Hodge-Deligne polynomials of character varieties of abelian groups

Let F be a finite group and X be a complex quasi-projective F-variety. For r in N, we consider the mixed Hodge-Deligne polynomials of quotients X^r/F, where F acts diagonally, and compute them for certain classes of varieties X with simple mixed Hodge structures. A particularly interesting case is when X is the maximal torus of an affine reductive group G, and F is its Weyl group. As an application, we obtain explicit formulae for the Hodge-Deligne and E-polynomials of (the distinguished component of) G-character varieties of free abelian groups. In the cases G=GL(n,C) and SL(n,C) we get even more concrete expressions for these polynomials, using the combinatorics of partitions.


Introduction
The study of the geometry, topology and arithmetic of character varieties is an important topic of contemporary research. Given a reductive complex algebraic group G, and a finitely presented group Γ, the G-character variety of Γ is the (affine) geometric invariant theory (GIT) quotient When the group Γ is the fundamental group of a Riemann surface (or more generally, a Kähler group), these spaces are homeomorphic to moduli spaces of G-Higgs bundles via the non-abelian Hodge correspondence (see, e.g. [1,2]) and have found interesting connections to important problems in Mathematical Physics in the context of mirror symmetry and the geometric Langlands correspondence.
Recently, some interesting formulas were obtained by Hausel, Letellier and Rodriguez-Villegas for the so-called E-polynomial of smooth GL n, ( )-character varieties of surface groups, by applying arithmetic harmonic analysis to their -models and proving these are polynomial count [3,4]. By computing indecomposable bundles on algebraic curves over finite fields, Schiffmann determined the Poincaré polynomial of the moduli spaces of stable Higgs bundles, hence of the corresponding GL n, ( )-character varieties of surface groups [5]. Other methods based on point counting were employed by Mereb [6] (the SL n, ( ) case) and  (the singular, small n case).
Moreover, geometric tools were developed by Lawton, Logares, Muñoz and Newstead to calculate the E-polynomials using stratifications of character varieties (over ) of surface groups, exploring directly the additivity of these polynomials [8,9]. This led to the development of a Topological Quantum Field Theory for character varieties by González-Prieto et al. [10,11].
In the present article, we deal instead with G-character varieties of free abelian groups, and with the determination of their mixed Hodge structures (MHSs) for a general complex reductive G. In particular, we explicitly compute the mixed Hodge polynomials of these varieties. The mixed Hodge polynomial μ X is a three variable polynomial μ μ t u v , , X X = ( ) defined for any (complex) quasi-projective variety X and encodes all numerical information about the MHS on the cohomology of X, generalizing both the Poincaré and the E-polynomials.
To present our main results, denote the G-character variety of the free abelian group Γ r ≅ , r ∈ , by: where // stands for the (affine) GIT quotient (see, e.g., [12,13]) for the natural G-action, by conjugation, on the space of representations Hom G , ). This later space consists of pairwise commuting r-tuples of elements of G and is of relevance in Mathematical Physics, namely, in the context of supersymmetric Yang-Mills theory [14]. When r is even, r is also a Kähler group (the fundamental group of a Kähler manifold) and the smooth locus of G m 2 ( ) is diffeomorphic to a certain moduli space of G-Higgs bundles over a m-dimensional abelian variety (see, for instance [15]).
The topology and geometry of character varieties of free abelian groups have been studied by Florentino-Lawton, Sikora, Ramras-Stafa, among others (see, e.g., [16][17][18][19]). It is known that the affine algebraic variety G r is not in general irreducible, but the irreducible component of the trivial r -representation, denoted G r 0 , has a normalization G r ⋆ isomorphic to T W r / ([17, Theorem 2.1]), where T G ⊂ is a maximal torus and W is the Weyl group, acting diagonally on T r (hence also on its cohomology). Thus, the varieties G r ⋆ are singular orbifolds of dimension r T dim with a special kind of MHSs, called balanced or of Hodge-Tate type and they satisfy the analogue of Poincaré duality for MHS. When r 2 = , Thaddeus proved that G 2 ⋆ are of crucial importance in mirror symmetry and Langlands duality and computed their orbifold E-polynomials [20]. Here, we obtain the following explicit formula for mixed Hodge polynomials of G r ⋆ .
