# Mixed Hodge structures on character varieties of nilpotent groups

Carlos Florentino  Departamento de Matemática, Faculdade de Ciências, Univ. de
Lisboa, Edf. C6, Campo Grande 1749-016 Lisboa, Portugal<EMAIL_ADDRESS>,  Sean Lawton  Department of Mathematical Sciences, George Mason University,
4400 University Drive, Fairfax, Virginia 22030, USA<EMAIL_ADDRESS>and
Jaime Silva  Departamento Matemática, ISEL - Instituto Superior de Engenharia
de Lisboa, Rua Conselheiro Emídio Navarro, 1, 1959-007 Lisboa, Portugal
<EMAIL_ADDRESS>
###### Abstract.

Let $\mathsf{Hom}^{0}(\Gamma,G)$ be the connected component of the identity of
the variety of representations of a finitely generated nilpotent group
$\Gamma$ into a connected reductive complex affine algebraic group $G$. We
determine the mixed Hodge structure on the representation variety
$\mathsf{Hom}^{0}(\Gamma,G)$ and on the character variety
$\mathsf{Hom}^{0}(\Gamma,G)/\\!\\!/G$. We obtain explicit formulae (both
closed and recursive) for the mixed Hodge polynomial of these representation
and character varieties.

###### Key words and phrases:

character variety, mixed Hodge structures, nilpotent group, Hodge polynomial

###### 2010 Mathematics Subject Classification:

Primary 32S35; Secondary 14L30

###### Contents

  1. 1 Introduction
  2. 2 Character Varieties and Mixed Hodge Structures
  3. 3 Irreducible Components
  4. 4 Mixed Hodge Structure on $\mathsf{Hom}^{0}(\Gamma,G)$
  5. 5 Mixed Hodge Structure on $\mathsf{Hom}^{0}(\Gamma,G)/\\!\\!/G$
  6. 6 Exotic Components

##  1\. Introduction

Let $K$ be a connected compact Lie group, and $\Gamma$ be a finitely generated
nilpotent group. The topology of the space of representations
$\mathsf{Hom}(\Gamma,K)$ and of its conjugation quotient space
$\mathsf{Hom}(\Gamma,K)/K$ was considered by Ramras and Stafa in [Sta, RS],
who obtained explicit formulae for the Poincaré polynomials of their identity
components $\mathsf{Hom}^{0}(\Gamma,K)$ and $\mathsf{Hom}^{0}(\Gamma,K)/K$.

Let $G$ be the complexification of $K$, and consider now the affine algebraic
varieties $\mathcal{R}_{\Gamma}G:=\mathsf{Hom}(\Gamma,G)$ and the geometric
invariant theoretic quotient by conjugation
$\mathcal{M}_{\Gamma}G:=\mathcal{R}_{\Gamma}G/\\!\\!/G$. In this article we
determine the mixed Hodge structures on the identity components
$\mathcal{R}_{\Gamma}^{0}G\subset\mathcal{R}_{\Gamma}G$ and
$\mathcal{M}_{\Gamma}^{0}G\subset\mathcal{M}_{\Gamma}G$ and compute their
mixed Hodge polynomials, generalizing the formulas obtained in [RS] and in
[FS].

We now describe more precisely our main results. A finitely generated
nilpotent group $\Gamma$ is said to have abelian rank $r$ if the torsion free
part of $\Gamma_{Ab}:=\Gamma/[\Gamma,\Gamma]$ has rank $r$. A connected
reductive complex affine algebraic group $G$ will be called a reductive
$\mathbb{C}$-group, and $T$, $W$ will stand, respectively, for a fixed maximal
torus and the Weyl group of $G$.

Recall that the mixed Hodge numbers $h^{k,p,q}(X)$ of a quasi-projective
variety $X$ are the dimensions of the $(p,q)$-summands of the natural mixed
Hodge structure (MHS) on $H^{k}(X,\mathbb{C})$. We say that $X$ is of _Hodge-
Tate type_ if $h^{k,p,q}(X)=0$ unless $p=q$.

######  Theorem 1.

Let $\Gamma$ be a finitely generated nilpotent group with abelian rank $r\geq
1$, and $G$ a reductive $\mathbb{C}$-group. Then, both
$\mathcal{M}_{\Gamma}^{0}G$ and $\mathcal{R}_{\Gamma}^{0}G$ are of Hodge-Tate
type. More concretely, the MHS on $\mathcal{M}_{\Gamma}^{0}G$ coincides with
the one of $T^{r}/W$, where $W$ acts diagonally, and the MHS of
$\mathcal{R}_{\Gamma}^{0}G$ coincides with that of $G/T\times_{W}T^{r}$.

Above, $W$ acts on $G/T\times T^{r}$ via the standard action on the
homogeneous space $G/T$ and by simultaneous conjugation on $T^{r}$. The MHS on
$G/T$ is the natural one coming from the full flag variety $G/B$, where
$B\subset G$ is a Borel subgroup.

Now, consider the mixed Hodge polynomial of the algebraic variety $X$, defined
as:

$\mu_{X}(t,u,v)=\sum_{k,p,q\geq
0}h^{k,p,q}(X)\,t^{k}u^{p}v^{q}\in\mathbb{Z}[t,u,v].$

Knowing the MHS of $\mathcal{M}_{\Gamma}^{0}G$ and $\mathcal{R}_{\Gamma}^{0}G$
allows for the explicit computation of their mixed Hodge polynomials, as
follows. Let $\mathfrak{t}$ denote the Lie algebra of the maximal torus $T$,
and recall that $W$ acts naturally on its dual $\mathfrak{t}^{*}$, as a
reflection group.

######  Theorem 2.

Let $\Gamma$ be a finitely generated nilpotent group with abelian rank $r\geq
1$, and $G$ a reductive $\mathbb{C}$-group of rank $m$. Then, we have:

(1.1)
$\mu_{\mathcal{R}_{\Gamma}^{0}G}\left(t,u,v\right)=\frac{1}{|W|}\prod_{i=1}^{m}(1-(t^{2}uv)^{d_{i}})\sum_{g\in
W}\,\frac{\det\left(I+tuv\,A_{g}\right)^{r}}{\det\left(I-t^{2}uv\,A_{g}\right)}$

and

$\mu_{\mathcal{M}_{\Gamma}^{0}G}\left(t,u,v\right)=\frac{1}{|W|}\sum_{g\in
W}\,\det\left(I+tuv\,A_{g}\right)^{r},$

where $d_{1},\ldots,d_{m}$ are the characteristic exponents of $G$, and
$A_{g}$ is the action of $g\in W$ on
$H^{1}(T,\mathbb{C})\cong\mathfrak{t}^{*}$.

The proof of these results starts by reducing the problem to the free part of
$\Gamma_{Ab}$, which by assumption is $\mathbb{Z}^{r}$, using the main
theorems of [BS]. Let $K$ be a maximal compact subgroup of $G$. Considering
the deformation retractions obtained in [FL2] for
$\mathcal{M}_{\mathbb{Z}^{r}}G$, and in [PeSo] for
$\mathcal{R}_{\mathbb{Z}^{r}}G$, we are then reduced to describing the
cohomology of the compact spaces
$\mathcal{R}_{\mathbb{Z}^{r}}^{0}K:=\mathsf{Hom}^{0}(\Gamma,K)$ and
$\mathsf{Hom}^{0}(\Gamma,K)/K$.

A priori, there is no reason to think these compact spaces have MHSs on their
cohomology groups. In [Ba] the rational cohomology ring of
$\mathcal{R}_{\mathbb{Z}^{r}}^{0}K$ is shown to be the Weyl group invariants
of $K/T_{K}\times T_{K}^{r}$ where $T_{K}=T\cap K\subset K$ is a maximal
torus. $K/T_{K}\times_{W}T_{K}^{r}$ is a desingularization of
$\mathcal{R}_{\mathbb{Z}^{r}}^{0}K$, and is homotopic to the space
$G/T\times_{W}T^{r}$. Given the natural MHSs on $G/T$, $T$, and on the
classifying space $BT$, in the context of equivariant cohomology, we conclude
that both $\mathcal{R}_{\mathbb{Z}^{r}}^{0}G$ and
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ are of Hodge-Tate type.

The formula for $\mathcal{M}_{\Gamma}^{0}G$ then follows from the one in [FS].
To get the formula for $\mathcal{R}_{\Gamma}^{0}G$ we observe, as in [RS,
Section 5], that the graded cohomology ring of $\mathcal{R}_{\mathbb{Z}^{r}}K$
is a regrading of the cohomology ring of the torus $T^{r}$. Using
representation theory, analogous to what is done in [KT], we determine the
regrading explicitly to obtain Formula (1.1).

The main results are proved in Sections 4 and 5, and a brief review of
relevant facts about mixed Hodge structures and character varieties in Section
2. In Section 3, we show that although the path-component of the identity is a
union of algebraic components and the mixed Hodge structure is determined by
the torus component (irreducible by [Sik]), there is in fact only one
irreducible component through the identity. This follows by closely analyzing
the main proof in [FL2]. Moreover, we give a description of the singular locus
of this component in the cases of classical $G$ (expanding on work of [Sik]).
The last section applies our results to examples of character and
representation varieties with “exotic components” considered in [ACG]; here,
the group $G$ is of the form $\mathrm{SL}(p,\mathbb{C})^{m}/\mathbb{Z}_{p}$
for a prime $p$.

### Acknowledgments

We thank Dan Ramras for pointing out a useful reference. Florentino was
partially supported by project QuantumG PTDC/MAT-PUR/30234/2017, FCT,
Portugal. Lawton is partially supported by a Collaboration Grant from the
Simons Foundation, and thanks IHES for hosting him in 2021 when this work was
advanced.

##  2\. Character Varieties and Mixed Hodge Structures

###  2.1. Character Varieties

Let $G$ be a connected reductive complex affine algebraic group. As mentioned
earlier, we will say $G$ is a _reductive $\mathbb{C}$-group_ in abbreviation.
Let $\Gamma$ be a finitely generated group. The set of homomorphisms
$\rho:\Gamma\to G$ has the structure of an affine algebraic variety over
$\mathbb{C}$ (not necessarily irreducible); the generators of $\Gamma$ are
translated into elements of $G$ satisfying algebraic relations determined by
the relations of $\Gamma$. Since $G$ admits a faithful representation
$G\hookrightarrow\mathrm{GL}(n,\mathbb{C})$ for some $n$, we will sometimes
refer to $\rho$ as a $G$-representation of $\Gamma$, or simply a
representation of $\Gamma$ when the context is clear.

We have two naturally defined varieties: the $G$-representation variety of
$\Gamma$,

$\mathcal{R}_{\Gamma}G:=\mathsf{Hom}(\Gamma,G),$

and the $G$-character variety of $\Gamma$,

$\mathcal{M}_{\Gamma}G:=\mathsf{Hom}\left(\Gamma,G\right)/\\!\\!/G,$

which is the affine geometric invariant theoretic (GIT) quotient under the
conjugation action of $G$ on $\mathcal{R}_{\Gamma}G$.

