be a ﬁnite group and be a complex quasi-projective N X ... · moduli spaces. Moreover, by showing...

HODGE-DELIGNE POLYNOMIALS OFCHARACTER VARIETIES OF FREE ABELIAN GROUPS

CARLOS FLORENTINO AND JAIME SILVA

Abstract. Let F be a finite group and X be a complex quasi-projective F -variety. Forr ∈ N, we consider the mixed Hodge-Deligne polynomials of quotients Xr/F where Facts diagonally, and compute them for certain classes of varieties X with simple mixedHodge structures. A particularly interesting case is when X is the maximal torus of anaffine reductive group G, and F is its Weyl group. As an application, we obtain explicitformulae 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 ofpartitions.

1. Introduction

The study of the geometry, topology and arithmetic of character varieties is an impor-tant topic of contemporary research. Given a reductive complex algebraic group G, and afinitely presented group Γ, the G-character variety of Γ is the (affine) geometric invarianttheory (GIT) quotient

MΓG := Hom(Γ, G)//G.

When the group Γ is the fundamental group of a Riemann surface (or more generally,a Kahler group) these spaces are homeomorphic to moduli spaces of G-Higgs bundlesvia the non-abelian Hodge correspondence (see, for example, [Hi, Si]), and have foundinteresting connections to important problems in Mathematical Physics in the context ofmirror 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,C)-character varieties of surfacegroups, by applying arithmetic harmonic analysis to their Z-models, and proving theseare polynomial count [HRV, HLRV]. By computing indecomposable bundles on algebraiccurves over finite fields, Schiffmann determined the Poincare polynomial of the modulispaces of stable Higgs bundles, hence of the corresponding GL(n,C)-character varietiesof surface groups [Sc]. Other methods based on point counting were employed by Mereb[Me] (the SL(n,C) case) and Baraglia-Hekmati [BH] (the singular, small n case).

Moreover, geometric tools were developed by Lawton, Logares, Munoz and Newstead tocalculate the E-polynomials using stratifications of character varieties (over C) of surfacegroups, exploring directly the additivity of these polynomials [LMN, LM]. This leadto the development of a Topological Quantum Field Theory for character varieties byGonzalez-Prieto et al [GPLM, GP].

2020 Mathematics Subject Classification. 32S35, 20C30; 14L30.Key words and phrases. Free abelian group, character variety, mixed Hodge structures, Hodge-Deligne

polynomials, equivariant E-polynomials, finite quotients.The authors acknowledge support from the projects PTDC/MAT-PUR/30234/2017 and EXCL/MAT-

GEO/0222/2012, FCT, Portugal, and “RNMS: GEometric structures And Representation varieties”(the GEAR Network), U. S. National Science Foundation. J. Silva was supported by the FCT grantSFRH/BD/84967/2012.

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2 C. FLORENTINO AND J. SILVA

In the present article, we deal instead with G-character varieties of free abelian groups,and with the determination of their mixed Hodge structures (MHS) for a general complexreductive G. In particular, we explicitly compute the mixed Hodge polynomials of thesevarieties. The mixed Hodge polynomial µX is a 3 variable polynomial µX = µX(t, u, v)defined for any (complex) quasi-projective variety X and encodes all numerical informa-tion about the MHS on the cohomology of X, generalizing both the Poincare and theE-polynomials.

To present our main results, denote the G-character variety of the free abelian groupΓ ∼= Zr, r ∈ N, by:

MrG :=MZrG = Hom(Zr, G)//G,

where // stands for the (affine) geometric invariant theory (GIT) quotient (see e.g. [MFK,Mu]) for the natural G-action, by conjugation, on the space of representations Hom(Zr, G).This later space consists of pairwise commuting r-tuples of elements of G, and is ofrelevance in Mathematical Physics, namely in the context of supersymmetric Yang-Millstheory [KS]. When r is even, Zr is also a Kahler group (the fundamental group of aKahler manifold) and the smooth locus of MZ2m(G) is diffeomorphic to a certain modulispace of G-Higgs bundles over a m-dimensional abelian variety (see, for instance [BF]).

The topology and geometry of character varieties of free abelian groups has been studiedby Florentino-Lawton, Sikora, and Ramras-Stafa, among others (see, eg, [FL, Si2, RS, St]).It is known that the affine algebraic variety MrG is not in general irreducible, but theirreducible component of the trivial Zr-representation, denotedM0

rG, has a normalizationM?

rG isomorphic to T r/W , ([Si2, Thm 2.1]) where T ⊂ G is a maximal torus and W is theWeyl group, acting diagonally on T r (hence also on its cohomology). Thus, the varietiesM?

rG are singular orbifolds of dimension r dimT with a special kind of mixed Hodgestructures, called balanced or of Hodge-Tate type and they satisfy the analogue of Poincareduality for MHS. When r = 2, Thaddeus proved that M?

2G are of crucial importance inmirror symmetry and Langlands duality, and computed their orbifold E-polynomials [Th].Here, we obtain the following explicit formula for mixed Hodge polynomials of M?

rG.

Theorem 1.1. Let r ≥ 1, G be a complex reductive group with maximal torus T and Weylgroup W . Then,

(1) µM?rG(t, u, v) =

1

|W |∑g∈W

det (I + tuv Ag)r ,

where Ag is the automorphism induced on H1(T,C) by g ∈ W , and I is the identityautomorphism.

One consequence of this result is a formula for the (compactly supported) E-polynomialof the irreducible component M0

rG ⊂MrG, for every such G (Theorem 5.4).Our approach to Theorem 1.1 is based on working with equivariant mixed Hodge struc-

tures and their corresponding equivariant polynomials, defined for varieties with an actionof 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 techniquesintroduced in [DL] for dealing with equivariant weight polynomials.

For the groups G = GL(n,C) and SL(n,C), we have that MrG is an irreduciblenormal variety, and the formula in Theorem 1.1 can be made even more concrete, interms of partitions of n, and allows explicit computations of the Hodge-Deligne, E- andPoincare polynomials of the corresponding character varieties MrG = M?

rG. We statethe main results below in the compactly supported version, the one which is relevant inarithmetic geometry (see [HRV], Appendix).

Let Pn denote the set of partitions of n ∈ N. By n = [1a12a2 · · ·nan ] ∈ Pn we denotethe partition of n with aj ≥ 0 parts of size j = 1, · · · , n, so that n =

∑j j aj.

HODGE-DELIGNE POLYNOMIALS OF CHARACTER VARIETIES 3

Theorem 1.2. Let G = SL(n,C), and r ≥ 1. The compactly supported E-polynomial ofMrG is

EcMrG(x) =

1

(x− 1)r

∑n∈Pn

n∏j=1

(xj − 1)ajr

aj! jaj,

where n = [1a12a2 · · ·nan ] ∈ Pn.

Theorem 1.2 generalizes, to every r, n ≥ 1, some formulas recently obtained in [BH, LM](the cases n = 2 and n = 3) by different methods, which are only tractable for lowvalues of n: the approach in [LMN, LM] uses stratifications and fibrations to computeE-polynomials of character varieties of free groups, respectively, surface groups; the com-putations in [BH] apply representation theory of finite groups, and point counting ofvarieties over finite fields.

By substituting x = 1 in EcMrG

, we obtain the Euler characteristics of these modulispaces. Moreover, by showing thatMrG have very special mixed Hodge structures (MHS)(that we call round, see Definition 3.7) Theorems 1.2 and immediately provide explicitformulas for their mixed and Poincare polynomials (Theorem 5.13).

The GL(n,C) case is particularly symmetric, as the generating function of mixed Hodgepolynomials gives precisely the formula of J. Cheah [Ch] for the mixed Hodge numbers ofsymmetric products. On the other hand, by examining the action of W on the cohomologyof a maximal torus, our methods allow for the computation of µMrG for all the classicalcomplex semisimple groups G. These will be addressed in upcoming work.

We now outline the contents of the article. In section 2, we review necessary backgroundon MHS, quasi-projective varieties, etc, and define the relevant polynomials, providing ex-amples, and focusing on balanced varieties. In section 3, we study properties of specialMHS, related with notions defined in [DL], and paying special attention to round varieties,for which the knowledge of either the Poincare polynomial or the E-polynomial allows thedetermination of µ. Section 4 is devoted to equivariant MHS, character formulae and thecohomology of finite quotients. Finally, in Section 5 we prove our main Theorem andprovide explicit calculations of Hodge-Deligne and E-polynomials (and Euler characteris-tics) of character varieties of Zr, in particular for GL(n,C) and SL(n,C); in the GL(n,C)case the computations are related to MHS on symmetric products, thereby obtaining acurious combinatorial identity. In the Appendix, we present a proof, based on [DL], ofthe equivariant version of a theorem in [LMN, LM] on the multiplicative property of theE-polynomial for fibrations.