where A g is the automorphism induced on H T, 1 ( ) by g W ∈ , and I is the identity automorphism.
One consequence of this result is a formula for the (compactly supported) E-polynomial of the irreducible component G G r r 0 ⊂ , for every such G (Theorem 5.4). Our approach to Theorem 1.1 is based on working with equivariant MHSs and their corresponding equivariant polynomials, defined for varieties with an action of a finite group, and focusing on certain classes of balanced varieties. In particular, we generalize to the context of the equivariant E-polynomial, some of the techniques introduced in [21] for dealing with equivariant weight polynomials.
For the groups G GL n, = ( ) and SL n, ( ), we have that G r is an irreducible normal variety, and the formula in Theorem 1.1 can be made even more concrete, in terms of partitions of n, and allows explicit computations of the Hodge-Deligne, Eand Poincaré polynomials of the corresponding character varieties G G r r = ⋆ . We state the main results below in the compactly supported version, the one which is relevant in arithmetic geometry (see [3], Appendix). Let n denote the set of partitions of n ∈ . By n n 1 2 a a a n n where n n 1 2 a a a n n 1 2 Theorem 1.2 generalizes, to every r n , 1 ≥ , some formulas recently obtained in [7,9] (the cases n 2 = and n 3 = ) by different methods, which are only tractable for low values of n: the approach in [8,9] uses stratifications and fibrations to compute E-polynomials of character varieties of free groups respectively, surface groups; the computations in [7] apply representation theory of finite groups and point counting of varieties over finite fields.
By substituting x 1 = in E G c r , we obtain the Euler characteristics of these moduli spaces. Moreover, by showing that G r have very special MHSs (that we call round, see Definition 3.7), Theorems 1.1 and 1.2 immediately provide explicit formulas for their mixed and Poincaré polynomials (Theorem 5.13).
The GL n, ( ) case is particularly symmetric, as the generating function of mixed Hodge polynomials gives precisely the formula of J. Cheah [22] for the mixed Hodge numbers of symmetric products. On the other hand, by examining the action of W on the cohomology of a maximal torus, our methods allow for the computation of μ G r for all the classical complex semisimple groups G. These will be addressed in upcoming work. We now outline the contents of the article. In Section 2, we review necessary background on MHS, quasi-projective varieties, etc., and define the relevant polynomials, providing examples and focusing on balanced varieties. In Section 3, we study properties of special MHS, related to notions defined in [21], and pay special attention to round varieties, for which the knowledge of either the Poincaré polynomial or the E-polynomial allows the determination of μ. Section 4 is devoted to equivariant MHS, character formulas and the cohomology of finite quotients. Finally, in Section 5 we prove our main theorem and provide explicit calculations of Hodge-Deligne and E-polynomials (and Euler characteristics) of character varieties of r , in particular for GL n, ( ) and SL n, ( ); in the GL n, ( ) case, the computations are related to MHS on symmetric products, thereby obtaining a curious combinatorial identity. In the Appendix, we present a proof, based on [21], of the equivariant version of a theorem in [8,9] on the multiplicative property of the E-polynomial for fibrations.
A preliminary version of the main results has been announced in [23].

Preliminaries on character varieties and on MHSs
We start by recalling the relevant definitions and properties of character varieties and of mixed Hodge structures (MHSs) on quasi-projective varieties, which serves to fix terminology and notation.