We endow $\mathcal{R}_{\Gamma}G$ with the Euclidean topology coming from a
choice of $r$ generators of $\Gamma$ and the natural embedding
$\mathsf{Hom}(\Gamma,G)\hookrightarrow G^{r}\subset\mathbb{C}^{rn^{2}}$, for
appropriate $n$. Hence, $\mathcal{M}_{\Gamma}G$ is naturally endowed with a
Hausdorff topology, as the GIT quotient identifies orbits whose closures
intersect (see [FL2, Theorem 2.1] for a precise statement). However, when
speaking of irreducible components we refer to the Zariski topology.

We note that $\mathcal{M}_{\Gamma}G$ is homotopic to the non-Hausdorff
(conjugation) quotient space $\mathcal{R}_{\Gamma}G/G$ by [FLR, Proposition
3.4], and so any homotopy invariant property of either
$\mathcal{M}_{\Gamma}(G)$ or $\mathcal{R}_{\Gamma}G/G$ applies to the other.

###  2.2. Mixed Hodge Structures

In this subsection we summarize facts about mixed Hodge structures; details
can be found in [De1, De2, PS, Vo]. The singular cohomology of a complex
variety $X$ is endowed with a decreasing Hodge filtration $F_{\bullet}$:

$H^{k}\left(X,\mathbb{C}\right)=F_{0}\supseteq\cdots\supseteq F_{k+1}=0$

that generalizes the same named filtration for smooth complex projective
varieties. In general, the graded pieces of this filtration do not constitute
a pure Hodge structure. However, the rational cohomology of these varieties
admits an increasing Weight filtration:

$0=W^{-1}\subseteq\cdots\subseteq W^{2k}=H^{k}\left(X,\mathbb{Q}\right),$

and the Hodge filtration induces a pure Hodge structure on the graded pieces
of its complexification, denoted $W_{\mathbb{C}}^{\bullet}$. The triple
$\left(H^{k}\left(X,\mathbb{C}\right),F_{\bullet},W_{\mathbb{C}}^{\bullet}\right)$
constitutes a mixed Hodge structure (MHS) over $\mathbb{C}$, and we denote the
graded pieces of the associated decomposition by:

$\displaystyle H^{k,p,q}\left(X,\mathbb{C}\right)$ $\displaystyle=$
$\displaystyle
Gr_{F}^{p}Gr_{p+q}^{W_{\mathbb{C}}}H^{k}\left(X,\mathbb{C}\right).$

Their dimensions, called _mixed_ Hodge numbers
$h^{k,p,q}(X):=\dim_{\mathbb{C}}H^{k,p,q}(X,\mathbb{C})$, are encoded in the
polynomial:

$\mu_{X}(t,u,v)=\sum_{k,p,q\geq
0}h^{k,p,q}(X)\,t^{k}u^{p}v^{q}\in\mathbb{Z}[t,u,v],$

called the mixed Hodge polynomial of $X$. This polynomial reduces to the
Poincaré polynomial of $X$, by setting $u=v=1$. These constructions can also
be reproduced for compactly supported cohomology, yielding a similar
decomposition into pieces denoted $H_{c}^{k,p,q}(X,\mathbb{C})$.

When the variety $X$ is smooth and projective the Hodge structure on
$H^{*}(X,\mathbb{C})$ is pure, that is: $h^{k,p,q}\neq 0\implies k=p+q$. We
are also interested in two other types of MHS that can be read from its Hodge
numbers. We say that $X$ is balanced or of Hodge-Tate type if $h^{k,p,q}\neq
0\implies p=q$. For those varieties that further satisfy $h^{k,p,q}\neq
0\implies k=p=q$ we call them round (see [FS]).

Recall that MHSs satisfy the Künneth theorem, so that, for the cartesian
product $X\times Y$ of varieties, we have:

(2.1) $\mu_{X\times Y}=\mu_{X}\,\mu_{Y}.$

Also important for this paper is the behavior of these structures under an
algebraic action of a finite group. If $F$ is a finite group acting
algebraically on a complex variety $X$, the induced action on the cohomology
respects the mixed Hodge decomposition. Moreover, one can recover the mixed
Hodge structure on the quotient by:

(2.2) $\displaystyle H^{k,p,q}(X/F,\mathbb{C})$ $\displaystyle\cong$
$\displaystyle H^{k,p,q}(X,\mathbb{C})^{F}.$

Then, the types of mixed Hodge structures on the quotient $X/F$ have similar
properties to that of $X$. In particular, if $X$ is pure, balanced or round,
respectively, so is $X/F$. The situation is even easier when $G$ is an
algebraic group and $F$ is a finite subgroup acting by left translation.

######  Lemma 3.

Let $G$ be an algebraic group and $F$ a finite subgroup. Then the MHS on $G$
and on $G/F$ coincide.

###### Proof.

This follows from the fact that the $F$-action on the MHS of $G$ is trivial,
as shown in [DiLe, §6]. Intuitively, the idea is that the action of $F$
extends to the action of a connected group. ∎

Another important invariant related to the MHS of $X$ is the $E$-polynomial,
obtained by specializing $\mu_{X}$ to $t=-1$: $E_{X}(u,v):=\mu_{X}(-1,u,v)$.
Then the Euler characteristic of $X$ is obtained as $\chi(X)=\mu_{X}(-1,1,1)$.
We will also consider the compactly supported version of $E_{X}$, called the
_Serre polynomial_ :

$E_{X}^{c}(u,v):=\sum_{k,p,q\geq
0}\left(-1\right)^{k}h_{c}^{k,p,q}(X)\,u^{p}v^{q}\in\mathbb{Z}[t,u,v],$

where $h_{c}^{k,p,q}(X)=\dim H_{c}^{k,p,q}(X,\mathbb{C})$, given its
fundamental role in the arithmetic of character varieties (see Section 5.3).

Let $K\left(\mathcal{V}ar_{\mathbb{C}}\right)$ be the Grothendieck ring of
varieties over $\mathbb{C}$. Additively, this is a ring generated by
isomorphism classes of algebraic varieties modulo the excision relation: if
$Y\hookrightarrow X$ is a closed subvariety, then in
$K\left(\mathcal{V}ar_{\mathbb{C}}\right)$ we identify:

$\displaystyle\left[X\right]$ $\displaystyle=$
$\displaystyle\left[Y\right]+\left[X\backslash Y\right].$

The product in $K\left(\mathcal{V}ar_{\mathbb{C}}\right)$ is given by
cartesian product: $\left[X\right]\cdot\left[Y\right]:=\left[X\times
Y\right]$. When calculated in the compactly supported cohomology, the
$E$-polynomial and the Euler characteristic are examples of motivic measures
\- maps from the objects of $\mathcal{V}ar_{\mathbb{C}}$ to a ring that
factors through the Grothendieck ring of varieties.

##  3\. Irreducible Components

For many groups $\Gamma$, $\mathcal{R}_{\Gamma}G$ is _not irreducible_ and/or
_not path-connected_ , and so the same happens with $\mathcal{M}_{\Gamma}G$.
Recall that path-connected algebraic varieties need not be irreducible, and
that irreducible algebraic varieties (over $\mathbb{C}$) are necessarily path-
connected.

Path-components of $\mathcal{R}_{\Gamma}G$ are sometimes related to path-
components of $\mathcal{R}_{\Gamma}K$ for a maximal compact subgroup $K\subset
G$. For example, for a finitely generated free group $F_{r}$,
$\mathcal{R}_{F_{r}}G\cong G^{r}$ and $\mathcal{R}_{F_{r}}K\cong K^{r}$ and so
there is a $\pi_{0}$-bijection by the (topological) polar decomposition:
$G\cong K\times\mathbb{R}^{n}$, for $n=\dim_{\mathbb{R}}K$. Much more non-
trivially, there is a strong deformation retraction from
$\mathcal{R}_{\Gamma}G$ to $\mathcal{R}_{\Gamma}K$ for $\Gamma$ finitely
generated and nilpotent by [Be]; see [PeSo] for the abelian case. And thus,
there is a bijection between path-components in these cases as well.

######  Example 4.

Let $\Gamma=\mathbb{Z}^{2}$ and $K=\mathrm{SO}(3)$. Suppose
$\rho\in\mathsf{Hom}(\Gamma,K)$ is given by the pair of commuting matrices
$\mbox{diag}(1,-1,-1)$ and $\mbox{diag}(-1,-1,1)$. Then, these matrices cannot
be simultaneously conjugated, within $K$, to the same maximal torus of
$\mathrm{SO}(3)$. This implies that $\mathsf{Hom}(\Gamma,K)$ is not path-
connected, since the collection of pairs that can be simultaneously conjugated
into a given maximal torus forms a disjoint path-component. Thus, by the
discussion above, $\mathsf{Hom}(\mathbb{Z}^{2},\mathrm{PGL}(2,\mathbb{C}))$ is
also not connected, as
$\mathrm{PGL}(2,\mathbb{C})\cong\mathrm{SO}(3,\mathbb{C})$ is the
complexification of $\mathrm{SO}(3)$.

Let us denote by

$\mathcal{R}_{\Gamma}^{0}G:=\mathsf{Hom}^{0}(\Gamma,G),$

and by

$\mathcal{M}_{\Gamma}^{0}G:=\mathsf{Hom}^{0}\left(\Gamma,G\right)/\\!\\!/G,$

the path-connected components of the identity representation in
$\mathcal{R}_{\Gamma}G$ and in $\mathcal{M}_{\Gamma}G$, respectively. In some
cases, $\mathcal{R}_{\Gamma}^{0}G$ and $\mathcal{M}_{\Gamma}^{0}G$ are
irreducible varieties; but they are always a finite union of irreducible
varieties.

###  3.1. The Torus Component

An interesting case is that of a finitely presentable group $\Gamma$ whose
abelianization is free, that is

$\Gamma_{Ab}:=\Gamma/[\Gamma,\Gamma]\cong\mathbb{Z}^{r},$

for some $r\in\mathbb{N}$. Examples in this class of groups include “exponent
canceling groups” (see [LaRa]) which are those that admit presentations such
that in all relations the exponents of each generator add up to zero; such as
right angled Artin groups (abbreviated RAAGs), and fundamental groups of
closed orientable surfaces.

For these groups, since $\Gamma\to\Gamma_{Ab}\cong\mathbb{Z}^{r}$ is
surjective, we can consider the following sequence:

$T^{r}\cong\mathsf{Hom}(\mathbb{Z}^{r},T)\hookrightarrow\mathsf{Hom}(\Gamma_{Ab},G)\hookrightarrow\mathsf{Hom}(\Gamma,G)\twoheadrightarrow\mathcal{M}_{\Gamma}G.$

Let us denote by $\mathcal{M}_{\Gamma}^{T}G\subset\mathcal{M}_{\Gamma}G$ the
image of the composition above and call it the _torus component_. Then,
$\mathcal{M}_{\Gamma}^{T}G$ is an irreducible component of
$\mathcal{M}_{\Gamma}G$, being the image of $T^{r}$ under a morphism.

