text:
\begin{document}
\thispagestyle{empty}

\title{Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve}

\author{
 Tam\'as Hausel
\\ {\it Department of Mathematics}
\\ {\it University of Texas at Austin}
\\ {\it Austin TX 78712, USA}
\\{\tt hausel@math.utexas.edu}}
\maketitle

\begin{abstract}
This is a survey of results and
conjectures
on mirror symmetry phenomena in the non-Abelian Hodge theory of a
curve. We start with the
conjecture of Hausel--Thaddeus which claims that certain
Hodge numbers of moduli spaces of flat $SL(n,\C)$ and $PGL(n,\C)$-connections
on a smooth projective algebraic curve agree.
We then change our point
of view in the non-Abelian
Hodge theory of the curve, and concentrate on the $SL(n,\C)$ and
$PGL(n,\C)$ character varieties of
the curve. Here we discuss a recent
conjecture of Hausel--Rodriguez-Villegas which claims, analogously to
the above conjecture, that
certain Hodge numbers of these character varieties also agree. We
explain that for Hodge numbers of character varieties one can use
arithmetic methods, and thus we
end up explicitly calculating, in terms of Verlinde-type formulas,
the number of representations
of the fundamental group into the finite groups $SL(n,\F_q)$ and
$PGL(n,\F_q)$, by using the character tables of these finite groups of Lie type.
Finally we explain a conjecture
which enhances the previous result, and
gives a simple formula for the  mixed Hodge polynomials, and in
particular for the Poincar\'e
polynomials of these character varieties, and detail the relationship
to results of Hitchin, Gothen, Garsia--Haiman and Earl--Kirwan. One
consequence of this conjecture is a curious Poincar\'e
duality type of symmetry, which leads to a conjecture, similar to
Faber's conjecture on the moduli space of curves,
about a strong Hard Lefschetz theorem for the character variety, which
can be considered as a generalization
of both the Alvis--Curtis duality in
the representation theory of finite groups of Lie type and a
recent result of the author on the quaternionic geometry of matroids.

\end{abstract}

\section{Introduction} Non-Abelian Hodge theory (\cite{hitchin}, \cite{simpson})
of a genus $g$ smooth complex projective curve $C$
studies three moduli spaces  attached to $C$ and a
reductive complex algebraic group $G$, which in this paper
will be either $GL(n,\C)$ or $SL(n,\C)$ or $PGL(n,\C)$. They are
$\M_{Dol}(G)$, the moduli space of semistable
$G$-Higgs bundles on $C$; $\M_{DR}(G)$,
the moduli space of flat $G$-connections on $C$ and $\M_B(G)$ the character variety, i.e. the
moduli space of representations of $\pi_1(C)$ into $G$ modulo conjugation.
With some assumptions these  moduli spaces are smooth varieties (or orbifolds
when $G=PGL(n,\C)$) with the
underlying differentiable manifolds canonically identified, which carries a
natural hyperk\"ahler metric.

The cohomology of this underlying manifold
has mostly been studied from the perspective of $\M_{Dol}(G)$. Using a natural circle action
on it \cite{hitchin} and \cite{gothen} calculated the Poincar\'e polynomials for $G=SL(2,\C)$ and
$G=SL(3,\C)$ respectively; while \cite{HT1} and \cite{markman} found a simple set of generators for the cohomology ring for $G=PGL(2,\C)$ and $G=PGL(n,\C)$ respectively. The paper $\cite{HT2}$ then
calculated the cohomology ring explicitly for $G=PGL(2,\C)$. The techniques used in these papers
do not seem to generalize easily to higher $n$.

A new perspective for these investigations on the cohomology of $\M_{Dol}(G)$ and $\M_{DR}(G)$
was introduced in \cite{HT3} and \cite{HT4}. It was shown there that the hyperk\"ahler
metrics and the Hitchin systems \cite{hitchin2}
for $\M_{DR}(G)$ and $\M_{DR}(G^L)$, with $G=SL(n,\C)$ and Langlands dual $G^L=PGL(n,\C)$ provide
the geometrical setup suggested in \cite{SYZ} as a criteria for mirror symmetry.
Based on this observation \cite{HT4} conjectured that a version of the topological mirror symmetry
also holds, i.e. that certain Hodge numbers for $\M_{DR}(G)$ and
$\M_{DR}(G^L)$ agree. Using the above mentioned results of \cite{hitchin} and \cite{gothen} this conjecture
was checked for $G=SL(2,\C), SL(3,\C)$. This mirror symmetry conjecture motivates the study of
not just the cohomology but the mixed Hodge structure on the cohomology of the spaces $\M_{DR}(G)$,
$\M_{Dol}(G)$ and $\M_{B}(G)$. While it was shown in \cite{HT4} that the mixed Hodge structure
of $\M_{Dol}(G)$ and $\M_{DR}(G)$ agree, and can be shown to be pure as in Theorem~\ref{pure}
or \cite{mehta}, the mixed Hodge structure on $\M_{B}(G)$ has not been
studied until very recently.

An important theme of this survey paper is in fact the mixed Hodge structure on
the  character variety
$\M_{B}(G)$ or alternatively   the three variable
polynomial $H(x,y,t)$ the so-called mixed Hodge polynomial or shortly
$H$-polynomial which encodes the dimensions of the graded
pieces of the mixed Hodge
structure on
$\M_{B}(G)$.
 In a recent project \cite{HRV} an arithmetic method was used
to calculate the  $E$-polynomial (where the E-polynomial $E(x,y)$ for a smooth variety is defined as $x^ny^nH(1/x,1/y,-1)$, where $n$ is the
complex dimension of the variety)
of $\M_{B}(G)$. The idea of \cite{HRV} is to count
the rational points of $\M_B(G(\F_q))$, the variety $\M_{B}(G)$ over the finite field $\F_q$,
where $q$ is a prime power. This count then is possible due to a result of
\cite{mednykh}, and the resulting formula, which resembles the famous Verlinde formula \cite{verlinde},
is a simple sum over irreducible representations of the finite group of Lie type $G(\F_q)$.
Thus the representation theory behind the $E$-polynomial of the character variety is that of the
finite groups of Lie type, which could be considered as an
analogue
of Nakajima's principle \cite{nakajima}, which states
that the representation theory of Kac--Moody algebras are
encoded in the cohomology of the (hyperk\"ahler) quiver varieties.

The shape of the $E$-polynomials of the various character varieties
then made us conjecture \cite{HRV} that the mirror symmetry conjecture
also holds for the pair $\M_{B}(G)$ and $\M_{B}(G^L)$ in the case of
$G=SL(n,\C)$ at
least.
Due to our
improved ability to calculate these Hodge numbers via this number
theoretical method,
we could
check this conjecture in the cases when $n$ is $4$ or a prime. As the
two mirror symmetry
conjectures of \cite{HT4} and \cite{HRV} are equivalent on the level
of Euler characteristic, we
get a proof of the original mirror symmetry conjecture of \cite{HT4}
on the level of Euler characteristic in these cases.

Perhaps even more interestingly \cite{HRV} achieves explicit formulas, in terms of
a simple
generating function, for the $E$-polynomials
of the character variety $\M_B(GL(n,\C))$. In particular it can be deduced
from this that the Euler
characteristic  of $\M_B(PGL(n,\C))$ is $\mu(n) n^{2g-3}$, where $\mu$ is
the fundamental number theoretic function: the M\"obius function, i.e. the sum of all
primitive $n$th roots of unity. This result, which could not be
obtained with other methods, in itself hints
at an interesting link between number theory and the
topology of our hyperk\"ahler manifolds. Another consequence of our
formula, is that these $E$-polynomials turn out to be palindromic,
i.e. satisfy an unexpected Poincar\'e duality-type of symmetry. In
fact this symmetry can be traced back to the Alvis--Curtis duality
\cite{alvis,curtis} in the representation theory of finite
groups of Lie type.

Then we present a deformation of the formula for the $E$-polynomial of the character variety
$\M_B(PGL(n,\C))$, which conjecturally \cite{HRV} should agree with the
$H$-polynomial.
Moreover
we will later
modify this formula to obtain what conjecturally should be the $H$-polynomial of
the Higgs moduli space $P\M_{Dol}(GL(n,\C))$. We also explain how,
using,
as a guide, our mirror symmetry conjectures,
one could get conjectures for the corresponding
$H$-polynomials for the varieties associated to
$SL(n,\C)$.

These conjectures imply a conjecture on the Poincar\'e
polynomials (where the Poincar\'e polynomial is obtained from the
$H$-polynomial as $H(1,1,t)$) of our manifolds $\M_{Dol}(PGL(n,\C))$.
This conjecture is similar in flavour to
Lusztig's conjecture \cite{lusztig} on the Poincar\'e polynomials of  Nakajima's
quiver varieties, which is also a hyperk\"ahler manifold, similar to
the Higgs moduli space $\M_{Dol}(G)$. We should also mention Zagier's
\cite{zagier} formula for the Poincar\'e polynomial of the moduli space $\N$ of stable
bundles (the ``K\"ahler version'' of $\M_{Dol}(SL(n,\C))$),
where the formula is
a similar sum, but is parametrized by ordered partitions of $n$.

We discuss in detail many checks on these conjectures, by showing how our
conjectures imply results obtained by Hitchin \cite{hitchin}, Gothen
\cite{gothen} and
Earl--Kirwan
\cite{earl-kirwan}. Already the combinatorics of these formulas are
non-trivial, and  surprisingly
the calculus of Garsia--Haiman \cite{garsia-haiman} needs to be used to
check the  conjecture when $g=0$.

Since a curious Poincar\'e  duality type of symmetry is satisfied for
the conjectured Hodge numbers of  $\M_{B}(PGL(n,\C))$,
we also discuss the conjecture that a certain version of the Hard
Lefschetz theorem is satisfied
for our non-compact varieties. This is then explained to be a
generalization  of a result in
\cite{hausel3} on the quaternionic geometry of matroids, and an analogue of the Faber conjecture
\cite{faber}
on the moduli space of curves.