Character varieties
Given a finitely generated group Γ and a complex affine reductive group G, the G-character variety of Γ is defined to be the (affine) GIT quotient (see [12,13]; [24] for topological aspects): Note that Hom G Γ, ( ), the space of homomorphisms ρ G : Γ → , is an affine variety, as Γ is defined by algebraic relations, and it is also a G-variety when considering the action of G by conjugation on Hom G Γ, ( ). The aforementioned GIT quotient is the maximal spectrum of the ring Hom G Γ, Hom The GIT quotient does not parametrize all orbits, since some of them may not be distinguishable by invariant functions. In fact, it can be shown (see, e.g., [13]) that the conjugation orbits of two representations ρ ρ G , : Γ ′ → define the same point in Hom G G Γ, ( )/ / if and only if their closures intersect: G ρ ⋅ ∩ G ρ ⋅ ′ ≠ ∅ (in either the Zariski or the complex topology coming from an embedding Hom G Γ, N ( )↪ ). For detailed definitions and properties of general character varieties, we refer to [16,25].
In this article, we will be mostly concerned with the case when Γ is a finitely generated free abelian group, Γ r = for some natural number r, the rank of Γ. The corresponding G-character varieties: ) is of central importance in determining the so-called moduli space of vacua of supersymmetric gauge theories on a r-dimensional torus, as studied in [14,26] and others.

MHSs
On a compact Kähler manifold X the complex cohomology satisfies the Hodge decomposition H X, ) for all p q k 1 + = + . This notion can be generalized to quasi-projective algebraic varieties X over , possibly non-smooth and/or non-compact. Namely, the complex cohomology of any such variety is also endowed with a natural filtration, the Hodge filtration F, and moreover, there is a special second increasing filtration on the rational cohomology: (1) The specialization of μ X for u v 1 = = gives the Poincaré polynomial of X: ) being the Betti numbers of X. Note that the coefficients of μ X and of P X are non-negative integers, whereas E X lives in the ring u v , [ ].
(2) As mentioned earlier, there is an entirely parallel theory for the compactly supported cohomology.
Here, the associated Hodge numbers are denoted by h H X dim c k p q c k p q , , , , ≔ ( ). If X stands for one of the polynomials in the aforementioned definition, we will distinguish its compactly supported version by writing X c . (3) Comment on terminology: there are inconsistencies in the literature on the terminology used for these polynomials. Since h X k p q , , ( ) are generally called Hodge-Deligne (or mixed Hodge) numbers, we refer to μ X as Hodge-Deligne or mixed Hodge polynomial. To emphasize the distinction, the compactly supported E-polynomial E X c will also be called the Serre polynomial of X, since its crucial behavior, as a generalized Euler characteristic, was first used by Serre in connection with the Weil conjectures (see [30]). (4) Many specializations of the E-polynomial have been studied in the literature. There is, for example, the weight polynomial W y w X y 1 [21]). This is a specialization of the E-polynomial since W y E y y , X X ( ) = ( ). Also, Hirzebruch's χ y -genus and the signature σ of a complex manifold X are given, in terms of E u v , X ( ), as: respectively (see Hirzebruch [31]).
We now collect some well-known important properties of these polynomials, for later use.
Proposition 2.4. For a quasi-projective variety X, we have: (1) The polynomials μ X and E X are symmetric in the variables u and v; in particular, if h X 0 c is additive for stratifications of X by locally closed subsets, and its degree is equal to X 2 dim . (5) All polynomials μ X , P X and E X are multiplicative under Cartesian products.

1(4). □
A common feature of the varieties in this paper is that their MHS is "diagonal:" for each k, the only nonzero mixed Hodge numbers are h k p q , , with p q = .
Definition 2.5. A quasi-projective variety X is said to be balanced or of Hodge-Tate type if for every nonnegative integer k 0 ∈ , and all p q Example 2.6.
(1) If X is connected, H X, 0 ( ) ≅ has always a pure Hodge structure, with trivial decomposition H X, 0 ( ) = H X 0,0,0 ( ). Dually, when X is also smooth, the compactly supported cohomology is also a trivial decom- (4) Consider the total space X of the trivial line bundle over an elliptic curve X Λ ≅ ( / ) × , where Λ is a rank two lattice in , ( +). It is easy to see that X is real analytically isomorphic to 2 ( ) * (but not complex analytically or algebraically isomorphic). From the Künneth isomorphism and considerations analogous to Example 2.6, we get: (1) The last example is a very special case (the genus 1, rank 1 case) of the non-abelian Hodge correspondence mentioned in Section 1, which produces diffeomorphisms between (Zariski open subsets of) moduli spaces of flat connections and certain moduli spaces of Higgs bundles over a given Riemann surface. The fact that one diffeomorphism type is balanced (the flat connection side of the correspondence) and the other is pure is a general feature (see [3,4]). (2) If X is balanced, its E-polynomial depends only on the product uv, so it is common to adopt the change of variables x uv ≡ . When written in this variable,