Obviously, the identity representation ($\rho(\gamma)=e$ for all
$\gamma\in\Gamma$, $e\in G$ being the identity) belongs to
$\mathcal{M}_{\Gamma}^{T}G$ since it comes from the identity representation in
$\mathsf{Hom}(\Gamma,T)$. Since $\mathcal{M}_{\Gamma}^{T}G$ is path-connected
(being irreducible over $\mathbb{C}$), we conclude that
$\mathcal{M}_{\Gamma}^{T}G\subset\mathcal{M}_{\Gamma}^{0}G$. We observe that
there are pairs $(\Gamma,G)$ where the varieties $\mathcal{M}_{\Gamma}^{T}G$
and $\mathcal{M}_{\Gamma}^{0}G$ agree, and others where they do not.

######  Example 5.

When $G$ is abelian (we always assume connected), it is clear that
$\mathcal{M}_{\Gamma}^{T}G=\mathcal{M}_{\Gamma}^{0}G$. For an example where
they disagree, let $\Gamma=\Gamma_{\angle}$ be the “angle RAAG” associated
with a path graph with 3 vertices, considered in [FL4]. Then, even for a low
dimensional group such as $G=\mathrm{SL}(2,\mathbb{C})$ we have that
$\mathcal{M}_{\Gamma}^{0}G$ has 3 irreducible components, one being
$\mathcal{M}_{\Gamma}^{T}G$ and 2 extra ones. Moreover, for the case
$G=\mathrm{SL}(3,\mathbb{C})$ there are components in
$\mathcal{M}_{\Gamma}^{0}G$ which have higher dimension than the dimension of
$\mathcal{M}_{\Gamma}^{T}G$.

######  Remark 6.

One can also ask if the identity representation is contained in a single
irreducible component of $\mathcal{M}_{\Gamma}^{0}G$. This also fails for
$\mathcal{M}_{\Gamma_{\angle}}^{0}(\mathrm{SL}(2,\mathbb{C}))$, as shown in
[FL4].

###  3.2. The Free Abelian Case

As seen in Examples 4, 5 and Remark 6, the comparison between the varieties
$\mathcal{M}_{\Gamma}G$, $\mathcal{M}_{\Gamma}^{0}G$ and
$\mathcal{M}_{\Gamma}^{T}G$ for general $\Gamma$ and $G$ is far from being
trivial.

We now show that $\mathcal{M}_{\Gamma}^{0}G=\mathcal{M}_{\Gamma}^{T}G$ when
$\Gamma$ is free abelian, for all $G$. In this situation, we are dealing with
representations defined by elements of $G$ that pairwise commute. The
following theorem generalizes Remark 2.4 in [Sik], and completely answers a
question raised in [FL2, Problem 5.7].

######  Theorem 7.

For every $r\in\mathbb{N}$ and reductive $\mathbb{C}$-group $G$,
$\mathcal{M}_{\mathbb{Z}^{r}}^{T}G=\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$.

###### Proof.

Let $K$ be a fixed maximal compact subgroup of $G$, and let $T_{K}=T\cap K$.
Then $T_{K}\subset K$ is a maximal torus in $K$.

Tom Baird [Ba] considered the _compact character variety_

$\mathcal{N}_{\mathbb{Z}^{r}}K:=\mathsf{Hom}(\mathbb{Z}^{r},K)/K,$

and showed that:

  1. (1)

The (path) connected component of the identity
$\mathcal{N}_{\mathbb{Z}^{r}}^{0}K\subset\mathcal{N}_{\mathbb{Z}^{r}}K$
coincides with the space of conjugation classes of representations

$\mathsf{Hom}^{T_{K}}(\mathbb{Z}^{r},K)/K,$

where $\mathsf{Hom}^{T_{K}}(\mathbb{Z}^{r},K)$ denotes the representations
$\rho$ whose $r$ evaluations $\rho(\gamma_{i})$ can be simultaneously
conjugated into the maximal torus $T_{K}$.

  2. (2)

$\mathcal{N}_{\mathbb{Z}^{r}}^{0}K$ is homeomorphic to the quotient
$T_{K}^{r}/W$, where $W=N_{K}(T_{K})/T_{K}$ is the Weyl group associated to
$T_{K}$.

By [FL2, Theorem 1.1], there is a strong deformation retraction from
$\mathcal{M}_{\mathbb{Z}^{r}}G$ to $\mathcal{N}_{\mathbb{Z}^{r}}K$ which (by
continuity) restricts to one from $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ to
$\mathcal{N}_{\mathbb{Z}^{r}}^{0}K$.

Let $[\rho]\in\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$. Then there exists a
commuting tuple $(g_{1},\ldots,g_{r})$ in $G^{r}$ such that
$[\rho]=[(g_{1},...,g_{r})]$. By [FL2, Proposition 3.1] we can assume that
each element $g_{i}\in G$ is semisimple. The strong deformation retraction
(SDR), which is $G$-conjugate equivariant, provides a path $\rho_{t}$ from
this tuple to a commuting tuple in $G_{K}^{r}$ where $G_{K}=\\{gkg^{-1}\ |\
g\in G,\ k\in K\\}$. With respect to an embedding
$G\hookrightarrow\mathrm{SL}(n,\mathbb{C})$, which preserves semisimplicity,
this SDR is given by the eigenvalue retraction defined by deforming $\ell
e^{i\theta}$ to $e^{i\theta}$ by sending $\ell$ to 1. By [FL2, Lemma 3.4],
there exists a single element $g_{0}$ in $G$ that will conjugate the resulting
$r$-tuple in $G_{K}^{r}$ to be in $K^{r}$. And by Baird’s result, and the fact
that we remain in the identity component by continuity, we know there is a
single element $k_{0}$ in $K$ that we can conjugate the resulting tuple in
$K^{r}$ so it is in $T_{K}^{r}$. Let $h_{0}:=k_{0}g_{0}$, and consider the
conjugated reverse path $\psi_{t}:=h_{0}\rho_{1-t}h_{0}^{-1}$. This path
begins in $T_{K}^{r}$ and ends (by definition) in $T^{r}$. Since
$\psi_{1}=h_{0}\rho_{0}h_{0}^{-1}=(h_{0}g_{1}h_{0}^{-1},\ldots
h_{0}g_{1}h_{0}^{-1})$, we conclude that

$[\rho]=[(g_{1},...,g_{r})]=[(h_{0}g_{1}h_{0}^{-1},\ldots
h_{0}g_{1}h_{0}^{-1})]=[\psi_{1}]$

is in $T^{r}/W$, as required. ∎

##  4\. Mixed Hodge Structure on $\mathsf{Hom}^{0}(\Gamma,G)$

Here we prove the statements in Theorems 1 and 2 that concern the connected
component $\mathcal{R}_{\Gamma}^{0}G$ of the trivial representation in the
representation variety $\mathcal{R}_{\Gamma}G=\mathsf{Hom}(\Gamma,G)$.

Let us first describe the situation for $\Gamma\cong\mathbb{Z}^{r}$. Consider,
as in the proof of Theorem 7, the compact character variety

$\mathcal{N}_{\mathbb{Z}^{r}}K=\mathsf{Hom}(\mathbb{Z}^{r},K)/K,$

where $K$ is a fixed maximal compact subgroup of $G$. Recall our convention
that $T_{K}=T\cap K$ where $T$ is a maximal torus in $G$. Baird [Ba] showed
that the isomorphism $\mathcal{N}_{\mathbb{Z}^{r}}^{0}K\cong T_{K}^{r}/W$, is
part of a natural $K$-equivariant commutative diagram:

$\textstyle{(K/T_{K})\times_{W}T_{K}^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{K}}$$\scriptstyle{\varphi_{K}}$$\textstyle{\mathsf{Hom}^{0}(\mathbb{Z}^{r},K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{K}}$$\textstyle{T_{K}^{r}/W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathcal{N}_{\mathbb{Z}^{r}}^{0}K,}$

where $\varphi_{K}$ is a desingularization of
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},K)$ which induces an isomorphism in
cohomology, and the vertical maps are the quotient maps by $K$-conjugation.

Passing to the complexification, there is an analogous $G$-equivariant
commutative diagram:

$\textstyle{(G/T)\times_{W}T^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G}}$$\scriptstyle{\varphi_{G}}$$\textstyle{\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G}}$$\textstyle{T^{r}/W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\chi}$$\textstyle{\mathcal{M}_{\mathbb{Z}^{r}}^{0}G.}$

There are some notable differences from the compact case:

  1. (1)

$\chi$ is bijective, birational, and a normalization map (Corollary 22),

  2. (2)

$\chi$ is not generally known to be an isomorphism, although it is when $G$ is
of classical type (Corollary 23),

  3. (3)

$\varphi_{G}$ is not even surjective, let alone a desingularization map (we
will say more about this below),

  4. (4)

$\pi_{G}$ is the GIT quotient map (with respect to the $G$-conjugation
action).

Despite these differences, we will show that the mixed Hodge structures of
$T^{r}/W$ and $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ coincide, as do those of
$(G/T)\times_{W}T^{r}$ and $\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)$.

###  4.1. Mixed Hodge Structures on a Smooth Model of
$\mathcal{R}_{\mathbb{Z}^{r}}^{0}G$

The above discussion suggests to consider the smooth irreducible algebraic
variety:

$S_{r}G:=(G/T)\times_{W}T^{r},$

whose MHS we now determine. The natural MHS on $G/T$ is the one of the full
flag variety $G/B$, where $B$ is a Borel subgroup, which has well-known
cohomology. Indeed, it is a classical fact that there is an identification
$K/T_{K}\cong G/B$. On the other hand, $K/T_{K}\hookrightarrow G/T$ is a
strong deformation retraction (see for example [BFLL, Theorem 10]), which
provides isomorphisms of cohomology spaces:

$H^{*}(G/B)\cong H^{*}(K/T_{K})\cong H^{*}(G/T).$

Since $T$ is contained in a certain Borel subgroup, there is a surjective
algebraic map $\varphi:G/T\to G/B$ which upgrades the above isomorphism to an
isomorphism of MHSs: $H^{*}(G/B)\cong H^{*}(G/T)$.

######  Theorem 8.

Let $m$ be the rank of $G$ and $d_{1},\ldots,d_{m}$ be its characteristic
exponents. The variety $S_{r}G$ is of Hodge-Tate type and its mixed Hodge
polynomial is given by:

$\mu_{S_{r}G}\left(t,u,v\right)=\frac{1}{|W|}\prod_{i=1}^{m}(1-(t^{2}uv)^{d_{i}})\sum_{g\in
W}\,\frac{\det\left(I+tuv\,A_{g}\right)^{r}}{\det\left(I-t^{2}uv\,A_{g}\right)}.$

###### Proof.

Since mixed Hodge structures respect the Künneth formula, from
$H^{*}(G/B)\cong H^{*}(G/T)$, we get an isomorphism:

$H^{*}(S_{r}G)\cong[H^{*}(G/T)\otimes H^{*}(T^{r})]^{W}\cong[H^{*}(G/B)\otimes
H^{*}(T^{r})]^{W},$

of mixed Hodge structures, where the superscript means that we are considering
the $W$-invariant subspace. Since the full flag variety $G/B$ is smooth and
projective, its cohomology has a pure Hodge structure. Moreover, there is an
isomorphism

$H^{*}(G/B)\cong H^{*}(BT)_{W}$

where $BT\cong(BS^{1})^{m}$ is the classifying space of $T$, and
$H^{*}(BT)_{W}$ is the algebra of co-invariants under the $W$-action on
$H^{*}(BT)$. Also, $H^{*}(BT)$ is a polynomial ring
$\mathbb{C}[x_{1},\ldots,x_{m}]$ where each $x_{i}$ has triple grading
$(2,1,1)$, since $BT$ can be identified with $(\mathbb{C}P^{\infty})^{m}$ (in
particular, it has pure cohomology). By a classical theorem of Borel (see [Re]
for a modern treatment), there is an isomorphism:

$H^{*}(BT)_{W}\cong\mathbb{C}[x_{1},\ldots,x_{m}]/(\sigma_{1},\ldots,\sigma_{m}),$

where the $\sigma_{i}$ are the homogeneous generators of the ring of
$W$-invariants $H^{*}(BT)^{W}$, with degrees $(2d_{i},d_{i},d_{i})$.