\paragraph{\bf Acknowledgment} This survey paper is based on the author's talk at the
``Geometric Methods in Algebra and Number Theory'' conference at the
University of Miami in December 2003. I would
like to thank the organizers for the invitation
and for the memorable conference.
Also most of the results and conjectures
surveyed here have recently been obtained in joint projects with
Michael Thaddeus \cite{HT3,HT4} and
with Fernando Rodriguez-Villegas \cite{HRV}.
The research described in this paper was partially supported by the
NSF grant DMS-0305505.

\section{Abelian and non-Abelian Hodge theory}

This section gives some basic definitions on Abelian and
non-Abelian Hodge theory which will be used in the paper later.
For details on them consult the sources indicated below.

\vskip.5cm

\begin{parag}{Hodge--De Rham theory} Fix a
smooth complex algebraic variety $M$. There are various
cohomology theories
which associate a graded anti-commutative ring to the variety $M$. First we
consider the singular, or Betti, cohomology $H^*_B(M,\C)$ of $M$
with complex coefficients.
This in fact can be defined for any reasonable
topological space.
The dimension of $H^k_B(M,\C)$ is called the $k$-th Betti number
and denoted $b_k(M)$. The Poincar\'e polynomial
is then formed from these numbers as coefficients: $$P(t;M)=\sum_k b_k(M) t^k.$$

Next we consider the De Rham cohomology
$H^*_{DR}(M,\C)$, which is the space of closed differential forms modulo
exact ones. This can be defined on any differentiable manifold. The De Rham
theorem then shows that these two cohomologies are naturally isomorphic:
\beq \label{derham} H^*_B(M,\C)\cong H^*_{DR}(M,\C).\eeq Now we assume that our variety is
projective. Then we have the Dolbault cohomology $H^*_{Dol}(M,\C)$, which
is defined as $$H^k_{Dol}(M,\C)=\bigoplus_{p+q=k} H^q(M,\Omega^p_M).$$
The Hodge theorem then implies that there is a natural isomorphism
\beq \label{hodge}H^k_{DR}(M,\C)\cong H^k_{Dol}(M,\C).\eeq The above two isomorphisms
then imply the Hodge decomposition theorem:
\beq H^k_{B}(M,\C)\cong \bigoplus_{p+q=k} H^{p,q}(M), \label{decomp}\eeq where
$H^{p,q}(M)$  denotes $H^p(M,\Omega^q_M)$. The dimension of $H^{p,q}(M)$ is denoted $h^{p,q}(M)$ and called the Hodge numbers of the variety $M$. From these numbers we form a two variable polynomial the
Hodge polynomial: $$H(x,y;M):=\sum_{p,q} h^{p,q}(M) x^p y^q.$$

For more details on these cohomology theories see \cite{griffiths-harris}.

\end{parag}
\vskip.5cm
\begin{parag}{Mixed Hodge structures}
Deligne \cite{deligne2} generalized the Hodge decomposition theorem (\ref{decomp}) to any complex variety $M$, not necessarily smooth or projective, by introducing a so called {\em mixed Hodge structure} on $H_B^*(M,\C)$.
This implies a
decomposition\footnote{In fact what one gets from a mixed Hodge structure are two filtrations on the cohomology,
and the decomposition in question
is the associated graded.} $$H_B^k(M,\C)\cong \bigoplus_{p,q} H^{p,q;k}(M), $$
where $p+q$ is called the weight of $H^{p,q;k}(M)$.
In the case of a smooth projective variety we have
$H^{p,q;p+q}(M)=H^{p,q}(M)$, i.e. that the weight of $H^{p,q;k}(M)$
is always $k$, this weight is called {\em pure weight}.
However in general
we could have other weights appear in the mixed Hodge structure of a
complex algebraic variety; indeed we will see such examples later.
The dimensions of $H^{p,q;k}(M)$ are
denoted by $h^{p,q;k}(M)$ and are called mixed Hodge numbers.
From them we form the three variable polynomial
\beq H(x,y,t;M):=\sum_{p,q,k} h^{p,q;k}(M) x^p y^q t^k.\label{mixedhodge}\eeq

Similarly, Deligne \cite{deligne2} constructs a mixed Hodge structure
on the compactly supported $H^*_{B,cpt}(M,\C)$ singular cohomology
of our complex algebraic variety $M$. This yields the decomposition
 $$H_{B,cpt}^k(M,\C)\cong \bigoplus_{p,q} H_{cpt}^{p,q;k}(M), $$ and leads
to the compactly supported mixed Hodge numbers $h^{p,q;k}_{cpt}(M)$,
which is defined as the dimension of $H_{cpt}^{p,q;k}(M)$. Then one
can introduce the $e$-numbers $e^{p,q}(M) = \sum_{k} (-1)^k h_{cpt}^{p,q;k}(M)$ from which we get the $E$-polynomial: \begin{eqnarray}E(x,y;M):=\sum_{p,q} e^{p,q}(M) x^p y^q.\label{epolynomial}\end{eqnarray}
Clearly for a smooth projective variety $E(x,y)=H(-x,-y)$. Moreover for a smooth variety Poincar\'e duality implies
that $$E(x,y)=(xy)^n H(1/x,1/y,-1),$$ where $n$ is the complex dimension of $M$.
The significance of the $E$-polynomial is that it is additive
for decompositions and multiplicative for Zariski locally trivial
fibrations.

For more details see the original \cite{deligne2} or \cite{batyrev-dais}
for more on the $E$-polynomials.
\end{parag}
\vskip.5cm
\begin{parag}{Stringy cohomology} \label{stringy} Suppose a finite group $\Gamma$ acts on our variety $M$. Then by the naturality of the mixed Hodge structure $\Gamma$ will act on
$H^{p,q,k}(M)$ and we have $$H^{p,q;k}(M/\Gamma)\cong \left(H^{p,q;k}(M)\right)^\Gamma.$$ However for a Calabi--Yau $M$ and $\Gamma$ preserving the Calabi--Yau structure
 string theorists \cite{vafa, zaslow} introduced different Hodge
numbers on the Calabi--Yau orbifold $M/\Gamma$: the so-called stringy Hodge numbers, which are the right
Hodge numbers for mirror symmetry. Their mathematical
significance is highlighted by a theorem of Kontsevich \cite{kontsevich}
that the stringy Hodge numbers agree with the ordinary
Hodge numbers of any
crepant resolution.
Following \cite{batyrev-dais} we can define the stringy
$E$-polynomials:
$$\Est(x,y;M/\Ga) = \sum_{[\gamma]} E(x,y;M^\gamma)^{C(\gamma)}
(xy)^{F(\gamma)}.$$
Here the sum runs over the conjugacy classes of
$\Ga$; $C(\ga)$ is the centralizer of $\ga$; $M^\ga$ is the subvariety
fixed by $\ga$; and $F(\ga)$ is an integer called the {\em fermionic
shift}, which is defined as follows.
The group element $\ga$ has finite order, so it acts on $TM|_{M^\ga}$
as a linear automorphism with eigenvalues $e^{2 \pi i w_1}, \dots,
e^{2 \pi i w_n}$, where each $w_j \in [0,1)$.  Let $F(\ga) = \sum
w_j$; this is an integer since, by hypothesis, $\ga$ acts trivially on
the canonical bundle.

The last cohomology theory we will need is the stringy cohomology
of a Calabi--Yau orbifold twisted by a $B$-field.
Following \cite{hitchin3} we let
$B\in H^2_\Gamma (M, U(1))$ i.e. an isomorphism class of a $\Ga$-equivariant flat unitary
gerbe. For any element $\ga\in \Gamma$ this $B$-field
induces a $C(\ga)$-equivariant local system $[L_{B,\gamma}]\in H^1_{C(\gamma)} (M^\gamma,U(1))$ on the fixed point set $M^\gamma$. Using
this we can twist the stringy $E$-polynomial of $M/\Gamma$ to get:
\beq \Est^B (x,y;M / \Ga) = \sum_{[\ga]}
E(x,y;M^\ga;L_{B,\ga})^{C(\gamma)}(xy)^{F(\ga)}.\label{bfieldstringy}\eeq

For more information on the mathematics of stringy cohomology see \cite{batyrev-dais}, for twisting with a $B$-field see \cite{HT4}.
\end{parag}
\vskip.5cm
\begin{parag}{Non-Abelian Hodge theory} The starting
point of non-Abelian Hodge theory is the identification of the space $H_B^1(M,\C^\times)$ with the space of homomorphisms from
$\pi_1(M)\to \C^\times$; the space $H_{DR}^1(M,\C^\times)$ with
algebraic local systems on $M$ and the space $$H_{Dol}(M,\C^\times)\cong
H^1(M,{\cal O}^\times)\oplus H^0(M,\Omega^1)$$ with pairs of
a holomorphic line bundle and a holomorphic one form.

This then can be generalized to any
non-Abelian complex reductive group $G$. We define $H_B^1(M,G)$
to be conjugacy classes of representations of $\pi_1(M)\to G$.
I.e. $$H_B^1(M,G):={\rm Hom}(\pi_1(M),G)// G,$$ which is the affine GIT quotient of the affine variety ${\rm Hom}(\pi_1(M),G)$ by the conjugation action by $G$. This is sometimes called the {\em character variety}.
The space $H_{DR}^1(M,G)$ can be identified as the moduli space of algebraic $G$-local systems on $M$. Finally $H_{Dol}^1(M,G)$ is defined as the moduli space of certain semistable $G$-Higgs bundles on $M$. We will
give precise definition in the case of a curve below.
The identification between  $H_B^1(M,G)$ and $H_{DR}^1(M,G)$,
which is analogous to the De Rham map (\ref{derham}),  is given by
the Riemann--Hilbert correspondence \cite{deligne1,simpson4},
while the identification
between  $H_{DR}^1(M,G)$ and $H_{Dol}^1(M,G)$, which is analogous to
the Hodge decomposition (\ref{hodge}), is given in \cite{corlette, simpson2}
by the theory
of harmonic bundles, which is the non-Abelian generalization of
Hodge theory.