Separably pure, elementary and round varieties
In this section, we collect many properties of MHS that are necessary later on. We also describe the types of Hodge structures that allow the recovery of the mixed Hodge polynomial given the Eor the Poincaré polynomial (Theorem 3.6), and concentrate on the case of round varieties, which are the Hodge types of our character varieties. We tried to be self-contained for the benefit of researchers in the field of character varieties or Higgs bundles that may not be familiar with MHS.

Elementary and separably pure varieties
The MHSs on the cohomology of a given quasi-projective variety X may be trivial, i.e., the decomposition of every H X, k ( ) is the trivial one, and many such examples are considered here. When this happens, the only non-zero h X k p q , , ( ) satisfy q p = (by Proposition 2.4(1)) and much of what can be said about the cohomology can be transported to MHSs. Adapting some notions from [21] (who worked with the weight polynomial), we introduce the following terminology.
Definition 3.1. Let X be a quasi-projective variety. X (or its cohomology) is called elementary if its MHSs are trivial decompositions of the cohomology, so that for every k ∈ there is only one p ∈ such that h X 0 . X is said to be separably pure if the MHS on each H X, k ( ) is in fact pure of total weight w k , and such that w w j k ≠ for every j k ≠ .
In this case, A general weight function is not enough to recover μ X from the weight or the E-polynomials (different degrees of cohomology may have equal total weights). However, this can be done (see Theorem 3.6) if the weight function k p k ↦ is injective, in which case the equality (2) takes the stronger form: (2) In a pure Hodge structure of total weight k on H X, k ( ) the only non-zero weight summand is Gr H X, ). So, a pure total cohomology is separably pure, but not conversely, as the case * shows (Example 2.6).
(3) When X is separably pure, instead of the weight function, one can define a degree function p q , Noting that, in fact, the degree k only depends on the total weight p q + (being separably pure) we can write this as p q k , In this article, most varieties are both separably pure and balanced, and an alternative characterization follows. Lemma 3.3. A quasi-projective variety X is separably pure and balanced if and only if it is elementary and its weight function k p k ↦ is injective.
Proof. If X is separably pure, the total weight in each H X, k ( ) has to be constant. But if X is also balanced, given k, all h X k p q , , ( ) vanish except for a unique pair p q p p , , k k ( ) = ( ), so we have an assignment k p k ↦ proving that X is elementary. Moreover, since the total weights are different for distinct k, the weight function is injective. The converse statement is easy since an elementary variety is Hodge-Tate and an injective weight function implies injectivity for total weights. □   (writing x uv = , see Remark 2.7(2)). So, GL 3, ( ) is elementary (hence balanced) but not separably pure: both degrees 4 and 5 have associated total weight 6 (the terms with x 3 ), so GL n, ( ) is not separably pure, for n 3 ≥ . Moreover, the same argument readily shows that GL n, ( ) is not elementary for n 5 ≥ .
The aforementioned examples show that this "yoga of weights," as alluded by Grothendieck, is very useful in understanding general properties of certain classes of varieties. When we know that a particular variety X has a degree or a weight function as above, we can determine the full collection of triples k p q , , In Figure 1, the shaded area illustrates Lemma 3.3; for the definition of round, see Section 3.2. The next result shows that elementary and separably pure are indeed the correct notions to be able to determine the mixed Hodge polynomial from the Poincaré or the E-polynomial, respectively. Theorem 3.6. Let X be a quasi-projective variety of dimension n. Then: (1) If X is elementary, with known weight function, its Poincaré polynomial determines its Hodge-Deligne polynomial.
(2) If X is separably pure, with known degree function, its E-polynomial determines its Hodge-Deligne polynomial.