From the above, and the fact that $\sigma_{1},\ldots,\sigma_{m}$ are
$W$-invariants, we obtain:

$H^{*}(S_{r}G)\cong[H^{*}(G/B)\otimes H^{*}(T^{r})]^{W}\cong[H^{*}(BT)\otimes
H^{*}(T^{r})]^{W}/(\sigma_{1},\ldots,\sigma_{m}).$

Now, the mixed Hodge polynomial $\mu_{X}(t,u,v)$ of a variety $X$ is the
Hilbert series of its cohomology $H^{*}(X)$ with the triple grading given by
its mixed Hodge structure. Denote by $\mathfrak{H}(A)$ the Hilbert series of a
graded algebra $A$, in the variable $x$. It is a standard result that, if
$a\in A$ is not a zero divisor, then

$\mathfrak{H}(A/(a))=\mathfrak{H}(A)\,(1-x^{d}),$

where $d$ is the degree of $a$. Applied to our case, and since
$\sigma_{1},\ldots,\sigma_{m}$ are not zero divisors, we get the equality of
Hilbert series in the three variables $t,u,v$:

$\mathfrak{H}(H^{*}(S_{r}G))=\mathfrak{H}([H^{*}(BT)\otimes
H^{*}(T^{r})]^{W})\,\prod_{i=1}^{m}(1-(t^{2}uv)^{d_{i}}).$

The result thus follows from:

(4.1) $\mathfrak{H}([H^{*}(BT)\otimes
H^{*}(T^{r})]^{W})=\frac{1}{|W|}\sum_{g\in
W}\,\frac{\det\left(I+tuv\,A_{g}\right)^{r}}{\det\left(I-t^{2}uv\,A_{g}\right)}.$

Formula (4.1) is obtained by applying Corollary 11 below with
$V_{0}=H^{2,1,1}(BT)$ and $V_{1}=\cdots=V_{r}=H^{1,1,1}(T)$, since
$H^{*}(BT)=S^{\bullet}V_{0}$ and $T^{r}$ has round cohomology generated in
degrees $(1,1,1)$:
$H^{*}(T^{r})=\wedge^{\bullet}H^{1,1,1}(T^{r})\cong(\wedge^{\bullet}H^{1,1,1}(T))^{\otimes
r}$. ∎

######  Remark 9.

The characteristic exponents referenced in Theorem 8 coincide with the ones
for a maximal compact $K\subset G$. Therefore, these are well-known for all
simple $G$ (see [RS, Table 1] or the table in [KT, Page 7]).

Recall some definitions and facts from the theory of representations of finite
groups. If $V=\oplus_{k\geq 0}V^{k}$ is a graded $\mathbb{C}$-vector space
(possibly infinite dimensional, but with finite dimensional summands), and
$g:V\to V$ is a linear map that preserves the grading, define the _graded-
character_ of $g$ by:

$\chi_{g}(V):=\sum_{k\geq 0}\text{tr}(g|_{V^{k}})\,x^{k}\in\mathbb{C}[[x]].$

It is additive and multiplicative, under direct sums and tensor products,
respectively:

$\chi_{g}(V_{1}\oplus
V_{2})=\chi_{g}(V_{1})+\chi_{g}(V_{2}),\quad\quad\chi_{g}(V_{1}\otimes
V_{2})=\chi_{g}(V_{1})\chi_{g}(V_{2}).$

A linear map $g:V\to V$ induces linear maps on the direct sums of all
symmetric powers $S^{\bullet}V:=\oplus_{j\geq 0}S^{j}V$, and of all exterior
powers $\wedge^{\bullet}V:=\oplus_{j\geq 0}^{\dim V}\wedge^{j}V$. Note that
$S^{\bullet}V$ is graded, with elements of $S^{j}V$ and $\wedge^{j}V$ having
degree $j\delta$, when $V$ is pure of degree $\delta$. We need to consider two
important cases, whose proofs are standard (see, for instance, [JPSer, page
69]).

######  Lemma 10.

Let $V$ be a vector space, whose elements are all of degree $\delta$, and
$g:V\to V$ a linear map. Then, we have:

$\chi_{g}(S^{\bullet}V)=\frac{1}{\det(I-x^{\delta}g)},\quad\quad\chi_{g}(\wedge^{\bullet}V)=\det(I+x^{\delta}g).$

Now, suppose that a finite group $F$ acts on $V$ preserving the grading.
Recall that the Hilbert-Poincaré series of the graded vector space $V^{F}$, of
$F$-invariants in $V$, can be computed as:

(4.2) $\mathfrak{H}(V^{F})=\frac{1}{|F|}\sum_{g\in F}\chi_{g}(V).$

Since all the above constructions are valid for triply graded vector spaces,
and characters are multiplicative under tensor products, the following is an
immediate consequence of Lemma 10.

######  Corollary 11.

If $V_{i}$, $i=0,\ldots,r$ are finite dimensional representations of a finite
group $F$, then the triply graded Hilbert-Poincaré series in $t,u,v$ is:

$\mathfrak{H}([S^{\bullet}V_{0}\otimes\wedge^{\bullet}V_{1}\otimes\cdots\otimes\wedge^{\bullet}V_{r}]^{F})=\frac{1}{|F|}\sum_{g\in
F}\frac{\prod_{i=1}^{r}\det(I+t^{a_{i}}u^{b_{i}}v^{c_{i}}g)}{\det(I-t^{a_{0}}u^{b_{0}}v^{c_{0}}g)},$

where each $V_{i}$ has pure triple degree $(a_{i},b_{i},c_{i})$,
$i=0,\ldots,r$.

###  4.2. Mixed Hodge Structure on $\mathcal{R}_{\Gamma}^{0}G$ for Nilpotent
$\Gamma$.

The lower central series of a group $\Gamma$ is defined inductively by
$\Gamma_{1}:=\Gamma$, and $\Gamma_{i+1}:=[\Gamma,\Gamma_{i}]$ for $i>1$. A
group $\Gamma$ is nilpotent if the lower central series terminates to the
trivial group.

Let $\Gamma$ be a finitely generated nilpotent group. It is a general theorem
that all finitely generated nilpotent groups are finitely presentable (and
residually finite); see [Hi].

Recall that the abelianization of $\Gamma$:

$\Gamma_{Ab}:=\Gamma/[\Gamma,\Gamma],$

can be written as $Ab(\Gamma)\simeq\mathbb{Z}^{r}\oplus F$ where
$r\in\mathbb{N}_{\geq 0}$ is the abelian rank of $\Gamma$, and $F$ is a finite
abelian group. We now generalize some of the previous results to nilpotent
groups.

######  Theorem 12.

Let $\Gamma$ be a finitely generated nilpotent group with abelian rank $r\geq
1$. Then, the algebraic variety
$\mathcal{R}_{\Gamma}^{0}G=\mathsf{Hom}^{0}(\Gamma,G)$ has dimension $\dim
G+(r-1)\dim T$ and its MHS coincides with the MHS on $(G/T)\times_{W}T^{r}$.

###### Proof.

By [BS], we have
$\mathsf{Hom}^{0}(\Gamma_{Ab},K)\cong\mathsf{Hom}^{0}(\Gamma,K)$ and
consequently, from [Be], $\mathsf{Hom}^{0}(\Gamma_{Ab},G)$ is homotopic to
$\mathsf{Hom}^{0}(\Gamma,G)$. From [Ba], we know that

$\varphi_{K}:(K/T_{K})\times_{W}T_{K}^{r}\to\mathsf{Hom}^{0}(\Gamma_{Ab},K)$

is a birational surjection (in fact, a desingularization) that induces an
isomorphism in cohomology. This map is defined by
$[(gT,t_{1},...,t_{r})]_{W}\mapsto(gt_{1}g^{-1},...,gt_{r}g^{-1})$, and we can
likewise define $\varphi_{G}$ in the complex situation.

These maps come together to form the following commutative diagram:

(4.3)
$\textstyle{(G/T)\times_{W}T^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{G}}$$\textstyle{\mathsf{Hom}^{0}(\Gamma_{Ab},G)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathsf{Hom}^{0}(\Gamma,G)}$$\textstyle{(K/T_{K})\times_{W}T_{K}^{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{K}}$$\textstyle{\mathsf{Hom}^{0}(\Gamma_{Ab},K)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\mathsf{Hom}^{0}(\Gamma,K).\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Since the bottom row induces isomorphisms in cohomology, by commutativity, all
maps induce isomorphisms in cohomology. Since the upper row is formed by
algebraic maps, these induce isomorphisms of mixed Hodge structures of the
respective cohomologies. The dimension formula is clear since $\dim G/T=\dim
G-\dim T$. ∎

######  Remark 13.

Let $\Gamma_{Ab}\cong\mathbb{Z}^{r}$ with free abelian generators
$\gamma_{1},...,\gamma_{r}$. We note some properties of the map
$\varphi_{G}:(G/T)\times_{W}T^{r}\to\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)$. Let
$G_{ss}$ be the set of semisimple elements of $G$ (elements in $G$ with closed
conjugation orbits), and
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},G_{ss}):=\\{\rho\in\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)\
|\ \rho(\gamma_{i})\in G_{ss},\ 1\leq i\leq r\\}$. It is shown in [FL2] that
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},G_{ss})$ is exactly the set of
representations with closed conjugation orbits. In the identity component,
these are exactly the representations whose image can be conjugated to a fixed
maximal torus. Hence, the image of $\varphi_{G}$ is exactly the set
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},G_{ss})$. So we see that $\varphi_{G}$ is not
surjective and $\mathsf{Hom}^{0}(\mathbb{Z}^{r},G_{ss})$ is a constructible
set (not obvious a priori). Now from [PeSo] we know that
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)$ is homotopic to
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},G_{ss})$, and since $G_{ss}$ is dense in $G$
we deduce that $\mathsf{Hom}^{0}(\mathbb{Z}^{r},G_{ss})$ is dense in
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)$. Thus, $\varphi_{G}$ is dominant,
homotopically surjective, and induces a cohomological isomorphism (from
Diagram (4.3)). Since $W$ acts freely, $(G/T)\times_{W}T^{r}$ is smooth
although $\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)$ is generally singular. By Remark
28 below, the Zariski dense representations in
$\mathsf{Hom}^{0}(\mathbb{Z}^{r},G)$ are smooth points. It is easy to see that
$\varphi_{G}^{-1}(\rho)$ is a point if $\rho$ is Zariski dense (a generic
condition). Hence, $\varphi_{G}$ is birational, although, unlike its compact
analogue $\varphi_{K}$, it is not a desingularization.