For an introduction to non-Abelian Hodge theory see \cite{simpson}, and
\cite{katzarkov-pantev}[Section 3], for more details on the construction of the spaces
appearing in non-Abelian Hodge theory and the maps between them
see \cite{simpson2,simpson3,simpson4}.
\end{parag}
\vskip.5cm
\begin{parag}{The case of a curve} We now fix a smooth
projective complex curve $C$ of genus $g$
and specify our spaces in the case when $M= C$ and $G=GL(n,\C)$.
According to the above definitions we have:
\bes{\cal M}_B(GL(n,\C))&:=&H^1_B(C,GL(n,\C))= \\ &=&
\{A_1,B_1,\dots,A_g,B_g\in GL(n,\C) | [A_1,B_1]\cdot \dots \cdot
[A_g,B_g]=Id \}// GL(n,\C).\ees
There is a natural way to twist these varieties, and because they
will be needed for $PGL(n,\C)$ we introduce these twisted varieties
here. For an integer $d$ we also consider:
\bes\M_B(GL(n,\C)):= \{A_1,B_1,\dots,A_g,B_g\in GL(n,\C) | [A_1,B_1]
\cdot \dots \cdot [A_g,B_g]=e^{\frac{2\pi i d}{n}}Id \}// GL(n,\C).
\ees

The De Rham space looks like
\bes{\mathcal M}_{DR}(GL(n,\C))&:=&H^1_{DR}(C,GL(n,\C))= \\ &=&
\{\mbox{moduli
space of flat $GL(n,\C)$-connections on $C$}\}\ees
and in the twisted case we need to fix a point $p\in C$, and define
\bes {\mathcal M}^d_{DR}(GL(n,\C)) := \left\{ \begin{array}{c} \mbox{
moduli space of flat $GL(n,\C)$-connections on $C\setminus\{ p \}$},
\\
\mbox{with holonomy $e^{\frac{2\pi i d}{n}}Id$ around $p$}
\end{array}\right\}.\ees

Finally the Dolbeault spaces are:
\bes{\mathcal M}_{Dol}(GL(n,\C))&:=&H^1_{Dol}(C,GL(n,\C))= \\ &=&
\{\mbox{moduli
space of semistable  rank $n$ degree $0$ Higgs bundles on $C$}\},\ees
where a rank $n$ Higgs bundle is a pair $(E,\phi)$ of a rank $n$
algebraic vector bundle $E$ on $C$, with degree $0$ and Higgs field
$\phi\in H^0(C,K{\rm End} E)$. A Higgs bundle is called
semistable
if for any Higgs subbundle $(F,\psi)$ (i.e. a subbundle with compatible Higgs fields) we have $\frac{{\rm deg}(F)}{{\rm rank}(F)}\leq \frac{{\rm deg}(E)}{{\rm rank}(E)}=0$.
The twisted version of ${\mathcal M}_{Dol}(GL(n,\C))$ is defined:
\bes{\mathcal M}^d_{Dol}(GL(n,\C)):=\{\mbox{ moduli
space of semistable  rank $n$ degree $d$ Higgs bundles on $C$}\}.\ees

The varieties defined above for $GL(n,\C)$ are all of dimension $n^2(2g-2)+2$.
The Betti space is affine, while the De Rham space is
analytically isomorphic, via the Riemann--Hilbert correspondence,
to the Betti space but not algebraically, so the De Rham space is
a Stein manifold as a complex manifold
but not an affine variety as an algebraic variety. Finally the Dolbeault
space is a quasi projective variety with large projective subvarieties.

From now on we will consider only the case when $(n,d)=1$; so we
fix such a $d$. In this
case the corresponding twisted spaces are additionally smooth, have a diffeomorphic underlying manifold $\M(GL(n,\C))$ which carries
a complete hyperk\"ahler metric \cite{hitchin}.
The complex
structures of $\M_{Dol}(GL(n,\C))$ and $\M_{DR}(GL(n,\C))$
appear in the hyperk\"ahler structure.

We started this subsection by determining these spaces for $GL(1,\C)
\cong \C^\times$.
By the identifications explained there we see that \beq {\M}_B(GL(1,\C))
&\cong& (\C^\times)^{2g},\nonumber \\
\M_{Dol}(GL(1,\C))&\cong& T^* Jac^d(C)\label{gl1}\eeq and
$\M_{DR}$ is a certain affine bundle over $Jac^d(C)$. Interestingly for $d=0$
they are all
algebraic groups and they act on the corresponding
spaces for $GL(n,\C)$ and any $d$ by tensorization.

We can consider the
map  $$\begin{array}{cccc} \lambda_{Dol}:&\M_{Dol}(GL(n,\C))&\to& \M_{Dol}(GL(1,\C))\\ &(E,\Phi)&\mapsto &({\det}(E),{\rm tr}(\phi)).\end{array}$$
The fibres of this map can be shown to be isomorphic using the
above tensorization action. So up to isomorphism it is irrelevant which fibre we take, but we usually take a point $(\Lambda,0)\in  \M_{Dol}(GL(1,\C))$ and define $$\M_{Dol}(SL(n,\C)):=\lambda_{Dol}^{-1}((\Lambda,0)),$$ for the other two spaces we have:

\bes {\mathcal M}^d_{DR}(SL(n,\C)) = \left\{ \begin{array}{c} \mbox{
moduli space of flat $SL(n,\C)$-connections on $C\setminus\{ p \}$}
\\
\mbox{with holonomy $e^{\frac{2\pi i d}{n}}Id$ around $p$}
\end{array}\right\},\ees
and
\bes\M_B(SL(n,\C))= \{A_1,B_1,\dots,A_g,B_g\in SL(n,\C) | [A_1,B_1]
\cdot \dots \cdot [A_g,B_g]=e^{\frac{2\pi i d}{n}}Id \}// SL(n,\C).
\ees

 The varieties $\M_{B}(SL(n,\C))$,   $\M_{DR}(SL(n,\C))$ and $\M_{DR}(SL(n,\C))$ are smooth of dimension $(n^2-1)(2g-2)$, with diffeomorphic
underlying manifold $\M(SL(n,\C))$. The Betti space
is affine, and the Betti and De Rham spaces are again analytically, but not
algebraically, isomorphic.

Moreover we see that a finite subgroup namely
$Jac[n]\cong \Z_n^{2g}\subset {\mathcal M}_{Dol}(GL(1,\C))$ preserves the
fibration $\lambda_{Dol}$ and thus acts on $\M_{Dol}(SL(n,\C))$. The quotient then is:
\bes \M_{Dol}(PGL(n,\C)):=\M_{Dol}(SL(n,\C))/{\rm Jac}[n]\ees
and similarly
\bes \M_{DR}(PGL(n,\C)):=\M_{DR}(SL(n,\C))/{\rm Jac}[n], \ees
and \bes \M_{B}(PGL(n,\C))=\M_{B}(SL(n,\C))/\Z_n^{2g}.\ees This
shows that the spaces  $\M_{B}(PGL(n,\C))$, $\M_{DR}(PGL(n,\C))$ and
$\M_{Dol}(PGL(n,\C))$ are hyperk\"ahler orbifolds of dimension
$(n^2-1)(2g-2)$. As they are orbifolds
we can talk about their stringy mixed Hodge numbers as defined above
in 2.\ref{stringy}. Moreover there are natural orbifold $B$-fields on
them, which we construct now. First we consider
a  universal Higgs pair
$(\bee, \bphi)$ on $\M_\Dol(SL(n,\C))\times C$, it exists
because $(d,n)=1$.
Restrict $\bee$ to $\M_\Dol\times \{p\}$ to get the vector
bundle $\bee_p$ on $\M_{Dol}(SL(n,\C))$.
Now we can consider the projective bundle ${\mathbb P}\bee_p$ of
$\bee_p$ which is a $PGL(n,\C)$-bundle. The bundle $\bee_p$ is
a $GL(n,\C)$ bundle but not a $SL(n,\C)$-bundle, because it has
non-trivial determinant. The obstruction class to lift the $PGL(n,\C)$ bundle ${\mathbb P}\bee$
to an $SL(n,\C)$ bundle is a class
$B\in H^2(\M_{Dol}(SL(n,\C),\Z_n))\subset H^2(\M_{Dol}(SL(n,\C)),U(1))$, which gives us our $B$-field on $\M_{Dol}(SL(n,\C))$. Moreover $B$
has \cite{HT4}[Section 3]
a natural equivariant extension $\hat{B}\in H_{\Gamma}^2(\M_{Dol}(SL(n,\C)),U(1))$, giving us our $B$-field on $\M_{Dol}(PGL(n,\C))$. This
$B$-field will come handy for our mirror symmetry discussions below.

For non-Abelian Hodge theory on a curve, see \cite{hitchin}, which gives
a gauge theoretical approach, and yields the natural hyperk\"ahler metrics
on our spaces. \cite{goldman-xia} introduces to the $GL(1,\C)$ case in detail.
On the geometry and cohomology
of ${\mathcal M}^1_{Dol}(SL(2,\C))$ see \cite{hausel2}.
\end{parag}
\vskip.5cm
\begin{parag}{The mixed Hodge structure  on
non-Abelian Hodge cohomologies}
The main subject of this survey paper is the mixed Hodge polynomial
of the (sometimes stringy, sometimes with a $B$-field)
cohomology of the spaces $\M_{Dol}(G)$, $\M_{DR}(G)$ and
$\M_{B}(G)$,
for our three choices for $G=
GL(n,\C), PGL(n,\C)$ or $SL(n,\C)$. As a notational convenience we may omit
$G$ and simply write $\M_{B}$, $\M_{DR}$ and $\M_{Dol}$, when it is clear what
$G$ should be, or if $G$ could be any of our groups.