A preliminary version of the main results has been announced in [FNSZ].Acknowledgements. We would like to thank many interesting and useful conversations

with colleagues on topics around mixed Hodge structures, especially T. Baier, I. Biswas,P. Boavida, G. Granja, S. Lawton, M. Logares, V. Munoz and A. Oliveira; and theanonymous referee for the careful and thorough reading, and suggestions leading to severalimprovements. Thanks are also due to the Simons Center for Geometry and Physics, andthe hosts of a 2016 workshop on Higgs bundles, where some of the ideas herein startedto take shape. The first author dedicates this article to E. Bifet, for his mathematicalenthusiasm on themes close to this one, which had a long lasting influence.

2. Preliminaries on Character Varieties and on Mixed Hodge Structures

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


2.1. Character varieties. Given a finitely generated group Γ and a complex affine reduc-tive group G, the G-character variety of Γ is defined to be the (affine) Geometric InvariantTheory (GIT) quotient (see [MFK, Mu]; [Sw] for topological aspects):

MΓ(G) = Hom(Γ, G)//G.

Note that Hom(Γ, G), the space of homomorphisms ρ : Γ → G, is an affine variety, as Γis defined by algebraic relations, and it is also a G-variety when considering the action ofG by conjugation on Hom(Γ, G).

The GIT quotient above is the the maximal spectrum of the ring C[Hom(Γ, G)]G ofG-invariant regular functions on Hom(Γ, G):

Hom(Γ, G)//G := Specmax(C[Hom(Γ, G)]G

).

The GIT quotient does not parametrize all orbits, since some of them may not be dis-tinguishable by invariant functions. In fact, it can be shown (see, for example [Mu])that the conjugation orbits of two representations ρ, ρ′ : Γ → G define the same pointin Hom(Γ, G)//G if and only if their closures intersect: G · ρ ∩ G · ρ′ 6= ∅ (in either theZariski or the complex topology coming from an embedding Hom(Γ, G) ↪→ CN). Fordetailed definitions and properties of general character varieties, we refer to [FL, Si1].

In this article, we will be mostly concerned with the case when Γ is a finitely generatedfree abelian group, Γ = Zr for some natural number r, the rank of Γ. The correspondingG-character varieties:

MZrG = Hom(Zr, G)//G,

have many interesting properties, as representations in Hom(Zr, G) can be naturallyidentified with r-tuples of group elements (A1, · · · , Ar) ∈ Gr that pairwise commute:AiAj = AjAi, for all i, j = 1, · · · , n.

WhenK is a compact Lie group, the analogous space of representations Hom(Zr, K) is ofcentral importance in determining the so-called moduli space of vacua of supersymmetricgauge theories on a r-dimensional torus, as studied in [KS, BFM] and others.

2.2. Mixed Hodge structures. On a compact Kahler manifold X the complex cohomologysatisfies the Hodge decomposition Hk(X,C) ∼=

⊕p+q=kH

p,q(X) which verifies Hq,p(X) ∼=Hp,q(X). This decomposition of Hk(X,C), a pure Hodge structure of weight k, can bedescribed, equivalently, by a decreasing filtration:

Hk(X,C) = F0 ⊇ F1 ⊇ . . . ⊇ Fk+1 = 0

satisfying Fp ∩ Fq = 0 and Fp � Fq = Hk(X,C) for all p+ q = k + 1.This notion can be generalized to quasi-projective algebraic varieties X over C, possibly

non-smooth and/or non-compact. Namely, the complex cohomology of any such varietyis also endowed with a natural filtration, the Hodge filtration F , and moreover, there is aspecial second increasing filtration on the rational cohomology:

0 = W−1 ⊆ . . . ⊆ W 2k = Hk (X,Q) ,

the weight filtration W , satisfying a compatibility condition with respect to the Hodgefiltration: the latter induces a filtration on the weighted graded pieces of the former,that needs to be a pure Hodge structure. The vector space Hk(X,Q), together with thefiltrations F and W , is the prototype of a mixed Hodge structure (MHS). We denote thegraded pieces of the associated decomposition by

Hk,p,q(X) := GrpFGrWCp+qH

k(X,C),

where WC stands for the complexified weight filtration. Notice that, even though differentfiltrations may lead to isomorphic graded pieces, for convenience, we sometimes refer to


the collection of these Hk,p,q(X) as the MHS of X. For background and proofs we referto [PS] and the original articles by Deligne [De1, De2].

The above constructions can be reproduced for the compactly supported cohomologygroups Hk

c (X,C), yielding an analogous decomposition:

Hk,p,qc (X) := GrpFGr

WCp+qH

kc (X,C).

MHS in the compactly supported context have interesting connections to number theoryas illustrated, for example, in the Appendix of [HRV] by N. Katz.

Mixed Hodge structures satisfy some nice properties, as follows.

Proposition 2.1. Let X and Y be complex quasi-projective varieties. Then,

(1) For all k, p, q, we have Hk,q,p(X) ∼= Hk,p,q(X);(2) The weight and Hodge filtrations are preserved by algebraic maps. Therefore, so

are mixed Hodge structures;(3) The Hodge and weight filtrations are preserved by the Kunneth isomorphism. There-

fore, so are mixed Hodge structures;(4) The mixed Hodge structures are compatible with the cup product:

Hk,p,q (X) ^ Hk′,p′,q′ (X) ↪→ Hk+k′,p+p′,q+q′ (X) ;

(5) If X is smooth of complex dimension n, mixed Hodge structures are compatiblewith Poincare duality:

Hk,p,q (X) ∼=(H2n−k, n−p, n−qc (X)

)∗.

Proof. All of these statements are standard. For convenience, we point to appropriatereferences. The proof of (1) follows from the purity of the Hodge structure on the gradedpieces of the weight filtration.

The other proofs are found in chapters 4 to 6 of the book [PS]. Specifically, for (2) see[PS, Proposition 4.18]; (3) and (4) appear in [PS, Theorem 5.44, Corollary 5.45] and (5)in [PS, Proposition 6.19]. �

2.3. Hodge polynomials and balanced varieties. The spaces Hk,p,q(X) are holomorphicinvariants and encode important geometric information (diffeomorphic complex manifoldsmay have non-isomorphic mixed Hodge structures, as in Example 2.6).

The mixed Hodge numbers of X are the complex dimensions of the MHS pieces

hk,p,q(X) := dimCHk,p,q(X),

and are typically assembled in a polynomial. By definition, for a pure Hodge structure,hk,p,q(X) 6= 0 unless k = p+ q.

Definition 2.2. Let X be a complex quasi-projective variety of complex dimension d. Themixed Hodge polynomial of X (also called Hodge-Deligne polynomial) is the three variablepolynomial of degree ≤ 2d

µX (t, u, v) :=∑

k.p,q≥0

hk,p,q(X) tkupvq.

Its specialization for t = −1

EX (u, v) :=∑

k.p,q≥0

hk,p,q(X) (−1)k upvq

is called the E-polynomial of X.


Remark 2.3. (1) The specialization of µX for u = v = 1 gives the Poincare polynomial ofX:

PX(t) :=∑k≥0

bk(X) tk,

with bk(X) := dimCHk(X,C) being the Betti numbers of X. Note that the coefficients

of µX and of PX are non-negative integers, whereas EX lives in the ring Z[u, v].(2) As mentioned before, there is an entirely parallel theory for the compactly supportedcohomology. Here, the associated Hodge numbers are denoted by hk,p,qc := dimCH

k,p,qc (X).

If PX stands for one of the polynomials in the above definition, we will distinguish itscompactly supported version by writing PcX .(3) Comment on terminology: There are inconsistencies in the literature on the termi-nology used for these polynomials. Since hk,p,q(X) are generally called Hodge-Deligne (ormixed Hodge) numbers, we refer to µX as Hodge-Deligne or mixed Hodge polynomial. Toemphasize the distinction, the compactly supported E-polynomial Ec

X will also be calledthe 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 [Du]).(4) Many specializations of the E-polynomial have been studied in the literature. Thereis, for example, the weight polynomial WX (y) :=

∑k,p(−1)kwk,p(X) yp, using the graded

pieces of the weight filtration wk,p(X) := dimCGrWCp Hk(X,C) (see [DL]). This is a

specialization of the E-polynomial since WX(y) = EX(y, y). Also, Hirzebruch’s χy-genus and the signature σ of a complex manifold X are given, in terms of EX(u, v),as: χy(X) = EX(−y, 1) and σ(X) = EX(−1, 1), respectively (see Hirzebruch [Hir]).

We now collect some well known important properties of these polynomials, for lateruse.

Proposition 2.4. For a quasi-projective variety X, we have:

(1) The polynomials µX and EX are symmetric in the variables u and v; in particular,if hk,p,q(X) 6= 0 then hk,q,p(X) 6= 0;

(2) Let hk,p,q(X) 6= 0. Then p, q ≤ k. Moreover, if X is smooth then p + q ≥ k; ifX is projective then p+ q ≤ k. In particular, if X is a compact Kahler manifoldp+ q = k;

(3) The (topological) Euler characteristic χ(X) is given by χ(X) = EX(1, 1);(4) The Serre polynomial (compactly supported E-polynomial) Ec

X , is additive forstratifications of X by locally closed subsets, and its degree is equal to 2 dimCX.