######  Corollary 14.

Let $G$ be a reductive $\mathbb{C}$-group of rank $m$ and characteristic
exponents $d_{1},\ldots,d_{m}$. Let $\Gamma$ be a finitely generated nilpotent
group of abelian rank $r\geq 1$. The variety $\mathcal{R}_{\Gamma}^{0}G$ is of
Hodge-Tate type and its mixed Hodge polynomial is given by:

(4.4)
$\mu_{\mathcal{R}_{\Gamma}^{0}G}\left(t,u,v\right)=\frac{1}{|W|}\prod_{i=1}^{m}(1-(t^{2}uv)^{d_{i}})\sum_{g\in
W}\,\frac{\det\left(I+tuv\,A_{g}\right)^{r}}{\det\left(I-t^{2}uv\,A_{g}\right)}.$

###### Proof.

This follows immediately from Theorem 8 and Theorem 12. ∎

Corollaries 14 and 32 together establish Theorem 2 from the Introduction.

######  Corollary 15.

For every finitely generated nilpotent group $\Gamma$, and reductive
$\mathbb{C}$-group $G$, the Poincaré polynomial and $E$-polynomial of
$\mathcal{R}_{\Gamma}^{0}G$ are given, respectively, by:

$\displaystyle P_{t}\left(\mathcal{R}_{\Gamma}^{0}G\right)$ $\displaystyle=$
$\displaystyle\frac{1}{|W|}\prod_{i=1}^{m}(1-t^{2d_{i}})\sum_{g\in
W}\,\frac{\det\left(I+t\,A_{g}\right)^{r}}{\det\left(I-t^{2}\,A_{g}\right)}$
$\displaystyle E_{\mathcal{R}_{\Gamma}^{0}G}(u,v)$ $\displaystyle=$
$\displaystyle\frac{1}{|W|}\prod_{i=1}^{m}(1-(uv)^{d_{i}})\sum_{g\in
W}\,\det\left(I-uv\,A_{g}\right)^{r-1}$

and the Euler characteristic of $\mathcal{R}_{\Gamma}^{0}G$ vanishes.

###### Proof.

This follows by simply evaluating Formula (4.4) at $u=v=1$, for the Poincaré
polynomial, and at $t=-1$ for the $E$-polynomial. Finally, the Euler
characteristic is obtained as
$\chi(\mathcal{R}_{\Gamma}^{0}G)=E_{\mathcal{R}_{\Gamma}^{0}G}(1,1)=0$, as
$r\geq 1$. ∎

###  4.3. Some computations for classical groups

For certain classes of groups, such as $G=\mathrm{SL}(n,\mathbb{C})$ and
$G=\mathrm{GL}(n,\mathbb{C})$, the above formulas can be made more explicit.
These cases have Weyl group $S_{n}$, the symmetric group on $n$ letters. In
the $\mathrm{GL}(n,\mathbb{C})$ case, the action of a permutation $\sigma\in
S_{n}$ on the dual of the Cartan subalgebra of $\mathfrak{gl}_{n}$ can be
identified with the action on $\mathbb{C}^{n}$ by permuting the canonical
basis vectors. Therefore, $\det(I-\lambda
A_{\sigma})=\prod_{j=1}^{n}(1-\lambda^{j})^{\sigma_{j}}$, where $\sigma\in
S_{n}$ is a permutation with exactly $\sigma_{j}\geq 0$ cycles of size
$j\in\\{1,\ldots,n\\}$ (see, for example, [FS, Thm. 5.13]). The collection
$(\sigma_{1},\sigma_{2}\ldots,\sigma_{n})$ defines a partition of $n$, one
with exactly $\sigma_{j}$ parts of length $j$, and the number of permutations
$\sigma\in S_{n}$ with this cycle pattern is (see [St, 1.3.2]):

$m_{\sigma}=n!\,{\textstyle(\prod_{j=1}^{n}\sigma_{j}!\,j^{\sigma_{j}}})^{-1}.$

Since the characteristic exponents of $\mathrm{GL}(n,\mathbb{C})$ are exactly
$1,2,\ldots,n$, this leads to the following explicit formula:

$\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{GL}(n,\mathbb{C})}\left(t,u,v\right)=\prod_{i=1}^{m}(1-(t^{2}uv)^{i})\sum_{\pi\vdash
n}\,\prod_{j=1}^{n}\frac{(1-(-tuv)^{j})^{\pi_{j}r}}{{\textstyle\pi_{j}!\,j^{\pi_{j}}}(1-(t^{2}uv)^{j})^{\pi_{j}}},$

where $\pi\vdash n$ denotes a partition of $n$ with $\pi_{j}$ parts of size
$j$.

Moreover, in the $\mathrm{GL}(n,\mathbb{C})$ case, we can also derive a
recursion relation, which completely avoids the determination of partitions or
permutations. Since $\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{GL}(n,\mathbb{C})}$
depends only on $tuv$ and $t^{2}uv$, we use the substitutions $x=tuv$, and
$w=tx=t^{2}uv$.

######  Proposition 16.

Let $G=\mathrm{GL}(n,\mathbb{C})$ and write
$\mu_{n}^{r}(x,w):=\mu_{\mathcal{R}_{\Gamma}^{0}G}(t,u,v)$ for a nilpotent
group $\Gamma$, of abelian rank $r\geq 1$. Then, we have the recursion
relation:

(4.5)
$\mu_{n}^{r}(x,w)=\frac{1}{n}\sum_{k=1}^{n}f((-x)^{k},w^{k})\,c_{k}(w)\,\mu_{n-k}^{r}(x,w),$

with $f(x,w):=\frac{(1-x)^{r}}{1-w}$ and
$c_{k}(w):=\prod_{i=0}^{k-1}(1-w^{n-i})$.

###### Proof.

For fixed $r\in\mathbb{N}$, let $\phi_{n}(z,w)$ be the rational function in
variables $z,w$, defined by:

$\phi_{n}(z,w):=\frac{1}{n!}\sum_{g\in
S_{n}}\,\frac{\det\left(I-z\,A_{g}\right)^{r}}{\det\left(I-w\,A_{g}\right)},$

with $\phi_{0}(z,w)\equiv 1$. By [Fl, Thm 3.1], the generating series for
$\phi_{n}(z,w)$ is a so-called plethystic exponential:

$1+\sum_{n\geq
1}\phi_{n}(z,w)\,y^{n}=\mathsf{PE}(f(z,w)\,y):=\exp\left(\sum_{k\geq
1}f(z^{k},w^{k})\,\frac{y^{k}}{k}\right)$

with $f(z,w)=\frac{(1-z)^{r}}{1-w}$. Differentiating the above identity with
respect to $y$ we get:

$\sum_{n\geq 1}n\phi_{n}(z,w)\,y^{n-1}=\left(1+\sum_{m\geq
1}\phi_{m}(z,w)\,y^{m}\right)\left(\sum_{k\geq
1}f(z^{k},w^{k})\,y^{k-1}\right),$

which, by picking the coefficient of $y^{n}$, leads to the recurrence:

(4.6)
$\phi_{n}(z,w)=\frac{1}{n}\sum_{k=1}^{n}f(z^{k},w^{k})\,\phi_{n-k}(z,w).$

To apply this to $\mu_{n}^{r}(x,w)$ we use Equation (4.4) in the form:

$\phi_{n}(-x,w)=\frac{\mu_{n}^{r}(x,w)}{\prod_{i=1}^{n}(1-w^{i})},$

so the wanted recurrence follows by replacing $z=-x$ in Equation (4.6). ∎

In the $\mathrm{SL}(n,\mathbb{C})$ case, also with Weyl group $S_{n}$, the
action is the same permutation action, but restricted to the vector subspace
of $\mathbb{C}^{n}$ whose coordinates add up to zero. Hence, the formula for
$\det(I-\lambda A_{\pi})$ acting on dual of $\mathfrak{sl}_{n}$ is now:

(4.7) $\det(I-\lambda
A_{\pi}):=\frac{1}{1-\lambda}\prod_{j=1}^{n}(1-\lambda^{j})^{\pi_{j}},$

for a permutation $\sigma\in S_{n}$ with $\sigma_{j}$ cycles of size $j$.
Recalling that $\mathrm{SL}(n,\mathbb{C})$ is a group of rank $n-1$ with
characteristic exponents $2,3,\ldots,n$, we derive the following formula,
reflecting the fact that the
$\mathcal{R}_{\Gamma}^{0}\mathrm{GL}(n,\mathbb{C})$ and
$\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(n,\mathbb{C})$ cases only differ by a
torus.

######  Corollary 17.

Let $G=\mathrm{SL}(n,\mathbb{C})$ and $\Gamma$ be a finitely generated
nilpotent group of abelian rank $r\geq 1$. Then:

(4.8)
$\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(n,\mathbb{C})}\left(x,w\right)=\frac{1}{(1+x)^{r}}\,\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{GL}(n,\mathbb{C})}\left(x,w\right).$

######  Remark 18.

The recursion formulae in (4.5) and (4.8) have been implemented in a
Mathematica notebook available upon request.

######  Example 19.

From (4.5) and (4.8) we can quickly write down the first few cases for
$\mathrm{SL}(n,\mathbb{C})$. To obtain
$\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(n,\mathbb{C})}(t,u,v)$ one just
needs to substitute $x=tuv$ and $w=t^{2}uv$.

$\displaystyle\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(2,\mathbb{C})}$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2}}\left((1+w)(1+x)^{r}+(1-w)(1-x)^{r}\right).$
$\displaystyle\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(3,\mathbb{C})}$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{6}}(1+2w+2w^{2}+w^{3})(1+x)^{2r}+{\textstyle\frac{1}{2}}(1-w^{3})(1-x^{2})^{r}$
$\displaystyle+{\textstyle\frac{1}{3}}(1-w-w^{2}+w^{3})(1-x+x^{2})^{r}.$
$\displaystyle\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(4,\mathbb{C})}$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{24}}(1+w)(1+w+w^{2})(1+w+w^{2}+w^{3})(1+x)^{3r}$
$\displaystyle+{\textstyle\frac{1}{4}}(1+w+w^{2})(1-w^{4})(1+x)^{r}(1-x^{2})^{r}$
$\displaystyle+{\textstyle\frac{1}{8}}(1-w^{3})(1-w+w^{2}-w^{3})(1-x)^{r}(1-x^{2})^{2r}$
$\displaystyle+{\textstyle\frac{1}{3}}(1-w^{2})(1-w^{4})(1+x^{3})^{r}$
$\displaystyle+{\textstyle\frac{1}{4}}(1-w)(1-w^{2})(1-w^{3})(1+x+x^{2}+x^{3})^{r}.$

Putting $x=t$ and $w=t^{2}$ we recover the expressions for the Poincaré
polynomial in [Ba] and [RS].111Note that our first term for $n=3$ corrects the
corresponding term in [Ba, pg. 749]. Note that with $x=-1$, $w=1$ we confirm
the vanishing of the Euler characteristic. With $w=-x$ we get formulas for the
$E$-polynomial, and with $x=w=1$ (that is, $t=u=v=1$) we get:

$\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(n,\mathbb{C})}\left(1,1,1\right)=2^{(n-1)r},$

the dimension of the total cohomology of $T^{r}$, confirming that
$H^{*}(\mathcal{R}_{\Gamma}^{0}\mathrm{SL}(n,\mathbb{C}))$ is a regrading of
$H^{*}(T^{r})$.

