We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus g to GL n (C) with fixed generic semi-simple conjugacy classes at k punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of GL n (F q ). We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincaré polynomial of certain associated quiver varieties which we obtain using the character table of gl n (F q ). In the last main result we prove that the Poincaré polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of GL n (F q ). As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, typically wild, quivers and the representation theory of GL n (F q ).

We give a cohomological interpretation of both the Kac polynomial and the refined Donaldson-Thomasinvariants of quivers. This interpretation yields a proof of a conjecture of Kac from 1982 and gives a new perspective on recent work of Kontsevich-Soibelman. This is achieved by computing, via an arithmetic Fourier transform, the dimensions of the isoytpical components of the cohomology of associated Nakajima quiver varieties under the action of a Weyl group. The generating function of the corresponding Poincaré polynomials is an extension of Hua's formula for Kac polynomials of quivers involving Hall-Littlewood symmetric functions. The resulting formulae contain a wide range of information on the geometry of the quiver varieties. The main resultsLet Γ = (I, Ω) be a quiver: that is, an oriented graph on a finite set I = {1, . . . , r} with Ω a finite multiset of oriented edges. In his study of the representation theory of quivers, Kac [17] introduced A v (q), the number of isomorphism classes of absolutely indecomposable representations of Γ over the finite field F q of dimension v = (v 1 , . . . , v r ) and showed they are polynomials in q. We call A v (q) the Kac polynomial for Γ and v. Following ideas of Kac [17], Hua [16] proved the following generating function identity:where P denotes the set of partitions of all positive integers, Log is the plethystic logarithm (see [14, §2.3.3]), , is the pairing on partitions defined bywith m j (λ) the multiplicity of the part j in the partition λ, which implies positivity for certain refined DT-invariants of symmetric quivers with no potential.The goal of Kontsevich-Soibelman's theory is to attach refined (or motivic, or quantum) DonaldsonThomas invariants (or DT-invariants for short) to Calabi-Yau 3-folds X. The invariants should only depend on the derived category of coherent sheaves on X and some extra data; this raises the possibility of defining DTinvariants for certain Calabi-Yau 3-categories which share the formal properties of the geometric situation, but are algebraically easier to study. The simplest of such examples are the Calabi-Yau 3-categories attached to quivers (symmetric or not) with no potential (c.f. [12]).Denote by Γ = (I, Ω) the double quiver, that is Ω = Ω Ω opp , where Ω opp is obtained by reversing all edges in Ω. The refined DT-invariants of Γ (a slight renormalization of those introduced by Kontsevich and Soibelman [19]) are defined by the following combinatorial construction.In fact, as a consequence of Efimov's proof [10] of [20, Conjecture 1], DT v (q) actually has non-negative coefficients. We will give an alternative proof of this in (1.10) by interpreting its coefficients as dimensions of cohomology groups of an associated quiver variety. Remark 1.2. We should stress that we have restricted to double quivers for the benefit of exposition; our results extend easily to any symmetric quiver. We outline how to treat the general case in §3.2. The technical starting point in this paper is a common generalization of (1.3) and Hua'...

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on C × × C × , modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.Relation between Hilbert schemes and modular forms was first investigated by Göttsche [8].