(5) All polynomials µX , PX and EX are multiplicative under cartesian products.

Proof. (1) follows from item (1) of Proposition 2.1. Item (2) is proved in [PS, Proposition4.20] and [PS, Theorem 5.39]. Item (3) is immediate from the definition. The proof of (4)can be found in [PS, Corollary 5.57], and (5) follows directly from 2.1(4). �

A common feature of the varieties in this paper is that their mixed Hodge structure is“diagonal”: for each k, the only non-zero mixed Hodge numbers are hk,p,q with p = q.

Definition 2.5. A quasi-projective variety X is said to be balanced or of Hodge-Tate typeif for every non-negative integer k ∈ N0, and all p 6= q, hk,p,q(X) = 0. In other words, ifhk,p,q(X) 6= 0 then q = p. We call p+ q the total weight of Hk,p,q(X).

Example 2.6. (1) If X is connected, H0(X,C) ∼= C has always a pure Hodge structure,with trivial decomposition H0(X,C) = H0,0,0(X). Dually, when X is also smooth, thecompactly supported cohomology is also a trivial decompositionH2n

c (X,C) = H2n,n,nc (X) .

(2) X = Cn is a (non compact) Kahler manifold with cohomology only in degree zero. By


the above, it has trivial pure Hodge structure: H∗ (X,C) = H0 (X,C) = H0,0,0 (X) andso

µCn (t, u, v) = 1, µcCn(t, u, v) = t2nunvn,

where the compactly supported version follows from Poincare duality.(3) Let X = C∗ = C\{0}. Although Kahler, its cohomology has no pure Hodge structure,since dimCH

1(X,C) = 1. Being smooth, using Proposition 2.1(1)-(2), the only non-zeroh1,p,q is h1,1,1, so h0,0,0 = h1,1,1 = 1 and all other Hodge numbers vanish. We then get

µC∗ (t, u, v) = 1 + tuv,

and the only non-zero hk,p,qc are h2,1,1c = h1,0,0

c = 1, by Poincare duality. Thus, µcC∗(t, u, v) =t2uv + t and Ec

C∗(u, v) = uv − 1. Observe that this is compatible with the decompositioninto locally closed subsets C1 = C∗ t C0 as in Proposition 2.4(4). Hence, Cn and (C∗)nare examples of balanced varieties. For a simple example of a non-balanced variety wecan take an elliptic curve, or any compact Riemann surface of positive genus.(4) Consider the total space X of the trivial line bundle over an elliptic curve X ∼= (C/Λ)×C, where Λ is a rank two lattice in (C,+). It is easy to see that X is real analyticallyisomorphic to (C∗)2 (but not complex analytically or algebraically isomorphic). From theKunneth isomorphism and considerations analogous to Example 2.6 we get:

µ(C∗)2 (t, u, v) = (1 + tuv)2 = 1 + 2tuv + t2u2v2,

µX (t, u, v) = (1 + tu)(1 + tv) = 1 + t(u+ v) + t2uv.

Indeed, (C∗)2 is balanced whereas the cohomology of X is pure.

Remark 2.7. (1) The last example is a very special case (the genus 1, rank 1 case) ofthe non-abelian Hodge correspondence mentioned in the Introduction, which producesdiffeomorphisms between (Zariski open subsets) of moduli spaces of flat connections andcertain moduli spaces of Higgs bundles over a given Riemann surface. The fact that onediffeomorphism type is balanced (the flat connection side of the correspondence) and theother is pure is a general feature (see [HRV, HLRV]).(2) If X is balanced, its E-polynomial depends only on the product uv, so it is commonto adopt the change of variables x ≡ uv. When written in this variable, the degree ofEcX(x) is now equal to dimCX, instead of 2 dimCX.

3. Separably pure, elementary and round varieties

In this section, we collect many properties of MHS that are necessary later on. Wealso describe the types of Hodge structures that allow the recovery of the mixed Hodgepolynomial given the E- or the Poincare polynomial (Theorem 3.6), and concentrate onthe case of round varieties, which are the Hodge types of our character varieties. We triedto be self-contained for the benefit of researchers in the field of character varieties or Higgsbundles that may not be familiar with MHS.

3.1. Elementary and separably pure varieties. The mixed Hodge structures on the co-homology of a given quasi-projective variety X may be trivial, ie, the decomposition ofevery Hk(X,C) is the trivial one, and many such examples are considered here. Whenthis happens, the only non-zero hk,p,q(X) satisfy q = p (by Proposition 2.4(1)) and muchof what can be said about the cohomology can be transported to mixed Hodge structures.Adapting some notions from [DL] (who worked with the weight polynomial), we introducethe following terminology.

Definition 3.1. Let X be a quasi-projective variety. X (or its cohomology) is calledelementary if its mixed Hodge structures are trivial decompositions of the cohomology, sothat for every k ∈ N there is only one p ∈ N such that hk,p,p(X) 6= 0 (and hk,p,q(X) 6= 0


for q 6= p).X is said to be separably pure if the mixed Hodge structure on each Hk (X,C) is in factpure of total weight wk, and such that wj 6= wk for every j 6= k.

Remark 3.2. (1) Note that X is elementary if it is balanced and there is a weight functionk 7→ pk, (defined only for those k ∈ N0 with Hk(X,C) 6= 0) such that hk,p,q = 0 for everypair (p, q) not equal to (pk, pk). In this case:

(2) GrWC2pkHk(X,C) = Hk,pk,pk (X) = Hk(X,C).

A general weight function is not enough to recover µX 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 7→ pk is injective, in which case the equality(2) takes the stronger form:

⊕mGr

WC2pkHm(X,C) = Hk,pk,pk (X) = Hk(X,C).

(2) In a pure Hodge structure of total weight k on Hk (X,Q) the only non-zero weightsummand is GrW2kH

k (X,Q). So, a pure total cohomology is separably pure, but not con-versely, as the case C∗ shows (Example 2.6).(3) When X is separably pure, instead of the weight function, one can define a degreefunction (p, q) 7→ k = k(p, q) (defined only on pairs (p, q) such that hk,p,q(X) 6= 0). Not-ing that, in fact, the degree k only depends on the total weight p + q (being separablypure) we can write this as (p, q) 7→ kp+q.

In this article, most varieties are both separably pure and balanced, and an alternativecharacterization follows.

Lemma 3.3. A quasi-projective variety X is separably pure and balanced if and only if itis elementary and its weight function k 7→ pk is injective.

Proof. If X is separably pure, the total weight in each Hk(X,C) has to be constant. Butif X is also balanced, given k, all hk,p,q(X) vanish except for a unique pair (p, q) = (pk, pk),so we have an assignment k 7→ pk proving that X is elementary. Moreover, since the totalweights are different for distinct k, the weight function is injective. The converse statementis easy since an elementary variety is Hodge-Tate and an injective weight function impliesinjectivity for total weights. �

Example 3.4. A family of balanced and pure varieties are the smooth projective toricvarieties (hence these are elementary). Indeed, every such toric variety X, where the C∗action has dj orbits in (complex) dimension j = 0, · · · , dimCX, has (See [PS, Ex. 5.58]):

µX(t, u, v) =n∑j=0

dj(t2uv − 1)j,

and the weight function is 2j 7→ j (the weights being (j, j)). For example, the Hodge-Deligne polynomials of projective spaces are µPn

C(t, u, v) =

∑nj=0 t

2jujvj.

As in the case of the complex affine multiplicative group C∗, more general complexaffine algebraic groups are balanced, but not necessarily pure or separably pure.

Example 3.5. The Poincare polynomial of GL(n,C) is well known, given byPGL(n,C)(t) =

∏nj=1 (1 + t2j−1). Also, by [De2, Theorem 9.1.5], we have:

µGL(n,C) (t, u, v) =n∏j=1

(1 + t2j−1ujvj

).

For example, µGL(3,C) (t, x) = 1 + tx + t3x2 + t4x3 + t5x3 + t6x4 + t9x6, (writing x = uv,see Remark 2.7(2)). So, GL(3,C) is elementary (hence balanced) but not separably pure:


Figure 1. Venn diagram with several classes of MHS

both degrees 4 and 5 have associated total weight 6 (the terms with x3), so GL(n,C) isnot separably pure, for n ≥ 3. Moreover, the same argument readily shows that GL(n,C)is not elementary for n ≥ 5.

The above examples show that this “yoga of weights”, as alluded by Grothendieck, isvery useful in understanding general properties of certain classes of varieties. When weknow that a particular variety X has a degree or a weight function as above, we candetermine the full collection of triples (k, p, q), such that hk,p,q(X) 6= 0.

In Figure 3.1, the shaded area illustrates Lemma 3.3; for the definition of round, seesubsection 3.2, below. The next result shows that elementary and separably pure areindeed the correct notions to be able to determine the mixed Hodge polynomial from thePoincare 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 Poincare polynomial deter-mines its Hodge-Deligne polynomial.