######  Example 20.

Consider now the group $G=\mathrm{Sp}(2n,\mathbb{C})$ which has rank $n$ and
dimension $n(2n+1)$. Its Weyl group is the so-called _hyperoctahedral group_ :
the group of symmetries of the hypercube of dimension $n$, denoted $C_{n}$, of
order $|C_{n}|=2^{n}n!$. It can be described as the subgroup of permutations
of the set $S_{\pm n}:=\\{-n,\ldots,-1,1,\ldots,n\\}$ satisfying:

$\sigma\in C_{n}\subset S_{\pm
n}\quad\quad\Longleftrightarrow\quad\quad\sigma(-i)=-\sigma(i)\quad\forall
1\leq i\leq n.$

The action of $g\in C_{n}$ on the dual of the Lie algebra
$\mathfrak{sp}_{2n}\cong\mathbb{C}^{n}$ is the following natural action. If we
denote by $e_{1},\ldots,e_{n}$ the standard basis of $\mathbb{C}^{n}$, and let
$e_{-i}:=-e_{i}$, then $g\cdot e_{i}=e_{\sigma(i)}$, for all $1\leq i\leq n$,
where $g\in C_{n}$ corresponds to the permutation $\sigma\in S_{\pm n}$.

Given that $\mathrm{Sp}(2,\mathbb{C})\cong\mathrm{SL}(2,\mathbb{C})$, we
consider the next case: $n=2$. The Weyl group of $\mathrm{Sp}(4,\mathbb{C})$
is $C_{2}$, and is known to be isomorphic to the dihedral group of order $8$
(the symmetries of the square):

$C_{2}=\\{e,a,a^{2},a^{3},ba,ba^{2},ba^{3}\\},$

where $a$ acts by counter-clockwise rotation of $\frac{\pi}{2}$ (that is
$e_{1}\mapsto e_{2}\mapsto-e_{1}\mapsto-e_{2}\mapsto e_{1}$) and $b$ is the
reflection along the first coordinate axis ($e_{1}\mapsto e_{1}$ and
$e_{2}\mapsto-e_{2}$). Then, we have:

$a=\left(\begin{array}[]{cc}0&-1\\\ 1&0\end{array}\right),\quad\quad
b=\left(\begin{array}[]{cc}1&0\\\ 0&-1\end{array}\right)$

and simple computations give the following table, with
$p_{g}(\lambda)=\det(I-\lambda A_{g})$.

$g\in C_{2}$ $p_{g}(\lambda)$ $e$ $(1-\lambda)^{2}$ $a,a^{3}$ $1+\lambda^{2}$
$a^{2}$ $(1+\lambda)^{2}$ $b,ba,ba^{2},ba^{3}$ $1-\lambda^{2}$

From this, since the characteristic exponents of $\mathrm{Sp}(4,\mathbb{C})$
are $2,4$, we compute, using again $x=tuv$ and $w=t^{2}uv$:

$\displaystyle\mu_{\mathcal{R}_{\Gamma}^{0}\mathrm{Sp}(4,\mathbb{C})}$
$\displaystyle=$ $\displaystyle\frac{1}{2^{2}2!}(1-w^{2})(1-w^{4})\sum_{g\in
C_{2}}\,\frac{p_{g}(-x)^{r}}{p_{g}(w)}$ $\displaystyle=$
$\displaystyle{\textstyle\frac{1}{8}}(1-w^{2})(1-w^{4})\left({\textstyle\frac{(1+x)^{2r}}{(1-w)^{2}}+2\frac{(1+x^{2})^{r}}{1+w^{2}}+\frac{(1-x)^{2r}}{(1+w)^{2}}+4\frac{(1-x^{2})^{r}}{1-w^{2}}}\right)$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{8}}(1+w)(1+w+w^{2}+w^{3})(1+x)^{2r}+{\textstyle\frac{1}{4}}(1-w^{2})^{2}(1+x^{2})^{r}$
$\displaystyle+{\textstyle\frac{1}{8}}(1-w)(1-w+w^{2}-w^{3})(1-x)^{2r}+{\textstyle\frac{1}{2}}(1-w^{4})(1-x^{2})^{r}.$

Again, we note that with $x=t$ and $w=t^{2}$ we obtain the Poincaré
polynomial, with $w=x=1$ we obtain $2^{2r}$, and with $w=1=-x$ we obtain zero,
as expected.

###  4.4. $G$-Equivariant Cohomology of $\mathcal{R}_{\Gamma}^{0}(G)$

For a Lie group $G$, denote $G$-equivariant cohomology (over $\mathbb{C}$) by
$H_{G}$. We now resume our main setup: $G$ is a reductive $\mathbb{C}$-group,
$K$ is a maximal compact subgroup of $G$, $T$ is a maximal torus in $G$ and
$T_{K}$ is a compatible maximal torus in $K$ (so $T_{K}=T\cap K$). Again, let
$\Gamma$ be a finitely generated nilpotent group of abelian rank $r\geq 1$, so
the torsion free part of its abelianization is $\mathbb{Z}^{r}$.

Since $G$ and $K$ are homotopic, as are $\mathcal{R}_{\Gamma}^{0}(G)$ and
$\mathcal{R}_{\Gamma}(K)$, we conclude that

$H_{G}^{*}(\mathcal{R}_{\Gamma}^{0}(G))\cong
H_{K}^{*}(\mathcal{R}_{\Gamma}^{0}(K)).$

Then, from Baird’s thesis [BaT], precisely pages 39 and 55, and Corollary
7.4.4, the $G$-equivariant and $K$-equivariant maps in Diagram (4.3) imply we
have the following isomorphisms:

$\displaystyle H_{K}^{*}(\mathcal{R}_{\Gamma}^{0}(K))$ $\displaystyle\cong$
$\displaystyle H_{K}^{*}(\mathcal{R}_{\mathbb{Z}^{r}}^{0}(K))$
$\displaystyle\cong$ $\displaystyle H_{K}^{*}((K/T_{K})\times T_{K}^{r})^{W}$
$\displaystyle\cong$ $\displaystyle H_{T_{K}}^{*}(T_{K}^{r})^{W}$
$\displaystyle\cong$ $\displaystyle[H^{*}(T_{K}^{r})\otimes
H^{*}(BT_{K})]^{W}$ $\displaystyle\cong$ $\displaystyle[H^{*}(T^{r})\otimes
H^{*}(BT)]^{W}.$

We have already computed the Hilbert series of this latter ring:

$\mathfrak{H}([H^{*}(BT)\otimes H^{*}(T^{r})]^{W})=\frac{1}{|W|}\sum_{g\in
W}\,\frac{\det\left(I+tuv\,A_{g}\right)^{r}}{\det\left(I-t^{2}uv\,A_{g}\right)}.$

Thus we conclude:

######  Corollary 21.

There is a MHS on the $G$-equivariant cohomology of
$\mathcal{R}_{\Gamma}^{0}(G)$ and the $G$-equivariant mixed Hodge polynomial
is:

$\mu_{\mathcal{R}_{\Gamma}^{0}(G)}^{G}=\frac{1}{|W|}\sum_{g\in
W}\,\frac{\det\left(I+tuv\,A_{g}\right)^{r}}{\det\left(I-t^{2}uv\,A_{g}\right)}.$

##  5\. Mixed Hodge Structure on $\mathsf{Hom}^{0}(\Gamma,G)/\\!\\!/G$

Now we prove the statements in Theorems 1 and 2 on the connected component
$\mathcal{M}_{\Gamma}^{0}G$ of the trivial representation of the character
variety $\mathcal{M}_{\Gamma}G=\mathsf{Hom}(\Gamma,G)/\\!\\!/G$.

We start with the free abelian case, $\Gamma\cong\mathbb{Z}^{r}$, noting a
number of corollaries to Theorem 7.

######  Corollary 22.

$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is irreducible, and there exists a
birational bijection

$\chi:T^{r}/W\to\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$

which is the normalization map.

###### Proof.

As noted earlier, $\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$ is irreducible, and we
have shown that
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G=\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$. We also
know from [Sik] that there is a bijective birational morphism
$T^{r}/W\to\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$. The result follows since
$T^{r}/W$ is normal (since the GIT quotient of a normal variety is normal). ∎

We will say that a reductive $\mathbb{C}$-group $G$ is of classical type if
its derived subgroup $DG$ admits a central isogeny by a product of groups of
type $\mathrm{SL}(n,\mathbb{C})$, $\mathrm{Sp}(2n,\mathbb{C})$, or
$\mathrm{SO}(n,\mathbb{C})$ for varying $n$ (not necessarily all the same $n$
within the product).

######  Corollary 23.

If $G$ is of classical type, then $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is
normal and $\chi:T^{r}/W\to\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is an
isomorphism.

###### Proof.

Given $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G=\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$
this follows from [FS]. Here is a sketch of the result in [FS]. Sikora showed
the result for $\mathrm{SL}(n,\mathbb{C})$, $\mathrm{Sp}(2n,\mathbb{C})$, or
$\mathrm{SO}(n,\mathbb{C})$ in [Sik]. It is trivially true for tori. In
general, $\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G\times
H)\cong\mathcal{M}_{\mathbb{Z}^{r}}^{0}G\times\mathcal{M}_{\mathbb{Z}^{r}}^{0}H$
and also
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G/F)\cong(\mathcal{M}_{\mathbb{Z}^{r}}^{0}G)/F^{r},$
for finite central subgroups $F$. The result then follows from the central
isogeny theorem for reductive $\mathbb{C}$-groups and the facts that GIT
quotients of normal varieties are normal, and cartesian products of normal
varieties are normal. ∎

Since $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G=\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$ we
know for any $[\rho]\in\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ its image is
contained in some maximal torus which we may assume is $T$. We will say such a
representation is Zariski dense if its image is Zariski dense in $T$. We note
that every representation in the identity component is reducible; that is, its
image is contained in a proper parabolic subgroup of $G$. For many choices of
$\Gamma$, reducible representations are singular points; see for example [FL3,
GLR]. The next corollary is in contrast to this.

######  Corollary 24.

Assume $r\geq 2$, and that $[\rho]\in\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is
Zariski dense. Then

  1. (1)

$[\rho]$ is a smooth point, and

  2. (2)

the map $\chi:T^{r}/W\to\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is étale at
$[\rho]$.