First take $G=GL(1,\C)$. From (\ref{gl1})
we can easily calculate the mixed Hodge polynomials as follows:
\bes \begin{array}{c} H(x,y,t;\M_{B}(GL(1,\C)))=(1+xyt)^{2g}\\
H(x,y,t;\M_{Dol}(GL(1,\C)))= H(x,y,t;\M_{DR}(GL(1,\C)))=
(1+xt)^g(1+yt)^g.\label{glone}\end{array}\ees
It is remarkable that $H(x,y,t;\M_{B}(GL(1,\C)))\neq H(x,y,t;\M_{DR}(GL(1,\C)))$ even though the spaces are analytically isomorphic. Moreover
we can explicitly see that the mixed Hodge structure on
$H^k(\M_{Dol}(GL(1,\C)),\C)$ and
$H^k(\M_{DR}(GL(1,\C)),\C)$ are pure, while on $H^k(\M_{B}(GL(1,\C)),\C)$
it is not.

From a K\"unneth argument we also see that:
\bes H(x,y,t;\M_{Dol}(GL(n,\C)))= H(x,y,t;\M_{Dol}(PGL(n,\C))) H(x,y,t;\M_{Dol}(GL(1,\C))), \ees
and similarly for the other two spaces.
Thus the calculation for $GL(n,\C)$ is equivalent with the calculation
for $PGL(n,\C)$.

Now we list what we know about the cohomologies $H^*(\M,\C)$.
The Poincar\'e polynomials $P(t;{\mathcal M}^1(SL(2,\C)))$ and
$P(t;{\mathcal M}^1(PGL(2,\C)))$ were calculated in \cite{hitchin}, while
the polynomials  $P(t;{\mathcal M}^1(SL(3,\C)))$ and
$P(t;{\mathcal M}^1(PGL(3,\C)))$ have been calculated
in \cite{gothen}. In both
papers Morse theory for a natural $\C^\times$ action on $\M_{Dol}$ was used (acting by multiplication of the Higgs field). The idea
is to calculate the Poincar\'e polynomial of the various fixed point components of this action, and then sum them up with a
certain shift. The largest of the fixed point components,
when $\phi=0$,
is an important and well-studied space itself so we
define it here as \begin{multline} \N(SL(n,\C)):=\\ \left\{ {\mbox{the moduli space of stable vector bundles
of fixed determinant bundle of degree {d}}}\label{smallguy}
\right\}.\end{multline} The Poincar\'e
polynomial of this space was calculated in \cite{harder-narasimhan} by
arithmetic and in \cite{atiyah-bott} by gauge theoretical methods with
explicit formulas in \cite{zagier}. Thus its contribution to $P(t;\M(SL(n,\C)))$ is easy to handle. However the other components of the fixed point set of the natural circle action is
 more cumbersome to determine already
when $n=4$, and consequently this Morse theory approach
has not been completed
for $n\geq 4$.

As a running  example here we calculate from \cite{hitchin}
the Poincar\'e polynomial of ${\mathcal M}^1(PGL(2,\C))$ when $g=3$:
\begin{multline} P(t;{\mathcal M}^1(PGL(2,\C)))=
\\=3\,{t}^{12}+12\,{t}^{11}+18\,{t}^{10}+32\,{t}^{9}+18\,{t}^{8}+12\,{t}^
{7}+17\,{t}^{6}+6\,{t}^{5}+2\,{t}^{4}+6\,{t}^{3}+{t}^{2}+1
\label{p2t}
\end{multline}

The cohomology ring of ${\mathcal M}^1_{Dol}(PGL(2,\C))$ has been described
explicitly by
generators \cite{HT1} and relations \cite{HT2}. A result which
proved to be essential to produce our main Conjecture~\ref{mainconj}. Finally
Markman \cite{markman} showed that for $PGL(n,\C)$ the universal
cohomology classes do generate the cohomology ring.

Considering the mixed Hodge structure on the cohomology of our
spaces the following result first appeared in \cite{mehta} using
a construction of \cite{HT1}. Here we present a simple proof.

\begin{theorem} The mixed Hodge structure on $H^k(\M_{Dol},\C)$ is pure of weight $k$. \label{pure}
\end{theorem}

\begin{proof} Recall the compactification
$\overline{\mathcal M}^d_{Dol}$ of
 $\M_{Dol}$ constructed in \cite{hausel1}.
From that paper  it follows
   that  $\overline{\mathcal M}^d_{Dol}$ is a projective orbifold, so its mixed
  Hodge structure on $H^k(\overline{\mathcal M}^d_{Dol},\C)$ is pure of weight $k$. Now
  \cite{hausel1} also implies that the natural map
  $H^*(\overline{\mathcal M}^d_{Dol},\C)\to H^*(\M_{Dol},\C)$ is
surjective. The functoriality of mixed Hodge structures \cite{deligne2} completes the proof.
\end{proof}

One can similarly prove the same result for $\M_{DR}$.

\begin{theorem} The mixed Hodge structure on $H^k(\M_{DR},\C)$ is pure of
  weight $k$.
\end{theorem}

\begin{proof} As explained in \cite{HT4}[Theorem 6.2] one can
deform the complex structure of $\overline{\mathcal M}^d_{Dol}$ to
the projective orbifold
$\overline{\mathcal M}^d_{DR}$, which is the
compactification of $\M_{DR}$ given by
Simpson in \cite{simpson5}. Now this way we see that
the natural map
$H^*(\overline{\mathcal M}^d_{DR},\C)\to H^*(\M_{DR},\C)$ is
a surjection, getting our
result as in the previous proof.
\end{proof}

In fact the argument in \cite{HT4}[Theorem 6.2] shows that
\begin{theorem}[HT4]  The mixed
Hodge structure on $H^*(\M_{Dol},\C)$
is isomorphic with the mixed Hodge structure on
$H^*(\M_{DR},\C)$. \label{dr=dol}
\end{theorem}

The mixed Hodge structure on $\M_{B}$ however has not
been
studied in the literature. We
will start the study of it later in this paper, where we will see
that this Hodge structure will in fact be very much not pure.
But for now we explain our reason to be interested in these
mixed Hodge structures on the spaces $\M_{Dol}$, $\M_{DR}$ and
$\M_{B}$.
The reason is mirror symmetry:
\end{parag}

\section{Mirror symmetry conjectures}

Our starting point in \cite{HT4} was the observation that the pair
$\M_{DR}(SL(n,\C))$ together with the $B$-field $B^e$ and
${\mathcal M}^d_{DR}(PGL(n,\C))$ with the $B$-field $\hat{B}^d$
 satisfy the
geometric picture for mirror symmetry conjectured by
Strominger--Yau--Zaslow \cite{SYZ} and modified for $B$-fields
by Hitchin in \cite{hitchin3}. This geometric picture
requires the existence of a special Lagrangian fibration on both spaces, so that the fibres are dual. In fact in \cite{HT4} it is shown
that the so-called Hitchin map \cite{hitchin2} provides the required special Lagrangian
fibration on our spaces, with dual Abelian varieties as fibers. For
details on this see \cite{HT4}[Section 3].

Our focus in this survey is the topological implications of this
mirror symmetry. The following conjecture is what we call
the topological
mirror test for our SYZ-mirror partners.

\begin{conjecture}[\cite{HT4}]\label{mirrorderham}
For all $d,e \in \Z$, satisfying $(d,n)=(e,n)=1$, we have
$$\Est^{B^e}\Left(x,y;\M_\DR(SL(n,\C))\Right)
= \Est^{\hat B^d}\Left(x,y;{\mathcal M}^e_\DR(PGL(n,\C))\Right).$$

\end{conjecture}
\begin{conjecturemark}
Since $M^d_\DR(SL(n,\C))$ is smooth, the left-hand side actually equals the $E$-polynomial
$E\Left(x,y;M^d_\DR(SL(n,\C))\Right)$, which is independent of $e$. This
motivates the following:
\begin{conjecture}[HT4] For any two $d_1$ and $d_2$ as long as $(d_1,n)=(d_2,n)=1$ we have:  \beq E\Left(x,y;{\mathcal M}^{d_1}_{Dol}(SL(n,\C))\Right)=E\Left(x,y;{\mathcal M}^{d_2}_{Dol}(SL(n,\C))\Right).\eeq \label{coprime}
\end{conjecture}
This, if so, is quite interesting as the Betti numbers of
  $\N(SL(n,\C))$, the moduli space of stable vector bundles, with
fixed determinant of degree $d$, which can be considered
as the ``K\"ahler version'' of $\M_{Dol}(SL(n,\C))$, is known to depend on
$d$.
Already
when $n=5$, Zagier's explicit formula \cite{zagier}
for $P(t;{\mathcal N}^1(SL(5,\C)))$ and
$P(t;{\mathcal N}^3(SL(5,\C)))$ are different.
We will see strong support for this
Conjecture~\ref{coprime} later in Corollary~\ref{betticoprime}.
\end{conjecturemark}

\begin{conjecturemark} Conjecture~\ref{mirrorderham}
was proved for $n=2$ and $n=3$ in \cite{HT4}.
The proof proceeds by first transforming the calculation
to $\M_{Dol}$ via Theorem~\ref{dr=dol} and then uses
the Morse theoretic method of \cite{hitchin} and \cite{gothen}.
It is unclear, however, how this method can be extended
for $n\geq 4$.
\end{conjecturemark}

\begin{conjecturemark} An important ingredient of the proofs was
a modification of a result of Narasimhan--Ramanan
\cite{narasimhan-ramanan} to Higgs bundles.  It
describes the fixed points of the action by the elements of
$Jac[n]$ on $\M_{Dol}(SL(n,\C))$. The fixed point sets will be
some lower rank $m|n$ Higgs moduli
spaces $\M_{Dol}(SL(m,\C);\tilde{C})$ for a certain covering
$\tilde{C}$ of $C$. Their cohomology enters in the stringy
contribution to the right hand side of
Conjecture~\ref{mirrorderham} (recall (\ref{bfieldstringy})).
\end{conjecturemark}

\begin{parag}{Number theory to the rescue}
Although our mirror symmetry  Conjecture~\ref{mirrorderham}
is still open for $n\geq 4$, recently
some support for the validity of it has been achieved in
form of progress
on another related conjecture.

\begin{conjecture}[HRV] \label{mirrorbetti}
For all $d,e \in \Z$, such that $(d,n)=(e,n)=1$, we have
$$\Est^{B^e}\Left(x,y,\M_\B(SL(n,\C))\Right)
= \Est^{\hat B^d}\Left(x,y,{\mathcal M}^e_\B(PGL(n,\C))\Right).$$
\end{conjecture}

This conjecture has been proved \cite{HRV} when $n$ is a prime
and when $n=4$; which implies
Conjecture~\ref{mirrorderham} on the level of Euler
characteristic in these cases.
The method of proof is arithmetic. We count
rational
points of the variety ${\M_B}$ over a finite field $\F_q$, when
$n$ divides $q-1$, where $q=p^r$ is a prime power. Because this
count will turn out to be a polynomial in $q$ we have the following
 result, which is basically the Weil conjecture for our special
smooth affine varieties:

\begin{theorem}[\cite{HRV}] The $E$-polynomial of $\M_B$ has only $x^ky^k$ type terms, and the polynomial  $E(q):=E(\sqrt{q},\sqrt{q})$ agrees with the number of rational points of $\M_{B}(G)$
over $\F_q$:  \bes E(q)=\#(\M_{B}(G)(\F_q))  \ees
\end{theorem}

The count is then
possible because we only
need to count the solutions of the equation:
\bes [A_1,B_1]
\cdot \dots \cdot [A_g,B_g]=\xi_n, \ees
in the finite group of Lie type $G(\F_q)$, i.e.
so that $A_i,B_i\in G(\F_q)$, where $\xi_n\in G$ is a central element of order $n$.