(2) If X is separably pure, with known degree function, its E-polynomial determinesits Hodge-Deligne polynomial.

Proof. (1) Suppose the Poincare polynomial of X is PX (t) =∑

k bktk and the weight func-

tion is k 7→ (pk, pk). Then, since the only non-trivial mixed Hodge pieces are Hk,pk,pk(X),we get µX (t, u, v) =

∑k bkt

kupkvpk .(2) Similarly, writing EX(u, v) =

∑p,q ap,qu

pvq, and the degree function as (p, q) 7→ kp+q,since the total weights are in one-to-one correspondence with the degrees of cohomology,we obtain µX(t, u, v) =

∑k,p,q ap,q(−t)kp+qupvq. �

3.2. Round varieties. From Theorem 3.6, if a variety X is both balanced and separablypure, then µX can be recovered from either EX or PX , knowing their degree/weightfunctions. A specially interesting case is the following.

Definition 3.7. Let X be a quasi-projective variety. If the only non-zero Hodge numbersare of type hk,k,k (X), k ∈ {0, · · · , 2 dimCX}, we say that X is round.

In other words, a round variety is both elementary and separably pure and its onlyk-weights have the form (k, k). Round varieties are referred as “minimaly pure” balancedvarieties in Dimca-Lehrer (see [DL, Def. 3.1(iii)]).

Remark 3.8. In general, cartesian products of elementary varieties are not elementary,and similarly for separably pure varieties. For instance, using Example 3.5 with n = 2,we see that GL(2,C)×GL(2,C) is not separably pure. On the other hand, the followingresult holds for round varieties.


Proposition 3.9. Let X and Y be round varieties. Then:

(1) The Hodge-Deligne polynomial of X reduces to a one variable polynomial, and canbe reconstructed from either the E or the Poincare polyonmial:

µX (t, u, v) = PX (tuv) = EX (−tu, v) .

(2) The cartesian product X × Y is round.

Proof. (1) By definition, if X is round, we can write:

µX(t, u, v) =∑k≥0

hk,k,k(X) tkukvk,

so that µX is a polynomial in tuv. Moreover, its Betti numbers are bk(X) = hk,k,k(X)giving the first equality. The second follows from EX(x) =

∑k≥0(−1)khk,k,k(X)xk.

(2) This follows at once from (1) and from Proposition 2.4(5). �

Remark 3.10. (1) If X satisfies Poincare duality on MHS, and dimCX = n, one has

µcX(t, u, v) = (t2uv)nµX(1t, 1u, 1v), P c

X(t) = t2nPX(1t), Ec

X(u, v) = (uv)nEX( 1u, 1v).

In particular χ(X) = EcX(1, 1). If X is additionally round, analogously to Proposition

(3.9), µcX can be reconstructed from P cX and Ec

X as:

µcX(t, u, v) = (uv)−nP cX(tuv) = (−t)nEc

X(−tu, v).

(2) A sufficient condition for roundness is the following: If X is balanced and separablypure and its cohomology has no gaps, in the sense that for every k ∈ N, the conditionHk(X,C) 6= 0 implies Hk−1(X,C) 6= 0, then X is round. This is easy to see from Lemma3.3 and the restrictions on weights (Proposition 2.4(2)).

4. Cohomology and Mixed Hodge Structures 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 mixed Hodge structures of quotients ofthe form Xr/F , where F acts diagonally on the cartesian product Xr, for general r ≥ 1.Of special relevance is a formula, in Corollary 4.8, for the Hodge-Deligne polynomial ofXr/F for an elementary variety X whose cohomology is a simple exterior algebra.

4.1. Equivariant mixed Hodge structures. The MHS of the ordinary quotient X/F isrelated with 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 Proposition2.1(1), the mixed Hodge structures. Therefore, Hk,p,q (X) and GrWC

p+qHk (X,C) are also

F -modules. Denoting these by[GrWp+qH

k (X,C)]

and[Hk,p,q (X)

], and calling them equi-

variant MHS, one may codify this information in polynomials with coefficients belongingto the representation ring of F , R(F ) (cf. [DL]).

Definition 4.1. The equivariant mixed Hodge polynomial is defined as:

µFX (t, u, v) =∑k,p,q

[Hk,p,q (X)

]tkupvq ∈ R(F ) [t, u, v] .

Evaluating at t = −1, gives us the equivariant E-polynomial :

EFX (u, v) =

∑k,p,q

(−1)k[Hk,p,q (X)

]upvq ∈ R(F ) [u, v] .

As in the non-equivariant case, we adopt the change of variable x = uv when X isbalanced. As in Proposition 2.1, several simple properties can be deduced.


Proposition 4.2. Let X be a quasi-projective F -variety, for a finite group F , and let PFXbe one of the polynomials in Definition 4.1. Then

(1) PX is obtained by replacing each representation in PFX by its dimension;(2) The Kunneth formula and Poincare Duality, for X smooth, are compatible with

equivariant MHS:

PFX×Y = PFX � PFY[GrWC

p+qHk (X,C)

]=

[(GrWC

2n−(p+q)H2n−kc (X,C)

)∗][Hk,p,q (X)

]=

[(H2n−k,n−p,n−qc (X)

)∗],

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 the that the Kunneth and Poincare maps are also morphisms inthe category of F -modules, which is easily checked. �

4.2. Cohomology of finite quotients. We recall some known facts concerning the usualand the compactly supported cohomology of the quotient X/F . Consider its equivariantcohomology, defined on rational cohomology by

H∗F (X,Q) := H∗ (EF ×F X,Q) ,

where EF is the universal principal bundle over BF , the classifying space of F , and EF×FX is the quotient under the natural action, which admits an algebraic map EF ×F X

π−→X/F . Since F is finite, so is the stabilizer of any point for the F action, and the Vietoris-Begle theorem (see e.g. [Sp, page 344]) implies that the pullback π∗ : H∗ (X/F,Q) →H∗ (EF ×F X,Q) provides an isomorphism

(3) H∗ (X/F,Q) ' H∗F (X,Q) .

Moreover, the fibration

X // EF ×F X // BF

has an induced Serre spectral sequence satisfying (see [Mc], for example)

Ep,q2∼= Hp (BF, Hq (X)) ∼= Hp (F, Hq (X))⇒ Hp+q

F (X,Q) .

Since F is finite, one can deduce Hp (F, Hq (X)) = 0 for all p > 0 and all q, since sheafcohomology vanishes in degrees higher than dimF . Then the Serre spectral sequenceconverges at the second step, and this gives ([Br])1

(4) H∗F (X,Q) ' H∗ (X,Q)F .

Combining Equations (3) and (4), one gets an isomorphism of graded vector spaces:2

H∗ (X/F,Q) ∼= H∗ (X,Q)F .(5)

Proposition 4.3. Let F be a finite group and X a smooth quasi-projective F -variety. Then,the pullback of the quotient map π∗ : H∗ (X/F,Q) → H∗ (X,Q) is injective and has

H∗ (X,Q)F as its image.

1For polynomials, the superscript F means we are taking the equivariant version, for vector spaceswith an F action, the superscript denotes the fixed subspace.

2We thank Donu Arapura for a suggestion leading to a shorter proof of Proposition 4.3


Proof. Assume first that F acts freely on X. Then, X/F has a well defined manifoldstructure, and one can realize the pullback in cohomology by the pullback in differentialforms. In particular, this shows that the image of the pullback π∗ : H∗ (X/F,Q) →H∗ (X,Q) is given by H∗ (X,Q)F . Using (5), this means that the pullback map is bijective

onto H∗ (X,Q)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 exteriorpowers of the orbifold cotangent bundle (see [ALR]). The result then follows because, formanifolds such as X, the de Rham orbifold cohomology reduces to the usual de Rhamcohomology. �

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 mixed Hodge structures, we seethat this isomorphism respects MHS (see also [LMN])

(6) H∗,∗,∗ (X,Q)F ∼= H∗,∗,∗ (X/F,Q) .

Moreover, since orbifolds satisfy Poincare duality (see Satake [Sa], where these are calledV -manifolds), this isomorphism is also valid for the compactly supported cohomology.

Corollary 4.4. Let X be a smooth complex quasi-projective F -variety, for F a finite group,and PX (t, u, v) denote either the Poincare, Hodge-Deligne or E-polynomials, for the usualor the compactly supported cohomologies, Then, PX/F (t, u, v) equals the coefficient of thetrivial representation in PFX (t, u, v).

Proof. Given Equation (6), the cohomology of X/F coincides with the invariant partof the induced action F y H∗(X,C), and this is precisely the coefficient of the trivialrepresentation in each equivariant polynomial PFX . �

Corollary 4.5. Let X be a smooth complex quasi-projective F -variety, as above. If hk,p,q(X/F ) 6=0 then hk,p,q(X) 6= 0. Consequently, if X is balanced, separably pure or round, then thesame is true for X/F .