###### Proof.

Since $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G=\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$,
and [Sik, Theorem 4.1] shows that if
$[\rho]\in\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$ and is Zariski dense then (1)
holds on $\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$, (1) is also true for
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$. For (2), [Sik, Theorem 4.1] shows that
the map induces an isomorphism of tangent spaces on the torus component when
$\rho$ is Zariski dense; this implies the map is étale at $[\rho]$ by (1). ∎

######  Remark 25.

If $r=1$, then we have $G/\\!\\!/G\cong T/W$ and is smooth if $DG$ is simply-
connected by [Ste] and [Bo, Proposition 3.1]. The converse is not true
however, since
$\mathrm{PSL}(2,\mathbb{C})/\\!\\!/\mathrm{PSL}(2,\mathbb{C})\cong\mathbb{C}$
is smooth.

The map $\chi:T^{r}/W\to\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is the
normalization map in general and it is an open question whether or not it is
an isomorphism in general [Sik]. We note that $\chi$ is an isomorphism if and
only if $\chi$ is étale and that holds if and only if
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is normal.

######  Corollary 26.

We have equality of Grothendieck classes:
$[T^{r}/W]=[\mathcal{M}_{\mathbb{Z}^{r}}^{0}G]$.

###### Proof.

Since $\chi$ is a bijective map, the result follows from [BB, Page 115]. ∎

######  Corollary 27.

Let $G$ be of classical type. Then the singular locus of
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is of orbifold type; that is, consists
only of finite quotient singularities.

###### Proof.

In the case that $G$ is of classical type we know that $\chi$ is an
isomorphism since $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is normal. Thus, the
singular locus of $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ is exactly the singular
locus of $T^{r}/W$. Since $T^{r}/W$ is the finite quotient of a manifold, the
result follows. ∎

######  Remark 28.

From this point-of-view, we can see easily why the Zariski dense
representations are smooth. The Zariski dense representations are tuples
$(t_{1},...,t_{r})$ that generate a Zariski dense subgroup of $T$ (most
$t_{i}$’s do this by themselves). If $w\cdot\rho=\rho$ then
$w\cdot\rho(\gamma)=\rho(\gamma)$ for all $\gamma$. Since $\rho$ is Zariski
dense we conclude that $w\cdot t=t$ for all $t\in T$. We conclude $w=1$ and so
$W$ acts freely on the set of Zariski dense representation. This shows they
are smooth points and the singular locus is contained in the non-Zariski dense
representations.

######  Remark 29.

Assume $r\geq 2$. If $\rho$ is not Zariski dense, then the identity component
of $A:=\overline{\rho(\mathbb{Z}^{r})}$ is a proper subtorus of $T$. It seems
reasonable to suppose that $A$ is contained in the fixed locus of a non-
trivial $w\in W$. The fixed loci $(T^{r})^{w}$ for $w\not=1$ are of
codimension greater than 1 since $r\geq 2$ and $(T^{w})^{0}$ is a proper
subtorus. So, in light of the Shephard-Todd Theorem [ShTo], it appears likely
that the non-Zariski dense representations are exactly the singular locus (for
$r\geq 2$).

###  5.1. The MHS on $\mathcal{M}_{\Gamma}^{0}G$

Given an isomorphism of groups $\varphi:\Gamma_{1}\to\Gamma_{2}$ there exists
a (contravariant) biregular morphism
$\varphi^{*}:\mathcal{M}_{\Gamma_{2}}G\to\mathcal{M}_{\Gamma_{1}}G$ given by
$\varphi^{*}([\rho])=[\rho\circ\varphi]$ with inverse $(\varphi^{-1})^{*}$.
Consequently, the topology of $\mathcal{M}_{\Gamma}G$ and its mixed Hodge
structure (MHS) are independent of the presentation of $\Gamma$. Hence, the
same holds for $\mathcal{M}_{\Gamma}^{0}G$, for any $\Gamma$.

Let us start with the free abelian case, $\Gamma\cong\mathbb{Z}^{r}$, where we
know that
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G=\mathcal{M}_{\mathbb{Z}^{r}}^{T}G$.

######  Theorem 30.

Let $G$ be a reductive $\mathbb{C}$-group, $T$ a maximal torus, and $W$ the
Weyl group. Then, the MHS of $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$ coincides
with the one of $T^{r}/W$ and its mixed Hodge polynomial is given by:

(5.1) $\displaystyle\mu_{\mathcal{M}_{\mathbb{Z}^{r}}^{0}G}\left(t,u,v\right)$
$\displaystyle=$ $\displaystyle\frac{1}{|W|}\sum_{g\in
W}\left[\det\left(I+tuv\,A_{g}\right)\right]^{r},$

where $A_{g}$ is the automorphism induced on $H^{1}(T,\mathbb{C})$ by $g\in
W$, and $I$ is the identity automorphism.

###### Proof.

We have the following commutative diagram with vertical arrows being strong
deformation retractions from [FL2]:

Thus, $\chi$ induces isomorphisms in cohomology and since it is an algebraic
map, these isomorphisms preserve mixed Hodge structures.

Thus, the MHS on $T^{r}/W$ and on $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G$
coincide. The formula then follows immediately from [FS]. ∎

######  Theorem 31.

Let $\Gamma$ be a finitely generated nilpotent group of abelian rank $r\geq
1$. The MHS on $\mathcal{M}_{\Gamma}^{0}G$ coincides with the MHS on
$T^{r}/W$.

###### Proof.

This follows from [BS, Corollary 1.4], where they prove isomorphisms in
cohomology given by algebraic maps. ∎

######  Corollary 32.

Let $\Gamma$ be a finitely generated nilpotent group of abelian rank $r\geq
1$. Then, for all reductive $\mathbb{C}$-groups $G$ we have:

$\displaystyle\mu_{\mathcal{M}_{\Gamma}^{0}G}\left(t,u,v\right)$
$\displaystyle=$ $\displaystyle\frac{1}{|W|}\sum_{g\in
W}\left[\det\left(I+tuv\,A_{g}\right)\right]^{r}.$

###### Proof.

This follows directly from Theorem 30 and Theorem 31. ∎

######  Remark 33.

The $G$-equivariant cohomology of the moduli space
$\mathcal{M}_{\Gamma}^{0}(G)$ is the usual cohomology since the $G$-action is
trivial on $\mathcal{M}_{\Gamma}^{0}(G)$.

###  5.2. Examples for classical groups

As in the case of representation varieties, the character varieties for
$\mathrm{GL}(n,\mathbb{C})$ and $\mathrm{SL}(n,\mathbb{C})$ also allow closed
expressions in terms of partitions $\pi$ of $n$. In [FS, Thm 5.13], it was
shown that $\mathcal{M}_{\mathbb{Z}^{r}}\mathrm{GL}(n,\mathbb{C})$ has round
cohomology, so that $x=tuv$ is the only relevant variable, and that:

$\mu_{\mathcal{M}_{\mathbb{Z}^{r}}\mathrm{GL}(n,\mathbb{C})}\left(x\right)=\mu_{\mathcal{M}_{\mathbb{Z}^{r}}\mathrm{SL}(n,\mathbb{C})}\left(x\right)\,(1+x)^{r}.$

From the present analysis, the same formulas work also for the identity
components of the character varieties of any nilpotent group $\Gamma$ with
abelianization $\mathbb{Z}^{r}$. Moreover, we can also obtain a recurrence
relation as follows.

######  Proposition 34.

Let $G=\mathrm{GL}(n,\mathbb{C})$ and write
$\nu_{n}^{r}(x):=\mu_{\mathcal{M}_{\Gamma}^{0}G}(t,u,v)$ for a nilpotent group
$\Gamma$, of abelian rank $r$. Then, with $h(x):=(1-x)^{r}$, we have:

(5.3) $\nu_{n}^{r}(x)=\frac{1}{n}\sum_{k=1}^{n}h((-x)^{k})\,\nu_{n-k}^{r}(x).$

###### Proof.

As in Proposition 16, define $\psi_{n}(z)$ to be the rational function of $z$:

$\psi_{n}(z):=\frac{1}{n!}\sum_{g\in
S_{n}}\det\left(I-z\,A_{g}\right)^{r}=\nu_{n}^{r}\left(-z\right),$

with $\psi_{0}(z)\equiv 1$. By [Fl, Thm 3.1], the generating series for the
$\psi_{n}(z)$ is now the plethystic exponential $1+\sum_{n\geq
1}\psi_{n}(z)\,y^{n}=\mathsf{PE}(h(z)\,y)$ with $h(z)=(1-z)^{r}$. As before,
the derivative with respect to $y$ now gives:

$\sum_{n\geq 1}n\psi_{n}(z)\,y^{n-1}=\left(1+\sum_{m\geq
1}\psi_{m}(z)\,y^{m}\right)\left(\sum_{k\geq 1}h(z^{k})\,y^{k-1}\right),$

which, by picking the coefficient of $y^{n}$, leads to the recurrence:

(5.4) $\psi_{n}(z)=\frac{1}{n}\sum_{k=1}^{n}h(z^{k})\,\psi_{n-k}(z),$

so the proposition follows by replacing $z=-x$ in Equation (5.4). ∎

###  5.3. Polynomial Count and Compactly Supported E-Polynomials

A spreading out of an affine variety $X$ (over $\mathbb{C}$) is a scheme
$\mathcal{X}$ over a $\mathbb{Z}$-algebra $R$ with an inclusion
$\varphi:R\hookrightarrow\mathbb{C}$ such that the extension of scalars
satisfies $\mathcal{X}_{\varphi}\cong X$. $X$ is said to have polynomial count
if there exists $P_{X}(x)\in\mathbb{Z}[x]$ and a spreading out $\mathcal{X}$
such that for all homomorphims $\phi:R\to\mathbb{F}_{q}$ to finite fields (for
all but finitely many primes $p$ so $q=p^{k}$) we have
$\\#\mathcal{X}_{\phi}(\mathbb{F}_{q})=P_{X}(q)$. Katz showed in the appendix
of [HaRo] that if $X$ has polynomial count, then the Serre polynomial
$E_{X}^{c}(u,v)$ (compactly supported $E$-polynomial) of $X$ is equal to
$P_{X}(uv)$. We note that $X$ is polynomial count if and only if its
Grothendieck class $[X]$ can be decomposed as a $\mathbb{Z}$-linear
combination of affine spaces.

If $F$ is a group acting on a set $X$ and $g\in F$, let $X^{g}:=\\{x\in X\ |\
gx=x\\}$ be the $g$-fixed locus.

######  Lemma 35.

Let $X$ be a smooth complex quasi-projective variety $($admitting a model over
$\mathbb{Z})$ and $F$ a finite group acting rationally on $X$. If $X^{g}$ are
polynomial count for all $g\in F$, then both $X$ and $X/F$ are polynomial
count.