Proof.
(1) Suppose the Poincaré polynomial of X is P t b t , and the degree function as p q k , p q ( ) ↦ + , since the total weights are in one-to-one correspondence with the degrees of cohomology, we obtain μ t u v , ,

Round varieties
From Theorem 3.6, if a variety X is both balanced and separably pure, then μ X can be recovered from either E X or P X , knowing their degree/weight functions. A specially interesting case is the following.
Definition 3.7. Let X be a quasi-projective variety. If the only non-zero Hodge numbers are of type h X k k k , , ( ), k X 0, , 2 dim ∈ { … }, we say that X is round.
In other words, a round variety is both elementary and separably pure and its only k-weights have the form k k , ( ). Round varieties are referred to as "minimally pure" balanced varieties in Dimca-Lehrer (see [21,    (1) The Hodge-Deligne polynomial of X reduces to a one-variable polynomial, and can be reconstructed from either the E or the Poincaré polynomial: , .
(2) The Cartesian product X Y × is round. Proof.
(1) If X satisfies Poincaré duality on MHS, and X n dim = , one has If X is additionally round, analogously to Proposition 3.9, μ X c can be reconstructed from P X c and E X c as: , . (2) A sufficient condition for roundness is the following: if X is balanced and separably pure and its cohomology has no gaps, in the sense that for every k ∈ , the condition H X, , then X is round. This is easy to see from Lemma 3.3 and the restrictions on weights (Proposition 2.4(2)).

Cohomology and MHSs for finite quotients
Let F be a finite group and X a complex quasi-projective F-variety. In this section, we outline some results on the cohomology and MHSs of quotients of the form X F r / , where F acts diagonally on the Cartesian product X r , for general r 1 ≥ . Of special relevance is a formula, in Corollary 4.8, for the Hodge-Deligne polynomial of X F r / for an elementary variety X whose cohomology is a simple exterior algebra.

Equivariant MHSs
The MHS of the ordinary quotient X F / is related to the one of X and its F-action, as follows. Since F acts algebraically on X, it induces an action on its cohomology ring preserving the degrees, and, by Proposition 2.
, , .  (1) X is obtained by replacing each representation in X F by its dimension; (2) The Künneth formula and Poincaré duality, for X smooth, are compatible with equivariant MHS: where ⊗ means that we take tensor products of graded F -representations. Proof.
(1) This follows immediately from the definition of dimension of representation. For (2), it suffices to see that the Künneth and Poincaré maps are also morphisms in the category of F -modules, which is easily checked. □