Proof. By Corollary 4.4, Hk,p,q(X/F ) ∼= Hk,p,q(X)F ⊂ Hk,p,q(X), so the first sentence fol-lows. Since all the properties of being balanced, etc, are relations between the coefficientsof t and u, v, they survive to the quotient. �

4.3. Character formulae. For a F -variety X, is useful to consider the characters of therepresentations Hk,p,q(X), when this space is viewed as a F -module.

For this, let Ag ∈ Aut(Hk,p,q(X)) be the induced automorphism of Hk,p,q(X) given bythe action of an element g ∈ F . Given k-weights (p, q), denote by

χk,p,q : F → Cg 7→ tr(Ag)

the character of Hk,p,q(X). In general, if we denote the character of a F -module V by χV ,because of the properties of these with respect to direct sums, we have:

(7) χµFX(t,u,v)(g) =∑k,p,q

χk,p,q(g) tkupvq,

where µFX(t, u, v) is viewed as a F -module, and equivalently as a direct sum of modulesgraded according to the triples (k, p, q). Let |F | be the cardinality of F .

Theorem 4.6. Let X be a quasi-projective F -variety. Then

µX/F (t, u, v) =1

|F |∑g∈F

∑k,p,q

χk,p,q(g) tkupvq.


Proof. If V is a F -module, and V = ⊕iVi is a decomposition of V into irreducible sub-representations, then by the Schur orthogonality relations, the coefficient of the trivialone-dimensional representation 1 is given by:

〈χV , χ1〉 =1

|F |∑g∈F

χV (g)χ1 (g) =1

|F |∑g∈F

χV (g) .

Applying this to V = µFX(t, u, v) gives, in view of Corollary 4.4:

µX/F (t, u, v) =⟨χµFX(t,u,v), χ1

⟩=

1

|F |∑g∈F

χµFX(t,u,v) (g) ,

and the wanted formula follows from Equation (7). �

Example 4.7. Let X = P1C × P1

C and consider the natural permutation action of S2∼= Z2.

If S denotes the one dimensional sign representation, by describing the induced actionon cohomology, it is not difficult to show that [H0,0,0(X)] ∼= [H4,2,2(X)] ∼= 1 (the trivialone-dimensional representation) and that [H2,1,1(X)] ∼= 1⊕ S, giving:

µS2X (t, u, v) =

[H0,0,0(X)

]⊕[H2,1,1(X)

]t2uv ⊕

[H4,2,2(X)

]t4u2v2

= 1 ⊕ (1⊕ S) t2uv ⊕ 1 t4u2v2.(8)

Alternatively, writing S2 = {±1}, and taking the trivial characters χ0,0,0(g) = χ4,2,2(g) ≡1, for g ∈ S2, and χ2,1,1(1) = 2, χ2,1,1(−1) = 0, we can use Theorem 4.6 to get:

µX/S2 (t, u, v) =1

2(1 + 2t2uv + t4u2v2) +

1

2(1 + t4u2v2) = 1 + t2uv + t4u2v2,

which coincides with the coefficient of 1 in Equation (8). Naturally, this is the expectedpolynomial, because X/S2 = Sym2 (P1

C) ' P2C.

An interesting application of Theorem 4.6, is when the cohomology of X is an exterioralgebra. To be precise, we say that H∗(X,C) is an exterior algebra of odd degree k0 if:

Hk0l(X,C) ∼=∧lHk0(X,C), ∀l ≥ 0,

and all other cohomology groups are zero.

Corollary 4.8. Let X be an elementary F -variety whose cohomology is an exterior al-gebra of odd degree k0, and let Hk0(X,C) = Hk0,p0,p0(X) be its (trivial) mixed Hodgedecomposition, for some p0 ≤ k0. Then, for r > 0 and the diagonal action of F on Xr:

µXr/F (t, x) =1

|F |∑g∈F

det(I + tk0xp0 Ag

)r,

with x = uv, where Ag is the automorphism of Hk (X,C) corresponding to g ∈ F , and Iis the identity automorphism. In particular, if X is round:

µXr/F (t, x) =1

|F |∑g∈F

det(I + (tx)k0 Ag

)r.

Proof. First, let r = 1. Since X is elementary and tensor and exterior products preservemixed Hodge structures, we get for all l ≥ 0,

H lk0(X,C) =∧lHk0(X,C) =

∧lHk0,p0,p0(X) = H lk0,lp0,lp0(X).

Applying Theorem 4.6 to this case, using x = uv, we get

µX/F (t, x) =1

|F |∑

g∈F∑

l≥0 χlk0,lp0,lp0(g) tlk0xlp0 .(9)


Now, for a general F -module V , with g ∈ F acting as Vg ∈ Aut(V ), we have:∑l≥0

χ∧l V (g) sl = det(I + s Vg).

This can be seen by expanding the characteristic polynomial of Vg in terms of traces of∧l Vg (see for example [Se, pg. 69]). Substituting the last equality into (9), we get theresult with s = tk0xp0 . Now, for a general r ≥ 1, it follows from Proposition 4.2(2) thatfor the diagonal action µFXr =

(µFX)r

, so

µXr/F (t, x) =1

|F |∑

g∈F χ(µFX(t,x))r (g)

=1

|F |∑

g∈F

(χµFX(t,x) (g)

)r=

1

|F |∑

g∈F det(I + tk0xp0Ag

)r.

Finally, the round case follows by setting p0 = k0. �

5. Abelian character varieties and their Hodge-Deligne polynomials

In this section, we apply the previous formulae to the computation of the Hodge-Deligne, Poincare and E-polynomials, of the distinguished irreducible component of somefamilies of character varieties. The important case of GL(n,C)-character varieties leadsto the action of the symmetric group on a torus, and is naturally related to works of I.G. Macdonald [Ma] and of J. Cheah [Ch] on symmetric products.

5.1. Mixed Hodge polynomials of abelian character varieties. As in Section 2.1, let Gbe a connected complex affine reductive group. For simplicity, the G-character variety ofΓ = Zr, a rank r free abelian group, will be denoted by

MrG :=MZrG = Hom(Zr, G)//G.

In general, the varieties MrG (as well as Hom(Zr, G)) are not irreducible. But there isa unique irreducible subvariety containing the identity representation, that we call thedistinguished component and denote by M0

rG, which is constructed as the image underthe composition

(10) T r = Hom(Zr, T )ι↪→ Hom(Zr, G)

π→MrG,

where π is the GIT projection, and T is a fixed maximal torus of G. This image,

M0rG := (π ◦ ι)(Hom(Zr, T )),

is then a closed subvariety of MrG (see [Th]) that we call the distinguished component.Let W be the Weyl group of G, acting by conjugation on T . We quote the followingresult from [Si2]. As in the introduction, denote by M?

rG the normalization of M0rG as

an algebraic variety, so that there is a birational map ν :M?rG→M0

rG.

Proposition 5.1. [Si2, Theorem 2.1] Let G be a complex reductive group and r ≥ 1. Then,M0

rG is an irreducible component of MrG, and there is an isomorphism M?rG∼= T r/W .

We now prove Theorem 1.1.

Theorem 5.2. Let G be a complex reductive algebraic group. Then, M?rG is round and

(11) µM?rG (t, u, v) =

1

|W |∑g∈W

[det (I + tuv Ag)]r ,

where Ag is the automorphism of H1(T,C) given by g ∈ W , and I is the identity.


Proof. Since cartesian products of round varieties are round, and the maximal torus ofG is isomorphic to (C∗)n for some n, T is a round variety and has an algebraic action ofW . Then W also acts diagonally on T r = (C∗)nr, so T r/W is also round by Corollary4.5. Moreover, the cohomology of T is an exterior algebra of degree k0 = 1, so Corollary4.8 immediately gives the desired formula for T r/W . The Theorem follows from theisomorphism M?

rG∼= T r/W of Proposition 5.1. �

Remark 5.3. By Remark 3.10(1), we obtain, in the compactly supported case:

(12) µcM?rG

(t, u, v) =tr·dimT

|W |∑g∈W

[det (tuvI + Ag)]r ,

where dimT is the rank of G. We also obtain a formula for the Poincare and for the Serrepolynomial Ec

M0rG

of the distinguished component M0rG.

Theorem 5.4. For every complex reductive algebraic group G and r ≥ 1, the Poincarepolynomial (respectively the Serre polynomial) of M0

rG is given by substituting u = v = 1in Equation (11) (respectively t = −1 in (12)).