###### Proof.

Since $X^{e}=X$ our assumption implies that $X$ is polynomial count. By
standard GIT, $X/F$ exists as a quasi-projective variety and since $X$ has an
integral model, so does $X/F$. Let $(X/F)(\mathbb{F}_{q})$ be the
$\mathbb{F}_{q}$-locus of $X/F$. The Cauchy-Frobenius-Burnside Lemma says

$|(X/F)(\mathbb{F}_{q})|=\frac{1}{|F|}\sum_{g\in F}|X^{g}(\mathbb{F}_{q})|$

since $|F|<\infty$. Since each $X^{g}$ is polynomial count, we conclude that
$X/F$ is as well. ∎

######  Theorem 36.

Let $G$ be a reductive $\mathbb{C}$-group and $r\geq 1$. Then
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G)$ is polynomial count.

###### Proof.

By Corollary 22, $\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G)$ is bijective, via an
algebraic map, to $T^{r}/W$. Thus the counting functions of these two
varieties are equal and so we show that $T^{r}/W$ is polynomial count. Since
$T$ is a (split) algebraic torus and a product of polynomial count varieties
is polynomial count, $T^{r}\cong(\mathbb{C}^{*})^{N}$, where
$N=r\mathrm{Rank}(G)$, is polynomial count with counting polynomial
$(x-1)^{N}$. Clearly $T^{r}$ is smooth and admits an integral model. Since $W$
acts diagonally by conjugation on $T^{r}$, for every $w\in W$, the $w$-fixed
locus $(T^{r})^{w}$ is an (algebraic) abelian subgroup of $T^{r}$, and so is a
product of a (split) subtorus with a finite abelian group. We conclude that
$(T^{r})^{w}$ is polynomial count for all $w\in W$. The result now follows by
Lemma 35. ∎

######  Theorem 37.

The compactly supported mixed Hodge polynomial of
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G)$ is

$\mu_{\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G)}^{c}(t,u,v)=\frac{t^{r\dim
T}}{|W|}\sum_{g\in W}\left[\det(tuvI+A_{g})\right]^{r}.$

###### Proof.

From Theorem 30, the mixed Hodge structure of
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G)$ coincides with the one of its
normalization $T^{r}/W$. The result is therefore an immediate consequence of
the formula in [FS, Remark 5.3], which reflects the fact that $T^{r}/W$
satisfies Poincaré duality for MHSs. ∎

######  Corollary 38.

The counting polynomial of $\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G)$ is

$P_{\mathcal{M}_{\mathbb{Z}^{r}}^{0}(G)}(x)=\frac{(-1)^{r\dim
T}}{|W|}\sum_{g\in W}\left[\det(A_{g}-xI)\right]^{r}.$

###### Proof.

From Theorem 36, and the discussion preceding it, we know that for a
polynomial count variety $X$, $P_{X}(x)=E_{X}^{c}(\sqrt{x},\sqrt{x})$. By
definition, this is also equal to $\mu_{X}^{c}(-1,\sqrt{x},\sqrt{x})$, so the
formula follows immediately from Theorem 37. ∎

######  Remark 39.

Lemma 35 implies that if $F$ acts freely and $X$ is polynomial count, then
$X/F$ is polynomial count too (since in that case there is only one strata).
Thus, the smooth model for $\mathcal{R}_{\mathbb{Z}^{r}}^{0}(G)$, namely
$(G/T)\times_{W}T^{r}$, is also polynomial count since $W$ acts freely on
$(G/T)\times T^{r}$, the product of polynomial count varieties is polynomial
count, and $T^{r}$ and $G/T$ are polynomial count by [Br]. However, this does
not show $\mathcal{R}_{\mathbb{Z}^{r}}^{0}(G)$ is polynomial count since the
map relating $\mathcal{R}_{\mathbb{Z}^{r}}^{0}(G)$ and $(G/T)\times_{W}T^{r}$,
although a cohomological isomorphism, is neither injective nor surjective.

##  6\. Exotic Components

In this last section we describe the MHS on the full character varieties of
$\mathbb{Z}^{r}$ (not only the identity component) in special cases described
in [ACG]. Let $p$ be a prime integer, and $\mathbb{Z}_{p}$ be the cyclic group
or order $p$. We will use the same notation for center of $\mathrm{SU}(p)$
which is realized as the subgroup of scalar matrices with values $p$-th roots
of unity. Let $\Delta(p)$ be the diagonal of $\mathbb{Z}_{p}$ in
$(\mathbb{Z}_{p})^{m}=Z(\mathrm{SU}(p)^{m})$. Let
$K_{m,p}:=\mathrm{SU}(p)^{m}/\Delta(p)$. For example,
$K_{1,p}=\mathrm{PU}(p)$, the projective unitary group.

In [ACG], all the components in $\mathsf{Hom}(\mathbb{Z}^{r},K_{m,p})$ and
$\mathcal{M}_{\mathbb{Z}^{r}}K_{m,p}$ are described. In particular,
$\mathcal{M}_{\mathbb{Z}^{r}}K_{m,p}$ consists of the identity component

$\mathcal{M}_{\mathbb{Z}^{r}}^{0}K_{m,p}\cong(S^{1})^{(p-1)rm}/(S_{p})^{m}$

where $S_{p}$ is the symmetric group on $p$ letters, and

$N(p,m):=\frac{p^{(m-1)(r-2)}(p^{r}-1)(p^{r-1}-1)}{p^{2}-1}$

discrete points.

There is a one-to-one correspondence between the isolated points in
$\mathcal{M}_{\mathbb{Z}^{r}}K_{m,p}$ and non-identity path-components in
$\mathsf{Hom}(\mathbb{Z}^{r},K_{m,p})$. Each such path-component is isomorphic
to the homogeneous space $\mathrm{SU}(p)^{m}/(\mathbb{Z}_{p}^{m-1}\times
E_{p})$ where $E_{p}\subset\mathrm{SU}(p)$ is isomorphic to the quaternion
group $Q_{8}$ if $p$ is even and the group of triangular $3\times 3$ matrices
over the $\mathbb{Z}_{p}$, with 1’s on the diagonal when $p$ is odd (either
way it is “extra-special” of order $p^{3}$).

######  Corollary 40.

Let $G_{m,p}$ be the complexification of $K_{m,p}$:
$G_{m,p}\cong\mathrm{SL}(p,\mathbb{C})^{m}/\Delta(p)$. Then:
$\mu_{\mathcal{R}_{\mathbb{Z}^{r}}G_{m,p}}\left(t,u,v\right)=$

$=\left(\frac{1}{p!}\prod_{i=2}^{p}(1-(t^{2}uv)^{i})\sum_{g\in
S_{p}}\frac{\det(I+tuv\,A_{g})^{r}}{\det(I-t^{2}uv\,A_{g})}\right)^{m}+N(p,m)\prod_{j=2}^{p}\left(1+t^{2j-1}u^{j}v^{j}\right)^{m}.$

###### Proof.

The Weyl group $W$ of $G_{m,p}$ is a direct product of $m$ copies of $S_{p}$:
the Weyl group of $\mathrm{SL}(p,\mathbb{C})$, and its action on the (dual of
the) Lie algebra of maximal torus, is the product action. Therefore, using
again $x=tuv$ and $w=t^{2}uv$, for the identity component
$\mathcal{R}_{\mathbb{Z}^{r}}^{0}G_{m,p}$ we have:

$\displaystyle\mu_{\mathcal{R}_{\mathbb{Z}^{r}}^{0}G_{m,p}}\left(t,u,v\right)$
$\displaystyle=$
$\displaystyle\left(\frac{1}{p!}\prod_{i=2}^{p}(1-w^{i})\right)^{m}\\!\\!\\!\\!\\!\sum_{(g_{1},\cdots,g_{m})\in
W}\\!\\!\frac{\det(I_{p}+x\,A_{g_{1}})^{r}\cdots\det(I_{p}+x\,A_{g_{m}})^{r}}{\det(I_{p}-w\,A_{g_{1}})\cdots\det(I-w\,A_{g_{m}})}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{p!}\prod_{i=2}^{p}(1-w^{i})\right)^{m}\left(\sum_{g\in
S_{p}}\frac{\det(I+x\,A_{g})^{r}}{\det(I-w\,A_{g})}\right)^{m}.$

Now, since the MHS on
$\mathrm{SL}(p,\mathbb{C})^{m}/(\mathbb{Z}_{p}^{m-1}\times E_{p})$ coincides
with that of $\mathrm{SL}(p,\mathbb{C})^{m}$ by Lemma 3, each of the $N(p,m)$
components, other than $\mathcal{R}_{\mathbb{Z}^{r}}^{0}G_{m,p}$, contributes
$\mu(\mathrm{SL}(p,\mathbb{C})^{m})=\mu(\mathrm{SL}(p,\mathbb{C}))^{m}=\prod_{j=2}^{p}\left(1+t^{2j-1}u^{j}v^{j}\right)^{m}$.
∎

For the character variety, from the fact that each isolated point adds a
constant 1, we have the following corollary:

######  Corollary 41.

The mixed Hodge polynomial of $\mathcal{M}_{\mathbb{Z}^{r}}G_{m,p}$ is:

$\displaystyle\mu_{\mathcal{M}_{\mathbb{Z}^{r}}G_{m,p}}\left(t,u,v\right)$
$\displaystyle=$ $\displaystyle\left(\frac{1}{p!}\sum_{g\in
S_{p}}\det\left(I+tuv\,A_{g}\right)^{r}\right)^{m}+N(p,m).$

###### Proof.

The same argument of Corollary 40, implies that the identity component
verifies:
$\mu_{\mathcal{M}_{\mathbb{Z}^{r}}^{0}G_{m,p}}=(\mu_{\mathcal{M}_{\mathbb{Z}^{r}}\mathrm{SL}(p,\mathbb{C})})^{m}$,
so the formula is clear. ∎

######  Remark 42.

Given two reductive groups $G$ and $H$, both the $(G\times H)$-representation
varieties and the $(G\times H)$-character varieties are cartesian products of
the $G$ and $H$ varieties:

$\mathcal{R}_{\Gamma}(G\times
H)=\mathcal{R}_{\Gamma}G\times\mathcal{R}_{\Gamma}H,\quad\quad\mathcal{M}_{\Gamma}(G\times
H)=\mathcal{M}_{\Gamma}G\times\mathcal{M}_{\Gamma}H.$

From Corollaries 40 and 41, the mixed Hodge polynomial of the identity
components of these $G_{m,p}$-character varieties behaves multiplicatively,
even though $\mathcal{R}_{\mathbb{Z}^{r}}^{0}G_{m,p}$ and
$\mathcal{M}_{\mathbb{Z}^{r}}^{0}G_{m,p}$ are not cartesian products.

######  Remark 43.

Since $G$ is connected,
$\mathcal{R}_{\mathbb{Z}^{r}}^{0}G=\mathcal{R}_{\mathbb{Z}^{r}}G$ if and only
if $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G=\mathcal{M}_{\mathbb{Z}^{r}}G$. And by
[FL2], $\mathcal{M}_{\mathbb{Z}^{r}}^{0}G=\mathcal{M}_{\mathbb{Z}^{r}}G$ if
and only if (a) $r=1$, or (b) $r=2$ and $G$ is simply connected, or (c) $r\geq
3$ and $G$ is a product of $\mathrm{SL}(n,\mathbb{C})$’s and
$\mathrm{Sp}(n,\mathbb{C})$’s.

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