊗ Z Q(q, t).
And finally we have the plethystic operators Log and Exp (see for example [HLV1,§2.3.3.]).
and every irreducible representation of H (G, H) arises in this way, that is, D H restricts to a bijection D H : Irr(G : H) ∼ − → Irr H (G, H).Since C[G] and H (G, H) are semisimple, we can apply Theorem 3.5.1 to W = Ind G H ½ H .If we denote the set of irreducible representations of G by Irr G, we find that .6.6) which gives a map Rep L → Rep G. Let Rep(G : H) denote the set of isomorphism classes of (finite-dimensional) representations of G at least one of whose irreducible components lies in Irr(G : H).Then we claim that if V ∈ Irr(L : K), then R G L V ∈ Rep(G : H) and hence R G L yields a map R G L : Irr(L : K) → Rep(G : H).
but since | Irr(G : U)| = | Irr H (G, U)| = | Irr N| = |Q n | via the bijections D U and T U , we must have equality.
and KY d,n becomes isomorphic to K[C n d ⋊ S n ], being a deformation of C[C n d ⋊ S n ].The specializations χ C n d ⋊Sn Λ := d θ1 (X Y Λ )are all the irreducible characters of C n d ⋊ S n as in Section 3.7.2(invoking again [Se, § 8.2, Proposition 25]).Therefore, Y d,n (u) is split semisimple and the X Y Λ is the full list of irreducibles.