Proof. From [FL, Cor. 4.9], there is a strong deformation retraction from M0rG to

Hom0(Zr, K)/K, the path component of the space of commuting r-tuples of elementsin K, containing the trivial r-tuple, up to conjugation, where K is a maximal compactsubgroup of G. Hence, these spaces have the same Poincare polynomials. On the otherhand, the formula of [St, Thm. 1.4] is the same as Equation (11) with u = v = 1. Indeed,given the identification of H1(T,C) with t ∼= Cn, the Lie algebra of T :

H1(T,C) ∼= H1 ((C∗)n ,C) = H1 (C∗,C)n ∼= t,

and the fact that every cohomology class has a left invariant representative, the action ofthe Weyl group W = Sn on H1(T,C) coincides with the one used in [St], in the contextof compact Lie groups.3

As indicated in Proposition 2.4(4), the Serre polynomial (Ec-polynomial) is additive fordisjoint unions of locally closed subvarieties. Therefore, for every bijective normalizationmorphism between algebraic varieties f : X → Y the Ec-polynomials of X and of Ycoincide. In particular, the Ec-polynomials of M?

rG and M0rG coincide. �

5.2. Normality of the distinguished component. Given the equalities of both Poincareand Serre polynomials ofM0

rG andM?rG, it is interesting to check where there is also an

equality µM?rG = µM0

rG. To handle this question, we start by considering some sufficient

conditions for normality of M0rG.

Lemma 5.5. Let F ⊂ G be a finite subgroup of the center of G, and H = G/F . If M0rG

is normal then M0rH is normal.

Proof. Let Hom0(Zr, G) := π−1(M0rG). By definition of π in equation (10) this is the

variety of homomorphisms that can be conjugated, in G, to some representation insidethe maximal torus T ⊂ G (and similarly for πH : Hom0(Zr, H) →M0

rH). The fibrationof algebraic groups F → G→ H, induces the following commutative diagram:

Hom(Zr, F ) ↪→ Hom(Zr, T ) � Hom(Zr, T ′)f↓ ↓

Hom(Zr, F ) ↪→ Hom0(Zr, G)φ� Hom0(Zr, H)

↓ π ↓ πHM0

rG � M0rH,

3We thank S. Lawton for calling our attention to the recent preprint [St], where the Poincare polynomialof analogous spaces for compact Lie groups is computed by quite different methods.


where T ′ := T/F is a maximal torus of H, and the surjections on the two top rows arediscrete fibrations (and finite etale morphisms). Since F is central, conjugating by G orby H are equivalent, so that M0

rG∼= Hom0(Zr, G)/H, and the map φ is H-equivariant,

we obtain isomorphisms:

M0rH∼= Hom0(Zr, H)/H ∼=

(Hom0(Zr, G)/F r

)/H ∼=M0

rG/Fr

because the actions of F r ∼= Hom(Zr, F ) and conjugation by H on commute. Hence, asan algebraic quotient by a finite group, if M0

rG is normal, then so is M0rF . �

Remark 5.6. If the action of F r ∼= Hom(Zr, F ) on M0rG was free, the quotient M0

rG →M0

rH would also be etale, and the converse of Lemma (5.5) would be valid (see e.g.[Dr, Theorem 4.4 (i)]). However, this is not the case for the isogeny Spin(n,C) →SO(n,C) (Spin(n,C) is the complexification of the universal cover Spin(n)→ SO(n), ofthe compact group SO(n)). The normality ofM0

rSpin(n,C) and ofM0rG for exceptional

groups G is known to be a difficult problem (see Sikora [Si2, Problem 2.3]).

For a complex reductive group G, let DG = [G,G] denote its derived group, which isa semisimple group. Let us call classical semisimple group to a group G that is a directproduct of groups of the three classical families: SL(n,C), Sp(n,C) and SO(n,C), n ∈ N.

Lemma 5.7. If DG is a classical semisimple group, then M0rG is a normal variety.

Proof. Sikora proved that, when G = SL(n,C), Sp(n,C) or SO(n,C), there are algebraicisomorphisms M0

rG∼= T r/W [Si2]. So for G in these 3 families, M0

rG is normal. It isclear that if G = G1 × G2 then the maximal torus is also a product, and that M0

rG =M0

rG1 ×M0rG2. Thus, M0

rG is normal for any classical semisimple group G.Finally, the result follows from Lemma 5.5, by taking finite quotients. Indeed, by the

central isogeny theorem, any reductive group G is a finite central quotient of a productof its derived group DG with a torus T (and clearly MrT =M0

rT∼= T r). �

Since the hypothesis of Lemma 5.7 implies that M?G = M0rG, we have proved the

following.

Theorem 5.8. Let r ≥ 1, and let G be a reductive group whose derived group is a classicalgroup. Then, the mixed Hodge polynomial of M0

rG 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 M0rG.

5.3. The GL(n,C) and SL(n,C) cases. The case of G = GL(n,C) is instructive, wherethe Weyl group is just the symmetric group, denoted by Sn. If X is a variety, we denoteits n-fold symmetric product by X(n) or by Symn(X) = Xn/Sn. As a set, Symn(X) is theset of unordered n-tuples of (not necessarily distinct) elements of X.

Proposition 5.10. Let G = GL(n,C), and let T ∼= (C∗)n denote a maximal torus of G.Then MrG =M0

rG =M?rG and we have isomorphisms of affine algebraic varieties

MrG ∼= T r/Sn ∼= Symn(C∗)r.

Proof. In [FL, Cor 5.14], it was shown that MrG = Hom(Zr, GL(n,C))//GL(n,C) is anirreducible variety (It is also path connected, given the strong deformation retractionfrom MrG to the path connected compact space Hom(Zr, U(n))/U(n), see [KS, FL]).Since M0

rG is irreducible of the same dimension (by [Si2, Thm 2.1(1)]), MrG = M0rG.

Moreover,M0rG is normal, hence isomorphic toM?

rG, and soMrG ∼=M?rG∼= T r/W , by

Sikora’s results in [Si2, Thm 2.1(2)-(3)]). Since W ∼= Sn acts diagonally, we have finallyT r/W = ((C∗)n)r/Sn ∼= ((C∗)r)n/Sn = Symn(C∗)r. �


We now turn to the proof of Theorem 1.2, on the SL(n,C)-character variety of Γ = Zr,and start with the case G = GL(n,C). Let Mσ denote a n × n permutation matrix (insome basis) corresponding to σ ∈ Sn and let In be the n× n identity matrix.

Proposition 5.11. Let G = GL(n,C). Then, MrG is round and its mixed Hodge polyno-mial is given by

µMrG (t, u, v) =1

n!

∑σ∈Sn

[det (In + tuvMσ)]r .

Proof. This formula is a direct application of Proposition 5.2, since the maximal torus isT ∼= (C∗)n, W ∼= Sn acts by permutation, and |Sn| = n!. So, the automorphism Aσ onH1(T,C) ∼= Cn acts by permutation, given by the matrix Mσ. �

Remark 5.12. There is a strong deformation retract fromMrG ∼= Symn(C∗)r to (S1)r/Sn ∼=Symn(S1)r (see [FL]) which is the space of n (unordered) points on the compact r-torus(S1)r. So our results relate also to the study of cohomology of so-called configurationspaces on compact Lie groups.

We now provide an even more concrete formula, and better adapted to computer cal-culations, using the relation between conjugacy classes of permutations and partitions ofa natural number n, to compute the above determinants.

For this, we set up some notation. Let n ∈ N and Pn be the set of partitions of n. Wedenote by n a general partition in Pn and write it as

n = [1a12a2 · · ·nan ]

where aj ≥ 0 denotes the number of parts of n of size j = 1, · · · , n; then, of coursen =

∑nj=1 j aj.

Theorem 5.13. Let G = GL(n,C), x = uv and n = [1a12a2 · · ·nan ] ∈ Pn. The mixedHodge polynomial of MrG is given by

µMrG (t, x) =∑n∈Pn

n∏j=1

(1− (−tx)j

)ajraj! jaj

.

Proof. To compute the determinant in Proposition 5.11, recall that any permutationσ ∈ Sn can be written as a product of disjoint cycles (including cycles of length 1),whose lengths provide a partition of n, say n(σ) = [1a12a2 · · ·nan ]. Moreover, any twopermutations are conjugated if and only if they give rise to the same partition, so theconjugation class of σ uniquely determines the non-negative integers a1, · · · , an. If σ is afull cycle σ = (1 · · ·n) ∈ Sn, and Mσ a corresponding matrix, by computing in a standardbasis, we easily obtain the conjugation invariant expression det(In − λMσ) = 1− λn. So,for a general permutation σ ∈ Sn with cycles given by the partition n(σ) we have

det(In − λMσ) =n∏j=1

(1− λj)aj .

Now, let cn(σ) be the size of the conjugacy class of the permutation σ, as a subset of Sn.Then the formula of Proposition 5.11, with λ = −tuv = −tx, becomes:

µMrG (t, x) =1

n!

∑σ∈Sn

[det (In − (−tx)Mσ)]r

=1

n!

∑n∈Pn

cn

n∏j=1

(1− (−tx)j)ajr,


where we replaced the sum over permutations by the sum over partitions n (each repeatedcn times). The result then follows from the well known formula

cnn!