A simple modification of a theorem of Mednykh \cite{mednykh},
(which goes back to Frobenius--Schur \cite{frobenius-schur},
and has since  been obtained by many authors,
for example by Freed--Quinn
\cite{freed-quinn}[(5.19)]) then implies:

\begin{theorem} Let $G=SL(n,\C)$ or $G=GL(n,\C)$. Then the
number of rational points on $\M_{B}(G)$ over a finite field
$\F_q$, where $q=p^r$ is a prime power,
with $n|(q-1)$  is given by the character  formula:
\bes \#(\M_{B}(G)(\F_q))=\sum_{\chi \in {\rm Irr(G(\F_q))}}
\frac{|G|^{2g-2}}{\chi(1)^{2g-1}} \chi(\xi_n) ,  \ees
where the sum is over all irreducible characters of the
finite group $G(\F_q)$ of Lie type.
\end{theorem}

The two theorems above imply the following

\begin{corollary}[\cite{HRV}] The $E$-polynomial of the character variety
$\M_B(G)$ is given by the character formula:
\beq E(q)=\sum_{\chi \in {\rm Irr(G(\F_q))}}
\frac{|G|^{2g-2}}{\chi(1)^{2g-1}} \chi(\xi_n).\eeq \label{eqcount}
\end{corollary}

\begin{remark}  An immediate consequence of this formula is
the Betti analogue of Conjecture~\ref{coprime}.
This follows from Corollary~\ref{eqcount} as that character
formula transforms by a Galois automorphism when one changes
from $d_1$ to $d_2$. Moreover because our varieties ${\mathcal M}^{d_1}_{B}(G)$ and  ${\mathcal M}^{d_2}_{B}(G)$  are Galois
conjugate themselves, we can deduce
that their Betti numbers agree, and
presumably their mixed Hodge
structure should also agree. In summary we have

\begin{corollary}[\cite{HRV}]  For all $d_1,d_2\in \Z$ \label{betticoprime}
as long as $(d_1,n)=(d_2,n)=1$ we have \beq E\Left(x,y;{\mathcal M}^{d_1}_B(G)\Right)=E\Left(x,y;{\mathcal M}^{d_2}_B(G)\Right)\eeq and
\beq P\Left(t;{\mathcal M}^{d_1}_B(G)\Right)=
P\Left(t;{\mathcal M}^{d_2}_B(G)\Right).\eeq
\end{corollary}

Thus we get an affirmative answer for Conjecture~\ref{coprime} on the level of the Poincar\'e polynomials.
In general Galois conjugate varieties tend to be (although need not
be see e.g. \cite{serre})  homeomorphic over $\C$.

\begin{problem} Are ${\mathcal M}^{d_1}_B(G\Right)$ and ${\mathcal M}^{d_2}_B(G\Right)$ homeomorphic for $(n,d_1)=(n,d_2)=1$?
\end{problem}
\end{remark}
\begin{remark}
In order to calculate the character formula in Corollary~\ref{eqcount}, we will need to know the
values of irreducible characters of $G$ on central elements.
Fortunately for $GL(n,\F_q)$ this has been calculated by Green \cite{green} and for $SL(n,\F_q)$ the required information, i.e. the value of the characters on central elements, was obtained
by Lehrer in \cite{lehrer}. In the next section we will show
an explicit result for the character formula for $GL(n,\F_q)$.
\end{remark}

\begin{remark} Our mirror symmetry Conjecture~\ref{mirrorbetti} then can be
translated to a complicated formula which is valid for the
character tables of $PGL(n,\F_q)$ and $SL(n,\F_q)$.
In particular we believe that by introducing punctures for our
Riemann surfaces
a similar mirror symmetry conjecture would in fact capture the
exact difference between the full character tables
of $PGL(n,\F_q)$
and $SL(n,\F_q)$ (not just on central elements as above).
This way our mirror symmetry proposal could be
phrased as follows: {\em the differences between the
character tables
of  $PGL(n,\F_q)$ and its Langlands dual $SL(n,\F_q)$ are
governed by mirror symmetry}. In particular it is particularly
enjoyable
to see the effect of mirror symmetry on the differences between
the  character tables of
$GL(2,\F_q)$ and $SL(2,\F_q)$, which were first
calculated in 1907 by  Jordan
\cite{jordan} and by Schur \cite{schur}.

\end{remark}
\end{parag}
\section{Explicit formulas for the E-polynomials} Here we
calculate the $E$-polynomials of $\M_B(PGL(n,\C))$, which we
denote by $E_n(q)$.
We need to start with partitions.

 We write a
partition of $n$ as $\lambda =(\lambda_{1}\geq \lambda_{2}\geq \cdots \geq \lambda
_{l}>0)$, so that $\sum \lambda_i=n$.  The
{\it Ferrers diagram} $d(\lambda)$
of $\lambda $ is the set of lattice points
\begin{equation}\label{e:d(lambda)}\{(i,j)\in {\mathbb Z_{\leq 0}} \times
  {\mathbb N} : j <\lambda_{-i+1} \}.
\end{equation}
 The {\it arm length} $a(z)$ and
{\it leg length} $l(z)$ of a point $z\in d(\lambda )$ denote the number of points
strictly to the right of $z$ and below $z$, respectively, as indicated
in this example:

 \begin{eqnarray*}\begin{array}[c]{cccccc}
 \bullet &      \bullet &       \bullet &
\bullet &       \bullet  \\

\cline{2-5}
\bullet &       \multicolumn{1}{|c|}{\llap{${}_{z}$}\bullet } & \bullet &
    \bullet&     \multicolumn{1}{c|}{\bullet }& {\scriptstyle a(z)} \\
\cline{2-5}
\bullet &       \multicolumn{1}{|c|}{\bullet }& \bullet &       \bullet \\

\bullet &       \multicolumn{1}{|c|}{\bullet }& \bullet \\
\cline{2-2}
\bullet &       \hbox to 0pt{\hss $\scriptstyle l(z)$\hss }\\
\end{array}
\end{eqnarray*}
where $\lambda=(5,5,4,3,1)$, $z=(-1,1)$, $a(z)=3$ and $l(z)=2$. The {\em hook length} then is defined as $$h(z)=l(z)+a(z)+1.$$

Let $$V_n(q)=E_n(q)  {q^{(1-g)n(n-1)}(q-1)^{2g-2}},$$ and
$$Z_n(q,T)=\exp\left(\sum_{r\geq 1} V_n (q^r) \frac{T^r}{r}\right).$$  We define
the Hook polynomials for a  partition $\lambda$ as follows
:
$${\cal H}^{\lambda}(q)=
\prod_{z \in d(\lambda)}
{q^{-l(z)}}(1-q^{h(z)}).$$

We can now formulate
\begin{theorem}[\cite{HRV}] The $E$-polynomials of the character varieties
$\M_{B}(PGL(n,\C))$ for $n=1,2,3,\dots$ are given by the following
generating function :
\begin{eqnarray}
\prod_{n=1}^\infty Z_n(q,T^n) = \sum_{\lambda\in {\cal P}}
({\cal H}^\lambda (q))^{2g-2} T^{|\lambda|},\end{eqnarray}
where ${\cal P}$ is the set of all partitions. \label{enq}
\end{theorem}

One simple corollary of this gives a new topological result:

\begin{corollary}[\cite{HRV}] The Euler characteristic of $\M(PGL(n,\C))$ is $\mu(n) n^{2g-3}$, where $\mu$ is the
  M\"obius function, i.e. $\mu(n)$ is the sum of primitive $n$th root of unities. \label{euler}
\end{corollary}

Another interesting application of the theorem is the following:

\begin{corollary} The $E$-polynomial $E_n(q)=E(q; \M_{B}(PGL(n,\C)))$
  is palindromic, i.e. it satisfies, what we call, the {\em curious}
  Poincar\'e duality:
$$q^{2N} E_n(1/q)=E_n(q),$$ where $2N=(n^2-1)(2g-2)$ is the complex
  dimension of $\M_{B}(PGL(n,\C))$.
\end{corollary}

\begin{remark} In fact this result originates in the so-called
  Alvis--Curtis duality \cite{alvis,curtis} in the character theory of $GL(n,\F_{q})$,
  which is a duality between irreducible representations of
  $GL(n,\F_{q})$. In particular, if $\chi, \chi^\prime\in {\rm
  Irr}(GL(n,\F_q))$ are dual representations then the dimension
  $\chi(1)$ is a polynomial in $q$ which satisfies
  $$q^{\frac{n(n-1)}{2}} \chi(1)(1/q)=\chi^\prime(1)(q).$$
\end{remark}
For example when $n=2$ Theorem~\ref{enq} gives:
\beq E_2(q)=(q^2-1)^{2g-2}+q^{2g-2}(q^2-1)^{2g-2}-\frac{1}{2} q^{2g-2} (q-1)^{2g-2}-\frac{1}{2}q^{2g-2}(q+1)^{2g-2}\label{e2q},\eeq
when $g=3$ this gives
\beq E(x,y;{\mathcal M}^1_{B}(PGL(2,\C)))=
{q}^{12}-4\,{q}^{10}+6\,{q}^{8}-14\,{q}^{6}+6\,{q}^{4}-4\,{q}^{2}+1,
\label{e2qg3}\eeq
which is a palindromic polynomial indeed. Note also that there does not seem to be much in common with
the Poincar\'e polynomial (\ref{p2t}).