Cohomology of finite quotients
We recall some known facts concerning the usual and the compactly supported cohomology of the quotient X F / . Consider its equivariant cohomology, defined on rational cohomology by where EF is the universal principal bundle over BF, the classifying space of F, and EF X F × is the quotient under the natural action, which admits an algebraic map EF X X F F π × ⟶ / . Since F is finite, so is the stabilizer of any point for the F action, and the Vietoris-Begle theorem (see e.g. [ Proof. Assume first that F acts freely on X. Then, X F / has a well-defined manifold structure, and one can realize the pullback in cohomology by the pullback in differential forms. In particular, this shows that the image of the pullback π H X F H X : , , ( / ) → ( ) * * * is given by H X, F ( ) * . Using (5), this means that the pullback map is bijective onto H X, F ( ) * . If F does not act freely, the same argument can be reproduced for the de Rham orbifold cohomology, in which representatives of orbifold cohomology classes are sections of exterior powers of the orbifold cotangent bundle (see [35]). The result then follows because, for manifolds such as X, the de Rham orbifold cohomology reduces to the usual de Rham cohomology. □ The isomorphism of (5) can be obtained as the pullback of the algebraic map π X X F : → / . Given that pullbacks of algebraic maps preserve MHSs, we see that this isomorphism respects MHS (see also [8]) , . Moreover, since orbifolds satisfy Poincaré duality (see Satake [36], where these are called V -manifolds), this isomorphism is also valid for the compactly supported cohomology. Proof. Given equation (6)  In general, if we denote the character of an F-module V by χ V , because of the properties of these with respect to direct sums, we have: , , where μ t u v , , X F ( ) is viewed as an F-module, and equivalently as a direct sum of modules graded according to the triples k p q , , ( ) . Let F | | be the cardinality of F.
is a decomposition of V into irreducible sub-representations, then by the Schur orthogonality relations, the coefficient of the trivial one-dimensional representation 1 is given by: Applying this to V μ t u v , , X F = ( ) gives, in view of Corollary 4.4: and the wanted formula follows from equation (7). □ . An interesting application of Theorem 4.6 is when the cohomology of X is an exterior algebra. To be precise, we say that H X, ( ) * is an exterior algebra of odd degree k 0 if: and all other cohomology groups are zero. ≤ . Then, for r 0 > and the diagonal action of F on X r : where A g is the automorphism of H X, k ( ) corresponding to g F ∈ , and I is the identity automorphism. In particular, if X is round: Proof. First, let r 1 = . Since X is elementary and tensor and exterior products preserve MHSs, we get for all l 0 ≥ , Applying Theorem 4.6 to this case, using x uv = , we get 1 . Now, for a general F-module V , with g F ∈ acting as V Aut V g ∈ ( ), we have: = . Now, for a general r 1 ≥ , it follows from Proposition 4.2(2) that for the diagonal action μ μ Finally, the round case follows by setting p k 0 0 = . □

Abelian character varieties and their Hodge-Deligne polynomials
In this section, we apply the previous formulas to the computation of the Hodge-Deligne, Poincaré and E-polynomials, of the distinguished irreducible component of some families of character varieties. The important case of GL n, ( )-character varieties leads to the action of the symmetric group on a torus and is naturally related to work of I. G. Macdonald [38] and of J. Cheah [22] on symmetric products.

Mixed Hodge polynomials of abelian character varieties
As in Section 2.1, let G be a connected complex affine reductive group. For simplicity, the G-character variety of Γ r = , a rank r free abelian group, will be denoted by In general, the varieties G r (as well as Hom G , r ( )) are not irreducible. But there is a unique irreducible subvariety containing the identity representation that we call the distinguished component and denote by G r 0 , which is constructed as the image under the composition where π is the GIT projection, and T is a fixed maximal torus of G. This image, is then a closed subvariety of G r (see [20]) that we call the distinguished component. Let W be the Weyl group of G, acting by conjugation on T . We quote the following result from [17]. As in Section 1, denote by   (11) where A g is the automorphism of H T, 1 ( ) given by g W ∈ , and I is the identity.
Proof. Since Cartesian products of round varieties are round, and the maximal torus of G is isomorphic to n ( ) * for some n, T is a round variety and has an algebraic action of W . Then W also acts diagonally on T r n r = ( ) * , so T W r / is also round by Corollary 4.5. Moreover, the cohomology of T is an exterior algebra of degree k 1 0 = , so Corollary 4.8 immediately gives the desired formula for T W r / . The theorem follows from the isomorphism G T W  (1), we obtain, in the compactly supported case: where T dim is the rank of G. We also obtain a formula for the Poincaré and for the Serre polynomial E G ) coincides with the one used in [19], in the context of compact Lie groups.³ As indicated in Proposition 2.4(4), the Serre polynomial (E c -polynomial) is additive for disjoint unions of locally closed subvarieties. Therefore, for every bijective normalization morphism between algebraic varieties f X Y : → the E c -polynomials of X and of Y coincide. In particular, the E c -polynomials of G  Theorem 5.8. Let r 1 ≥ , and let G be a reductive group whose derived group is a classical group. Then, the mixed Hodge polynomial of G r 0 is given by formula (11).
This motivates the following conjecture.
Conjecture 5.9. For every r 1 ≥ and complex reductive G, formula (11) holds for G r 0 .