=∏n

j=11

aj ! jaj. �

Remark 5.14. SinceMrG is an orbifold of dimension nr, it satisfies the Poincare Dualityfor mixed Hodge structures (4.4(1)), and we compute:

µcMrG (t, x) = (−t)nr∑n∈Pn

n∏j=1

((−tx)j − 1)ajr

aj! jaj.(13)

We now obtain the mixed Hodge polynomial forMrSL(n,C), by relating it toMrGL(n,C).

Theorem 5.15. The mixed Hodge polynomials of the free abelian character varieties ofGL(n,C) and SL(n,C) are related by µMrGL(n,C) (t, x) = (1 + tx)r µMrSL(n,C) (t, x), giv-ing:

µMrSL(n,C) (t, x) =∑n∈Pn

1

(1 + tx)r

n∏j=1

(1− (−tx)j)ajr

aj! jaj.

Proof. In contrast to the projection GL(n,C) → PGL(n,C) := GL(n,C)/C∗ there is noalgebraic map GL(n,C)→ SL(n,C) that commutes with the Weyl group action. So, weneed to resort to the equivariant framework. We will actually prove a stronger equality:

µSnT r (t, x) = (1 + tx)r µSn

T r(t, x) ,

where T ∼= (C∗)n and T = {z ∈ T | z1 · · · zn = 1} are the maximal torus of GL(n,C),and of SL(n,C), respectively, and the Sn action is the natural permutaion action on thecoordinates. From Corollary 4.4(2), Proposition 5.1, and from the irreducibility of thesecharacter varieties, the theorem will follow. Using the multiplicativity of the equivariantpolynomials (Proposition 4.2(2)) it suffices to show this for r = 1. Consider the fibrationof quasi-projective varieties

T −→ Tπ−→ C∗,

where π (z1, . . . , zn) = z1 · · · zn. By considering the trivial action on C∗, this is a fibration

of Sn-varieties with trivial monodromy, since it is in fact a T -principal bundle (and Tis a connected Lie group). Then Theorem A.1 gives us an equality of the equivariantE-polynomials:

ESn(T ) = ESn(T )E(C∗) = ESn(T )(1− x).

Finally, the desired formula comes from the relations in Proposition 3.9, since all varietiesin consideration are round. �

Now, we turn to the computation of some Ec-polynomials, which relate to some formulasobtained in [LM].

Corollary 5.16. For r ∈ N and G = GL(n,C) we have

EcMrG(x) =

∑n∈Pn

n∏j=1

(xj − 1)ajr

aj! jaj,

with n = [1a1 · · ·nan ] ∈ Pn, and χ (MrG) = 0.

Proof. The formula for EcMrG

(x) follows directly from Equation (13) with t = −1. Giventhe previous Theorem, the vanishing of the Euler characteristic is clear, since all factorsxj − 1 in Ec

MrGvanish when x = 1, and χ(MrG) = Ec

MrG(1) by Remark 3.10(1). �

We now prove Theorem 1.2.


Theorem 5.17. Let G = SL(n,C). Then, we have χ(MrG) = nr−1 and

EcMrG(x) =

∑n∈Pn

1

δ(n)pn(x)r,

where δ(n) :=∏n

j=1 aj! jaj , and pn(x) = 1

x−1

∏nj=1(xj − 1)aj .

Proof. The formula for EcMrG

(x) follows immediately from Theorem 5.15 and Corollary5.16. For the Euler characteristic, we need to compute Ec

MrG(1). First note that pn(x)

is a polynomial and can be factorized as pn(x) = (x − 1)mh(x) with h ∈ Z[x] and m =(∑nj=1 aj

)− 1. So, pn(1) = 0 unless

∑nj=1 aj = 1. The only partition with

∑nj=1 aj = 1

is n = [n1] (just one part, of length n), which corresponds to a cyclic permutation suchas (1 · · ·n) ∈ Sn. The size of its conjugacy class is cn = (n− 1)! and we get

χ (MrG) = EcMrG(1) =

1

n!(n− 1)! lim

x→1

(xn − 1

x− 1

)r=

1

nnr = nr−1,

as we wanted to show. �

Remark 5.18. The Ec-polynomials ofMrG for SL(2,C) and SL(3,C) are already presentin [LM]. For n ≥ 4 these formulas are new, and can also be upgraded to mixed Hodgepolynomials by using Remark 3.10(1) and Poincare duality.

Example 5.19. The following table gives the explicit values of δ(n) and pn(x) up to n = 5(in each row, the ordering is preserved). All the formulas can be easily implemented inthe available computer software packages (in this paper, most of our calculations wereperformed with GAP). For simplicity, the notation [12] refers to a partition of n = 3 withtwo cycles: one of length 1, another of length 2 (not a cycle of length twelve).

n |Pn| n δ(n) pn(x)

2 2 [2]; [12] 2; 2 x+ 1; x− 1

3 3[3];

[12]; [13]3;

2; 6x2 + x+ 1;

x2 − 1; (x− 1)2

4 5[4];

[13]; [22];[122]; [14]

4;3; 8;4; 24

x3 + x2 + x+ 1;x3 − 1; (x2 − 1) (x+ 1) ;

(x− 1)2 (x+ 1) ; (x− 1)3

5 7[5]; [14];

[123]; [23];[122]; [132]; [15]

5; 4;6; 6;

8; 12; 120

x5−1x−1

; x4 − 1;(x3 − 1) (x− 1) ; (x3 − 1) (x+ 1) ;

(x2 − 1)2

; (x− 1)3 (x+ 1); (x− 1)4

For example, with n = 4, the table immediately gives

Ec(x) =1

4(x3+x2+x+1)r+

1

3(x3−1)r+

1

8(x2−1)r(x+1)r+

1

4(x−1)2r(x+1)r+

1

24(x−1)3r,

for any r ≥ 1.

5.4. Symmetric products and Cheah’s formula. So far, our approach to Hodge numbersfor the character varietiesMrG, for G = GL(n,C) and SL(n,C) is well adapted to fixingn ∈ N, and let r be arbitrary, as we can see from theorems 5.13 and 5.16. On the otherhand, since the GL(n,C)-character varieties of Zr are symmetric products

MrG =MrGL(n,C) = Symn(C∗)r,as in Proposition 5.10, we can apply a formula of J. Cheah [Ch] for the mixed Hodgenumbers of symmetric products. Indeed, this will lead to an “orthogonal” approach: byfixing small values for r ∈ N, we obtain simple formulas valid for all n ∈ N.


Let X be a quasi-projective variety with given compactly supported Hodge numbershk,p,qc (X). Cheah’s formula gives the generating function of the mixed Hodge polynomialsof all symmetric products X(n) = SymnX is (see [Ch]):

(14)∑n≥0

µcX(n)(t, u, v) zn =∏p,q,k

(1− (−1)kupvqtkz

)(−1)k+1hk,p,qc (X).

We start by observing that, for varieties satisfying Poincare duality, Cheah’s formulastays unaffected when passing from µc to µ and from hk,p,qc to hk,p,q.

Proposition 5.20. Let X satisfy Poincare duality. Then∑n≥0

µX(n)(t, u, v) zn =∏p,q,k


)(−1)k+1hk,p,q(X).

Proof. This is a simple calculation. Let X have complex dimension d. From the relationbetween µX and µcX , in Remark 3.10(1), Cheah’s formula (14) is equivalent to:∑

n≥0

µX(n)(t−1, u−1, v−1) (t2uv)ndzn =∏p,q,k


)(−1)k+1h2d−k,d−p,d−q(X)

Now, changing the indices (k, p, q) to (k′, p′, q′) = (2d − k, d − p, d − q), which preservesthe parity of k, we obtain∑n≥0

µX(n)

(1

t,

1

u,

1

v

)((t2uv)dz

)n=∏p′,q′,k′

(1− (−1)k

′u−p

′v−q

′t−k

′(t2dudvdz)

)(−1)k′+1hk

′,p′,q′ (X)

which is clearly equivalent to the desired formula, under the substitution:(t−1, u−1, v−1, t2dudvdz) 7→ (t, u, v, z). �

For round varieties, as before, all products reduce to a single index.

Proposition 5.21. Let X be a round variety of dimension d satisfying Poincare duality.Then ∑

n≥0

µX(n)(t, u, v) zn =∏k

(1− (−tuv)kz

)(−1)k+1hk,k,k(X).

Proof. This is immediate from Proposition (5.20), since the only non-zero Hodge numbersof a round variety X are hk,k,k(X), for some k. �

We are now ready to apply this formula toMrGL(n,C). Since this space is Symn(C∗)r,we should consider X = (C∗)r.

Corollary 5.22. Let G = GL(n,C). Then:∑n≥0

µMrG(t, u, v) zn =∏k≥0

(1− (−tuv)kz

)(−1)k+1(rk) =

∏k odd

(1 + (tuv)kz

)(rk)∏

k even (1− (tuv)kz)(rk).