\section{A conjectured formula for mixed Hodge polynomials}

Here we present the conjecture of \cite{HRV} on the $H$-polynomials of
the spaces $\M_B(PGL(n,\C))$.
As usual
we fix the curve $C$ and its genus $g$ and the group $PGL(n,\C)$ and
write $\M_B$ for $\M_B(PGL(n,\C))$ and $H_n(x,y,t)$ for $H(x,y,t;\M_B)$.

Now we introduce rational functions $H_n(q,t)$ in two variables,
via a generating function. We let $$V_n(q,t)=H_n(q,t)  \frac{(qt^2)^{(1-g)n(n-1)}(qt+1)^{2g}}{(qt^2-1)(q-1)},$$ and
$$Z_n(q,t,T)=\exp\left(\sum_{r\geq 1} V_n (q^r,-(-t)^r) \frac{T^r}{r}\right).$$  We define
the $t$-deformed Hook polynomials for genus $g$ and partition $\lambda$ as follows:
$${\cal H}_g^{\lambda}(q,t)=
\prod_{x\in d(\lambda)}
\frac{(qt^2)^{(2-2g)l(x)}(1+q^{h(x)} t^{2l(x)+1})^{2g}}{(1-q^{h(x)} t^{2l(x)+2})(1-q^{h(x)}t^{2l(x)})}.$$
The following generating function then defines our rational functions
$H_n(q,t)$: \begin{eqnarray} \label{main}
\prod_{n=1}^\infty Z_n(q,t,T^n) = \sum_{\lambda\in {\cal P}} {\cal H}^\lambda_g (q,t) T^{|\lambda|}.\end{eqnarray}
 Because for the character variety
we have that $h^{i,j;k}(\M_{B})=0$ provided that $i\neq j$ the following
conjecture describes $H_n(x,y,t)$ completely.

\begin{conjecture}[\cite{HRV}] \label{mainconj}
 The mixed Hodge polynomials of the varieties
$\M_{B}(PGL(n,\C))$ are given by the generating function (\ref{main}):
\bes H_n(\sqrt{q},\sqrt{q},t)=H_n(q,t)\ees
\end{conjecture}

Thus $H_n(q,t)$, which is a priori only a rational function,
is conjectured to be the $H$-polynomial of the character
variety, so in the next conjecture we formalize
our expectations from $H_n(q,t)$, with the addition of a curious,
Poincar\'e duality-type of symmetry, which was
in fact our most important guide to come up with these formulas:

\begin{conjecture} The rational functions $H_n(q,t)$ defined in the generating function of (\ref{main})
satisfy the following properties:
\label{comb}
\begin{itemize}
\item $H_n(q,t)$ is a polynomial in $q$ and $t$.
\item Both the $q$ degree and the $t$ degree of the polynomial $H_n(q,t)$ agree with
$2N=2(n^2-1)(g-1)$. In fact the largest degree monomial in both variables is $(qt)^{2(n^2-1)(g-1)}$.
\item All coefficients of $H_n(q,t)$ are non-negative integers. \label{combconj}
\item The coefficients of $H_n(q,t)=\sum h^i_j q^i t^j$ satisfy, what we call the {\em curious Poincar\'e duality}:
\begin{eqnarray}h^{i-j}_{{N}-j}=h^{i+j}_{{N}+j} \label{curious} \end{eqnarray}
\end{itemize}
\end{conjecture}
In the following
we list some checks and implications  of the above conjectures:
\vskip.4cm
\begin{conjecturemark}
Computer calculations with Maple gives
$H_n(q,t)$ from the above
generating function when $n=2,3,4$. In all these cases for small $g$
we do get a polynomial in $q$ and $t$ with
the expected degree and positive coefficients, satisfying the curious symmetry (\ref{curious}).
\end{conjecturemark}
\begin{conjecturemark}
Using the explicit description of the cohomology ring of
$\M_{B}$ in
\cite{HT2}, one can write
down a monomial basis for $H^*(\M_{B},\C)$ in the tautological
generators.
Then in turn one can
figure out the action of the Frobenius on these generators, which in turn provides a
formula for the mixed Hodge polynomial
of $\M_{B}$. This formula can be brought to the form
\begin{eqnarray} H_2(\sqrt{q},\sqrt{q},t)&=&
\frac{(q^2t^3+1)^{2g}}{(q^2t^2-1)(q^2t^4-1)}+
\frac{q^{2g-2}t^{4g-4}(q^2t+1)^{2g}}{(q^2-1)(q^2t^2-1)}-\nonumber \\
&-&\frac{1}{2}\frac{q^{2g-2}t^{4g-4}(qt+1)^{2g}}{(qt^2-1)(q-1)}-\frac{1}{2}
\frac{ q^{2g-2}t^{4g-4}(qt-1)^{2g}}{(q+1)(qt^2+1)},
\label{h2qt}\end{eqnarray}
which agrees with the
conjectured one through (\ref{main}), and clearly reduces to (\ref{enq}),
when $t=-1$. For example when $g=3$, this gives
\begin{multline}H(\sqrt{q},\sqrt{q},t;\M(PGL(2,\C)))=\\
= {t}^{12}{q}^{12}+{t}^{12}{q}^{10}+6\,{t}^{11}{q}^{10}+{t}^{12}{q}^{8}+
{t}^{10}{q}^{10}+6\,{t}^{11}{q}^{8}+16\,{t}^{10}{q}^{8}+6\,{t}^{9}{q}^
{8}+{t}^{10}{q}^{6}+{t}^{8}{q}^{8}+26\,{t}^{9}{q}^{6}+\\ +16\,{t}^{8}{q}^{
6}+6\,{t}^{7}{q}^{6}+{t}^{8}{q}^{4}+{t}^{6}{q}^{6}+6\,{t}^{7}{q}^{4}+
16\,{t}^{6}{q}^{4}+6\,{t}^{5}{q}^{4}+{t}^{4}{q}^{4}+{t}^{4}{q}^{2}+6\,
{t}^{3}{q}^{2}+{t}^{2}{q}^{2}+1, \label{h2qtg3}
\end{multline}
which is a common refinement of (\ref{p2t}) when $q=1$ and of
$(\ref{e2qg3})$
when $t=-1$. Note also how the curious Poincar\'e duality appears
when one refines the Poincar\'e polynomial (\ref{p2t}), which does not possess
any kind of symmetry, to the mixed Hodge
polynomial (\ref{h2qtg3}).

\end{conjecturemark}
\begin{conjecturemark} Note that $P_{n}(t)=H_{n}(1,t)$ should be the Poincar\'e polynomial
of the character
variety, which is the same as the Poincar\'e
polynomial of the diffeomorphic Higgs moduli space $\M_{Dol}$.
For $n=2$ Hitchin in \cite{hitchin} calculated the Poincar\'e
polynomial
of this latter space, and an easy calculation shows that if one substitutes $q=1$ into
(\ref{h2qt}) we get $P_2(t)=H_2(1,t)$, the Poincar\'e polynomial
of Hitchin. For $n=3$ Gothen in \cite{gothen} calculated $P_3(t)$.
Because it is so pleasant to work with a formula like (\ref{h2qt}), we also give
what our Conjecture~\ref{main} gives in the $n=3$ case:
\begin{eqnarray*} H_3(q,t)&=&{\frac { \left( {q}^{3}{t}^{5}+1 \right)
      ^{2\,g}
\left( {q}^{2}{t}^{3}+1 \right) ^{2\,g}}{ \left( {q}^{3}{t}^{6}-1 \right)  \left( {q}^{3}{t
}^{4}-1 \right)  \left( {q}^{2}{t}^{4}-1 \right)  \left( {q}^{2}{t}^{2
}-1 \right) }}+{\frac {{q}^{6\,g-6}{t}^{12\,g-12} \left( {q}^{3}t+1
 \right) ^{2\,g} \left( {q}^{2}t+1 \right) ^{2\,g}}{ \left( {q}^{3}{t}
^{2}-1 \right)  \left( {q}^{3}-1 \right)  \left( {q}^{2}{t}^{2}-1
 \right)  \left( {q}^{2}-1 \right) }}+\\
&+&{\frac {{q}^{4\,g-4}{t}^{8\,g-8}
 \left( {q}^{3}{t}^{3}+1 \right) ^{2\,g} \left( qt+1 \right) ^{2\,g}}{
 \left( {q}^{3}{t}^{4}-1 \right)  \left( {q}^{3}{t}^{2}-1 \right)
 \left( q{t}^{2}-1 \right)  \left( q-1 \right) }}+\frac{1}{3}\,{\frac {{q}^{6
\,g-6}{t}^{12\,g-12} \left(  \left( qt+1 \right) ^{2\,g} \right) ^{2}}
{ \left( q{t}^{2}-1 \right) ^{2} \left( q-1 \right) ^{2}}}-\\ &-&\frac{1}{3}\,{
\frac {{q}^{6\,g-6}{t}^{12\,g-12} \left( {q}^{2}{t}^{2}-qt+1 \right) ^
{2\,g}}{ \left( {q}^{2}{t}^{4}+q{t}^{2}+1 \right)  \left( {q}^{2}+q+1
 \right) }}-{\frac {{q}^{4\,g-4}{t}^{8\,g-8} \left( {q}^{2}{t}^{3}+1
 \right) ^{2\,g} \left( qt+1 \right) ^{2\,g}}{ \left( {q}^{2}{t}^{4}-1
 \right)  \left( {q}^{2}{t}^{2}-1 \right)  \left( q{t}^{2}-1 \right)
 \left( q-1 \right) }}-\\&-&{\frac {{q}^{6\,g-6}{t}^{12\,g-12} \left( {q}^{
2}t+1 \right) ^{2\,g} \left( qt+1 \right) ^{2\,g}}{ \left( {q}^{2}{t}^
{2}-1 \right)  \left( {q}^{2}-1 \right)  \left( q{t}^{2}-1 \right)
 \left( q-1 \right) }}.\end{eqnarray*}
Now it is a nice exercise to show that $H_3(1,t)$ does indeed produce
(the corrected version\footnote{One accidental
mistake in the calculation of
\cite{gothen} were pointed out in (10.3) of
\cite{HT4}.} of) Gothen's complicated looking formula in \cite{gothen}.