GL n, ( ) and SL n, ( ) cases
The case of G GL n, = ( ) is instructive, where the Weyl group is just the symmetric group, denoted by S n . If X is a variety, we denote its n-fold symmetric product by X n ( ) or by Sym X X S n n n ( ) = / . As a set, Sym X n ( ) is the set of unordered n-tuples of (not necessarily distinct) elements of X. Let M σ denote a n n × permutation matrix (in some basis) corresponding to σ S n ∈ and let I n be the n n × identity matrix. ( ) / ≅ ( ) (see [16]), which is the space of n (unordered) points on the compact r-torus S r 1 ( ) . So our results relate also to the study of cohomology of the so-called configuration spaces on compact Lie groups.
We now provide an even more concrete formula, and better adapted to computer calculations, using the relation between conjugacy classes of permutations and partitions of a natural number n, to compute the aforementioned determinants.
For this, we set up some notations. Let n ∈ and n be the set of partitions of n. We denote by n a general partition in n and write it as  Proof. To compute the determinant in Proposition 5.11, recall that any permutation σ S n ∈ can be written as a product of disjoint cycles (including cycles of length 1), whose lengths provide a partition of n, say n σ n 1 2 a a a n 1 2 [ ] ( ) = ⋯ . Moreover, any two permutations are conjugated if and only if they give rise to the same partition, so the conjugation class of σ uniquely determines the non-negative integers a a , , n 1 … . If σ is a full cycle σ n S 1 n = ( ⋯ ) ∈ , and M σ a corresponding matrix, by computing in a standard basis, we easily obtain the conjugation invariant expression I λM λ det 1 n σ n ( − ) = − . So, for a general permutation σ S n ∈ with cycles given by the partition n σ ( ) we have n 1 1 ( … ) = ⋯ . By considering the trivial action on * , this is a fibration of S n -varieties with trivial monodromy, since it is in fact a  T -principal bundle (and  T is a connected Lie group). Then Theorem A.1 gives us an equality of the equivariant E-polynomials: Finally, the desired formula comes from the relations in Proposition 3.9, since all varieties in consideration are round. □ Now, we turn to the computation of some E c -polynomials, which relate to some formulas obtained in [9]. ) are already present in [9]. For n 4 ≥ these formulas are new and can also be upgraded to mixed Hodge polynomials by using Remark 3.10(1) and Poincaré duality.
Example 5.19. The following table gives the explicit values of δ n ( ) and p x n ( ) up to n 5 = (in each row, the ordering is preserved). All the formulas can be easily implemented in the available computer software packages (in this paper, most of our calculations were performed with GAP). For simplicity, the notation [12] refers to a partition of n 3 = with two cycles: one of length 1, another of length 2 (not a cycle of length 12).  Proof. We now have ) is diffeomorphic to the cotangent bundle of the projective space n 1 − parametrizing semistable bundles over an elliptic curve of rank n and trivial determinant (see [15,40]).

A combinatorial identity
Appendix A Multiplicativity of the E -polynomial under fibrations In this appendix, we prove a multiplicative property of the E-polynomial under fibrations, used in Theorem 5.15. This is a consequence of the fact that the Leray-Serre spectral sequence is a spectral sequence of mixed Hodge structures. E u v E u v , , , . , where it is used to calculate the Serre polynomials of certain twisted character varieties. We detail the argument here, for the reader's convenience. First, assume that the F -action is trivial on the three spaces. The Leray-Serre spectral sequence of the fibration is a sequence of mixed Hodge structures ([27, Theorem 6.5]), and it is proved in [21, Theorem 6.1] that under the given assumptions, its second page E a b