Proof. Letting X = (C∗)r, d = r = dimX, and hk,k,k(X) =(rk

), 0 ≤ k ≤ r, the proof is

immediate from Proposition (5.21). �

Example 5.23. The simplest example of this formula is when r = 1 [For r = 0, Symn(C∗)0

is a single point]. In this case, X = C∗ and(

10

)=(

11

)= 1, so we expand the right

hand-side as a power series in z, as:

1 + tuvz

1− z= 1 +

∑n≥1

(1 + uvt) zn

In particular, µM1GL(n,C)(t, u, v) = 1 + tuv, for n ≥ 1, which agrees with the fact thatM1GL(n,C) = Symn(C∗) ∼= Cn−1 × C∗ has the same Hodge structure than C∗.


The case with r = 2 is an interesting result in itself.

Proposition 5.24. Let G = GL(n,C), r = 2 and n ≥ 1. Then:

µM2G(t, u, v) = (1 + tuv)2(1 + (tuv)2 + ...+ (tuv)2n−2),

PM2G(t) = (1 + t)2(1 + t2 + · · ·+ t2n−2) and χ(M2G) = 0.

Proof. We now have(

22

)=(

20

)= 1 and

(21

)= 2 so we expand the right hand-side of

Corollary 5.22 as a power series in z, writing λ = tuv for simplicity:

(1 + λz)2

(1− z)(1− λ2z)= 1 +

(1 + λ)2

1− λ2

(1

1− z− 1

1− λ2z

)= 1 + (1 + λ)2

∑n≥0

(1 + λ2 + · · ·+ λ2n−2

)zn.

This gives the desired formulas for M2GL(n,C), with λ = tuv, with uv = 1 for PM2G

and λ = −1 for the Euler characteristic. �

The next corollary follows immediately from Theorem 5.15.

Corollary 5.25. Let G = SL(n,C), r = 2 and n ≥ 1. Then:

µM2G(t, x) = 1 + (tx)2 + ...+ (tx)2n−2,

PM2G(t) = (1 + t2 + · · ·+ t2n−2) and χ(M2G) = n.

Remark 5.26. The equality of Poincare polynomials PM2SL(n,C) = PPn−1C

is not a coinci-

dence. In fact, by non-abelian Hodge correspondence, M2SL(n,C) is diffeomorphic tothe cotangent bundle of the projective space Pn−1

C parametrizing semistable bundles overan elliptic curve of rank n and trivial determinant (see [BF, Tu]).

5.5. A combinatorial identity. We finish the article with an interesting purely combina-torial identity. We could not find out whether this identity was noticed before. Recallthat In is the identity n× n matrix and Mσ a permutation matrix associated to σ ∈ Sn.

Theorem 5.27. Fix r ∈ N0. Then, for formal variables x, z (or considering x, z ∈ C in asmall disc around the origin) we have:∏

k≥0

(1− xkz

)(−1)k+1(rk) =

∑n≥0

∑σ∈Sn

zn

n!det(In − xMσ)r.

Proof. Putting together Corollary 5.22 and the formula for µMrGL(n,C) in Proposition 5.11we obtain ∑

n≥0

1

n!

∑σ∈Sn

[det (In + tuvMσ)]r zn =∏k≥0

(1− (−tuv)kz

)(−1)k+1(rk)

which becomes the desired identity, by setting x = −tuv. Note that for r = 0 the formulais still valid and reduces to the geometric series. The formula holds also for z, x ∈ Cwhere the series converges. We readily check that convergence holds whenever |x| < 1and |z| < 2−r, using the bound, valid for |x| < 1,∑

σ∈Sn

n∏j=1

(1− xj)ajr <∑σ∈Sn

2|σ|r < n! 2nr,

where |σ| =∑

j aj denotes the number of cycles of σ, with aj parts of size j, as in theproof of Theorem 5.13. �


Appendix A. Multiplicativity of the E-polynomial under Fibrations

In this appendix, we prove a multiplicative property of the E-polynomial under fi-brations, used in Theorem 5.15. This is a consequence of the fact that the Leray-Serrespectral sequence is a spectral sequence of mixed Hodge structures.

A.1. E-polynomials of fibrations. It is well known that for an algebraic fibration of alge-braic varieties

Z −→ Eπ−→ B

the Poincare polynomials do not behave multiplicatively, in general. Then, a fortiori, amultiplicative property is not expected for the Hodge-Deligne polyomial. But when allinvolved varieties are smooth and the associated monodromy is trivial, this property turnsout to be valid for the E-polynomials. Moreover, if there is a finite group F acting onthe three varieties, and the involved maps respect the action - an equivariant fibration ofF -varieties -, one gets a multiplicative formula for their equivariant polynomials, underthe assumption that the higher direct images sheaves Rjπ∗CE (associated to the presheafU 7→ Hj (π−1U,CE) for U ⊂ B) are constant, where CE is the constant sheaf on E.

Theorem A.1. Let F be a finite group and consider an algebraic fibration between smoothcomplex algebraic quasi-projective varieties

Z // Eπ // B

(not necessarily locally trivial in the Zariski topology). Suppose also that this is a fibra-tion of F -varieties (all spaces are F -varieties and the maps are F -equivariant). If Z isconnected and Rjπ∗CE are constant for every j, then

EFE (u, v) = EF

Z (u, v) � EFB (u, v) .

Proof. The non-equivariant version of this result is the content of Proposition 2.4 in [LMN],where it is used to calculate the Serre polynomials of certain twisted character varieties.We detail the argument here, for the reader’s convenience. Firstly, assume that the F -action is trivial on the three spaces. The Leray-Serre spectral sequence of the fibration is asequence of mixed Hodge structures ([PS, Theorem 6.5]) and it is proved in [DL, Theorem

6.1] that under the given assumptions, its second page Ea,b2 admits an isomorphism

Ea,b2 ' Ha (B) �Hb (Z) ,

which is actually an isomorphism of mixed Hodge structures. In particular, we get anequality between their respective graded pieces:

GrpFGrWp+qE

a,b2 =

⊕p′+p′′=p

⊕q′+q′′=q

Grp′

FGrWp′+q′H

a (B) �Grp′′

F GrWp′′+q′′H

b (Z)

=⊕

p′+p′′=p

⊕q′+q′′=q

Ha,p′,q′(B) �Hb,p′′,q′′(Z)(15)

Using once more the fact that this is a spectral sequence of mixed Hodge structures, weget one spectral sequence of vector spaces for each pair (p, q):

E(p, q)a,b2 := GrpFGrWp+qE

a,b2 ⇒ GrpFGr

Wp+qH

a+b (E) .

Now set

P(p,q) (t) :=∑k≥0

dim

(⊕a+b=k

E (p, q)a,b2

)tk.


Given that ⊕a+b=kE(p, q)a,b2 ⇒ ⊕a+b=kGrpFGr

Wp+qH

a+b (E), one has

P(p,q) (−1) =∑k

(−1)k dimGrpFGrWp+qH

k (E)

=∑k

(−1)k hk,p,q (E) .

So, by definition, EE (u, v) =∑

p,q P(p,q) (−1)upvq. On the other hand, using (15)

P(p,q) (−1) =∑k

∑a+b=k

∑p′+p′′=p

∑q′+q′′=q

(−1)a ha,p′,q′ (B) (−1)b hb,p

′′,q′′ (Z) ,

substituting into EE (u, v) =∑

p,q P(p,q) (−1)upvq, and switching summation order, one

gets EE (u, v) = EB (u, v)EZ (u, v), as wanted. Succinctly, the argument follows from thefact that the spectral sequence above can be seen as an equality between derived functorsand, thinking in terms of K-theory, passing to cohomology for obtaining the next sheet inthe sequence does not change an alternating sum.

Finally, to prove the equivariant version, suppose that all the cohomologies are F -modules. Then, since we have a fibration of F -varieties the associated Leray-Serre spec-tral sequence is a spectral sequence of F -modules. The associated sequences E (k,m)p,q2

are also spectral sequences of F -modules, since the graded pieces for the Hodge andweight filtration are so. To get the desired equality, it suffices to proceed as before: ineach step of the we substitute the dimension hk,p,q(·) by the corresponding F -module[GrpFGr

Wp+qH

k (·)]F

, and the operations performed in (Z,+,×) are replaced by those in

the ring (R (F ) [u, v] ,⊕,�). �

Remark A.2. The above proof follows the proof of Theorem 6.1 (ii) in [DL], where theversion for the equivariant weight polynomial is obtained (which is implied by TheoremA.1, since the weight polynomial is a specialization of the E-polynomial). We also remarkthat the study of multiplicative invariants under fibrations, goes back at least to work ofChern, Hirzebruch and Serre in the mid fifties [CHS], on the signature theorem.

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Departamento de Matematica, Faculdade de Ciencias, Univ. de Lisboa, Edf. C6, CampoGrande 1749-016 Lisboa, Portugal

Email address: [email protected]

Departamento de Matematica, ISEL - Instituto Superior de Engenharia de Lisboa, RuaConselheiro Emıdio Navarro, 1, 1959-007 Lisboa, Portugal

Email address: [email protected]

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