It is also worth noting that many of the individual terms in
$H_n(q,t)$ have poles at
$q=1$, however
according to our conjecture these poles somehow cancel each other.
\end{conjecturemark}

\begin{conjecturemark}
 When $g=0$, we know from the definitions that $H_1(x,y,t)=1$ and
$H_n(x,y,t)=0$ otherwise.
One can deduce the same
from Conjecture~\ref{mainconj} by applying
part f of Theorem 2.10 in \cite{garsia-haiman} to calculate the right
hand side of (\ref{main}). Moreover Conjecture~\ref{comb} has the
same flavour as the main conjecture in \cite{garsia-haiman} about
$q,t$ Catalan numbers, which was in turn proved
by Haiman in \cite{haiman}
using some subtle intersection theory on the
Hilbert scheme of $n$ points on $\C^2$. Apart from the fact that this
Hilbert scheme is also a hyperk\"ahler manifold, the similarities
between the two conjectures are rather surprising.
\end{conjecturemark}

\begin{conjecturemark}
\noindent When $g=1$ we have $H_n(x,y,t)=1$ for every $n$, but
this we
could not prove from (\ref{main}) for $H_n(q,t)$.
\end{conjecturemark}

\begin{conjecturemark}
Let us look at the conjecture (\ref{curious}). Recall that $H^2$
of our varieties are exactly one dimensional, generated by a class, call it
$[\omega]$, which is the K\"ahler class in the complex structure of
$\M_{Dol}$.
 This carries the weight $q^2t^2$ in the
mixed Hodge structure. We have the following hard Lefschetz
type of conjecture which enhances the curious
Poincar\'e duality of the conjecture of (\ref{curious}):

\begin{conjecture} If $L$ denotes the map by multiplication with
  $[\omega]$,
 then we conjecture that the map
$$ L^k: H^{{N}-k,N-k;i-k}(\M_{B}(PGL(n,\C)))\to H^{N+k,N+k;i+k}(\M_{B}(PGL(n,\C)))$$ is an isomorphism.
\end{conjecture}

Interestingly this conjecture implies a theorem of \cite{hausel3}
that the Lefschetz map $L^k:H^{N-k}\to H^{N+k}$ is injective for
$\M_{Dol}$,
and it is explained
there how this weak version of Hard Lefschetz, when applied to toric
hyperk\"ahler varieties, yields
new inequalities for the $h$-numbers of matroids. See also
\cite{hausel-sturmfels} for the original argument on toric hyperk\"ahler
varieties.
Furthermore this conjecture
can be proved when $n=2$ using the explicit description of the cohomology ring
in \cite{HT2}. The general case  can also be thought of as an analogue
of the Faber
conjecture \cite{faber} on the cohomology of the moduli space of curves,
which is another non-compact variety whose cohomology ring is conjectured
to satisfy a certain form of the Hard Lefschetz theorem.
\end{conjecturemark}

\begin{conjecturemark}
There are two subspaces of the cohomology $H^*(\M_{B},\C)$
 which are
 particularly interesting.
One is the middle dimensional cohomology $H^{2N}(\M_{B},\C)$,
 which is the
 top non-vanishing cohomology.
The mixed Hodge structure will break it into parts with respect to the $q$-degree. The
curious Poincar\'e dual (\ref{curious}) of these spaces are also
 interesting: it is
 easy to see that they are exactly the pure part of the mixed Hodge structure i.e. spaces of the form $H^{i,i;2i}$.  (Another significance of the pure part is that if there is a smooth projective
compactification of the variety  then its image is in this pure part.)
Thus it would already be interesting  to get the pure part of $H_n(q,t)$. In fact it is easy to identify the pure part in our
case with what we call the Pure ring, which is the subring of $H^*(\M,\C)$ generated by the
tautological classes $a_i\in H^{2i}(\M,\C)$ for $i=2,..,n$
(the other
 tautological classes, which generate the cohomology ring,
are not pure classes).

For example when $n=2$, it was known
\cite{hitchin}
that the middle degree cohomology of the Higgs moduli space
$\M_{Dol}(PGL(2,\C))$
is $g$ dimensional.
The Pure ring was determined in \cite{HT2},
and it was found to be
$g$ dimensional due to the relation $\beta^g=0$ (where
$\beta=a_2$).
Thus these two seemingly
unrelated observations are dual to each other via our curious Poincar\'e duality (\ref{curious}). To see this curious duality in action let us
recall the formula (\ref{h2qtg3}). The terms which contain the top degree
$12$ in $t$ are $t^{12}q^{12}$, $t^{12} q^{10}$ and $t^{12} q^8$, which
are curious Poincar\'e dual via (\ref{curious}) to the terms $1$, $t^4q^2$
and $t^8q^4$, which is exactly the ring generated by the degree four class
$\beta$, which has additive basis $1$, $\beta$ and $\beta^2$.

The analogous
ring, generated by the corresponding classes
$a_2,\dots,a_n\in H^*(\N,\C)$,
which a priori is a quotient of our Pure ring
(as $\N\subset \M_{Dol}$ naturally), was studied for
the moduli space $\N$ of rank $n$, degree $d$ stable bundles (with
 $(n,d)=1$) in \cite{earl-kirwan},
where they in particular found the top non-vanishing degree of this
ring to be $2n(n-1)(g-1)$.
Computer calculations for our conjecture for $n=2,3,4$ also show
that
our conjectured Pure ring has the same $1$-dimensional top
degree. This and the known situation for $n=2$ (see \cite{HT2}),
yields the following

\begin{conjecture} \label{pontrequ} The Pure rings of
$\M_{Dol}$
 and $\N$, i.e. the subrings of the cohomology rings generated
by the classes $a_2,\dots,a_n$
are isomorphic. In particular, unlike the whole cohomology ring of
 $\N$, it does not depend on $d$.
\end{conjecture}

Now we explain a
combinatorial consequence of this conjecture. First
we extract  a conjectured formula for $PP_n(t)$
the Poincar\'e polynomial  of the Pure ring. Indeed we only have
to deal with monomials in Conjecture~\ref{mainconj} whose
$t$-degree
is double of their $q$-degree.

Thus let $$PV_n(t)=PP_n(t)\frac{ t^{2(1-g)n(n-1)}}{(t^2-1)},$$ and
$$PZ_n(t,T)=\exp\left(\sum_{r\geq 1} PV_n (t^r) \frac{T^r}{r}\right).$$ We now  define
the pure part of the $t$-deformed Hook polynomials for genus
$g$ and
partition $\lambda$ as follows:
\begin{eqnarray*}{\cal P H}^{\lambda}_{g}(t)=
t^{4(1-g)n(\lambda^\prime)} \prod_{x\in d(\lambda); a(x)=0}
\frac{1}{(1-t^{2 h(x)})},\end{eqnarray*}
where $$n(\lambda^\prime):=\sum_{z\in d(\lambda)} l(z).$$
Thus we get the conjecture that $PP_n(t)$ is given by
\begin{eqnarray} \label{pontr}
\prod_{n=1}^\infty PZ_n(t,T^n) = \sum_{\lambda\in {\cal P}} {\cal P
  H}^\lambda_g (t) T^{|\lambda|}.\end{eqnarray}

Now combining the two conjectures above we can formulate a new conjecture:

\begin{conjecture} The rational functions $PP_{n}(t)$ defined in (\ref{pontr}) satisfy
  \begin{itemize} \item $PP_{n}(t)$ is a polynomial in $t$
 \item all coefficients of $PP_{n}(t)$ are non-negative integers
 \item The degree of $PP_{n}(t)$ is $2n(n-1)(g-1)$, and the coefficients of the leading term is $1$
\end{itemize}

\end{conjecture}

So for example when $n=3$ our conjecture gives for the Poincar\'e polynomial of the Pure ring:
$$PP_3(t)={\frac {1}{ \left( {t}^{6}-1 \right)  \left( {t}^{4}-1 \right) }}+t^{12\,g-12} -{
\frac {{t}^{8\,g-8}}{t^2-1}}+\frac{1}{3}\,{\frac {{t}^{12\,g-12}}{ \left( t^2-1
 \right) ^{2}}}-\frac{1}{3}\,{\frac {{t}^{12\,g-12}}{{t}^{4}+t^2+1}}-{\frac {{t}^{8\,g-8}}{ \left( {t}^{4}-1
 \right)  \left( t^2-1 \right) }}+{\frac {{t}^{
12\,g-12}}{t^2-1}}$$
\end{conjecturemark}

\begin{conjecturemark} Interestingly we can modify the formula of Conjecture~\ref{mainconj} to get a
   conjectured formula for the mixed Hodge polynomial of
   $\M_{Dol}$. Recall from Theorem~\ref{pure} that the
   mixed Hodge structure on $H^k(\M_{Dol},\C)$ is pure of
   weight $k$, thus this mixed Hodge polynomial is equivalent
with the $E$-polynomial.

We now introduce polynomials $H_n(q,x,y)$ of three variables.
Let $$V_n(q,x,y)=H_n(q,x,y)  \frac{(qxy)^{(1-g)n(n-1)}(qx+1)^{g} (qy+1)^{g}}{(qxy-1)(q-1)},$$ and
$$Z_n(q,x,y,T)=\exp\left(\sum_{r\geq 1} V_n (q^r,-(-x)^r,-(-y)^r) \frac{T^r}{r}\right).$$  We define
the $(x,y)$-deformed Hook polynomials for genus $g$ and partition $\lambda$ as follows:
\begin{eqnarray}{\cal H}_g^{\lambda}(q,x,y)=
\prod_{z\in d(\lambda)}
\frac{(qxy)^{(2-2g)l(z)}(1+q^{h(z)} y^{l(z)}
  x^{l(z)+1})^{g}(1+q^{h(z)} x^{l(z)} y^{l(z)+1})^{g}} {(1-q^{h(z)}
  (xy)^{l(z)+1})(1-q^{h(z)}(xy)^{l(z)})}.\label{xydefhook}\end{eqnarray} The following generating
function defines $H_n(q,x,y)$:
\begin{eqnarray}
\prod_{n=1}^\infty Z_n(q,x,y,T^n) = \sum_{\lambda\in {\cal P}} {\cal H}^\lambda_g (q,x,y) T^{|\lambda|}.\end{eqnarray}

Clearly we have $H_n(q,t,t)=H_n(q,t)$ which says that a specialization
of $H_n(q,x,y)$ gives the mixed Hodge polynomial $H_n(q,t)$ of
$\M_B$.  The following conjecture says that another
specialization gives the mixed Hodge polynomial of $\M_{Dol}$ and
$\M_{DR}$.

\begin{conjecture} $H_n(q,x,y)$ is a polynomial with non-negative
 integer coefficients
with specialization
$H_n(1,x,y)$ equal to the $E$-polynomial
  of  the Higgs moduli space  $\M_{Dol}(PGL(n,\C))$. \label{higgs}
\end{conjecture}

Thus we have a mysterious formula $H_n(q,x,y)$ which
 specializes, on one hand to the $H$-polynomial of the character
 variety, and on the other hand
to the mixed Hodge polynomial of the Higgs (or
 equivalently flat connection) moduli space. It would be very
 interesting to find a geometrical meaning for $H_n(q,x,y)$.

Checks on this Conjecture~\ref{higgs} include a proof for $n=2$ and
$n=3$, (one can easily modify Hitchin's and Gothen's argument to get
the Hodge polynomial instead of the Poincar\'e polynomial of the Higgs
moduli space) and
also computer checks that the shape of the polynomial $H_n(1,x,y)$ is the
expected one when $n=4$.

Consider now the specification $H_n(q,-1,y)$. Interestingly, the
corresponding specification of the $(x,y)$-deformed Hook
polynomials (\ref{xydefhook}) becomes a polynomial, showing that
at least $H_n(q,-1,y)$ is
a polynomial. We get an even nicer formula if we make the further
specification $H_n(1,-1,y)$ which by Conjecture~\ref{higgs}
should be the Hirzebruch $y$-genus
of the moduli space of Higgs bundles $\M_{Dol}$. Namely, for $g>1$, most of the $(x,y)$
deformed Hook polynomials vanish, when one substitutes first $x=-1$ and
then $q=1$. Indeed, the only partitions which will have a non-zero
contribution to the $y$-genus are the partitions of the form
 $n=1+1+\dots+1$; when $l(z)=0$ only for once.
This in turn
gives the following closed formula for the conjectured $y$-genus of
$\M_{Dol}$:

\begin{conjecture} The Hirzebruch $y$-genus of $\M_{Dol}(PGL(n,\C))$, for $g>1$,
equals $$ (1-y+\dots +(-y)^{n-1})^{g-1}\sum_{m | n} \frac{\mu(m)}{m} \left( (-y)^{n(n-n/m)}m
\prod_{i=1}^{n/m-1} (1-(-y)^{mi})^2 \right)^{g-1}$$
\end{conjecture}

In particular note that the term corresponding to $m=1$, is exactly
the known $y$-genus of $\N$ (see \cite{narasimhan-ramanan}). The rest
thus should be thought as contribution of the other fixed point
components of the circle action on $\M_{Dol}$. Of course this conjectured
formula gives the known specialization of Corollary~\ref{euler}
at $y=-1$, while the $y=1$ specialization gives $\mu(n) n^{g-2}$ when
$n$ is odd, and $0$ when $n$ is even. The specialization at $y=1$
can be
thought of as the signature of the pairing on the rationalized
circle equivariant
cohomology of $\M_{Dol}$ as defined in \cite{hausel-proudfoot}.
\end{conjecturemark}

\begin{conjecturemark} Finally we discuss how to obtain a conjecture
for the mixed Hodge polynomial of $\M_{B}(SL(n,\C))$. For the mixed Hodge polynomial of
$\M_{Dol}(SL(n,\C))$ the mirror symmetry conjecture~\ref{mirrorderham},
together with Conjecture~\ref{higgs} imply a conjecture. For $\M_{B}(SL(n,\C))$ the mixed Hodge polynomial contains more information than the $E$-polynomial. In order thus to have a conjecture on
$H_n(x,y,t;\M_{B}(SL(n,\C)))$
a mirror symmetry conjecture is needed on the level of the $H$-polynomial.
We finish by formulating such a conjecture, generalizing
Conjecture~\ref{mirrorbetti} for $H$-polynomials:

\begin{conjecture} For all $d,e \in \Z$, with $(d,n)=(e,n)=1$ we have
$$\Hst^{B^e}\Left(x,y,t;\M_\B(SL(n,\C))\Right)
= \Hst^{\hat B^d}\Left(x,y,t;{\mathcal M}^e_\B(PGL(n,\C))\Right),$$
where $\Hst^B$ is the stringy mixed Hodge polynomial twisted with a
$B$-field, which can be defined identically as $\Est^B$ is
defined in (\ref{bfieldstringy}).
\end{conjecture}

\end{conjecturemark}

config: arxiv


TITLE: Interesting finite-cardinality theorems
QUESTION [14 upvotes]: What are the most interesting results in mathematics that say there are only finitely many of something?
Examples:

If it's ever shown that there are only finitely many twin primes, that would fit here.
The compactness theorem for first-order logic is an example.  Suppose $S$ is a set of first-order sentences.  Suppose $\varphi$ is a first-order sentence that is true in every structure in which every sentence in $S$ is true.  Then for some finite $S_0\subseteq S$, $\varphi$ is true in every structure in which every sentence in $S_0$ is true.
The Robertson–Seymour theorem is an example.  If a class of finite graphs is closed under taking minors, then it is equal to the class of graphs that have no minor isomorphic to any of the members of a set of "forbidden minors".  The theorem says that that set is always finite.  (Among such classes of graphs are planar graphs, outer-planar graphs (embeddable in a plane with all vertices on a circle), graphs knotlessly embedable in $\mathbb R^3$, and graphs linklessly embedable in $\mathbb R^3$.  In some cases, finding the finite set of forbidden minors is challenging.
The Borel–Cantelli lemma considers an infinite sequence $E_1, E_2, E_3, \ldots$ of events in a probability space.  It says that if $\Pr(E_1)+\Pr(E_2)+\Pr(E_3)+\cdots<\infty$ then only finitely many of $E_1, E_2, E_3, \ldots$ occur. (Or maybe more precisely,: the probability that only finitely many occur is $1$.  The set of outcomes for which infnitely many of the events occur may be non-empty, but its measure is $0$.)

PS: Maybe two different sorts of results can fit here:

Results that say, for example, exactly eleven of something exist, and maybe lists them.  For example, there are only five regular polyhedra, and we all know which five.  If only finitely many twin primes exist, then perhaps a theorem would say exactly which ones they are.
Results that say that something must in every instance be finite, but do not and cannot give a finite upper bound.  The The Borel–Cantelli lemma is of that sort.  Even specifying which events are in the sequence and what their probabilities are does not make it possible to give a finite upper bound.  The compactness theorems of logic are also of that sort.  And maybe I should have mentioned compactness theorems of topology?  Tychonoff's theorem,  maybe?  The Robertson–Seymour theorem in general is of that sort, but some of its concrete instances are substantial theorems in their own right.  The one that says planar graphs are those that exclude two specified forbidden minors.  I seem to recall a simple example in which their were something like 30 forbidden minors – maybe it was triangulations of a torus or something like that.
Maybe in some cases the question of whether an example is of the first kind or the second might itself be a hard problem.

Despite my inclusion of the first bullet point above, somehow I hesitate to include uniqueness theorems in general.  There are zillions of those, and I don't think anyone would want to compile a comprehensive list of uniqueness theorems in mathematics unless maybe they're creating a reference book.

REPLY [1 votes]: This result in Universal Algebra is an obvious one but I find it very useful (and often used) in many branches of Algebra:
Let $A,B,C$ be algebras over a signature $\Omega$. Let $f:A\rightarrow C$ and $g:A\rightarrow B$ be epimorphisms such that $\ker g\subset\ker f$. Then there exist exactly one homomorphism $\bar{f}:B\rightarrow C$ such that $\bar{f}g=f$.
A related results in Algebra (theory of groups, rings, modules...) are better known as Isomorphism Theorems.
One more, rather obvious but useful, construction:
Let $X$ be a set and $f:X\rightarrow A$ a (set theoretic) function into the $\Omega$-algebra $A$. Then there exists exactly one homomorphic extension $\bar{f}:F_\Omega(X)\rightarrow A$ of the function $f$.
Again, the uniqueness of the homomorphic extension is rather obvious, but the whole construction is of great importance.


TITLE: L is normal extension if it's splitting field
QUESTION [0 upvotes]: I am trying to understand this proof That if L is splitting field of some polynomial then it's normal extension .

I got the part till we get that $L(\alpha)/K(\alpha)$ and $L(\beta )/K(\beta)$ are isomorphic as they are splitting field of same irreducible polynomial over two isomorphic fields ( right ? ) , on second page .
But after it how are we writing
$$[L(\alpha):K]=[L(\beta ):K]$$ ?
Why didn't we directly wrote $[L(\alpha):L]=[L(\beta ):L]$ ,as both $L(\alpha)$ and$L(\beta)$ are isomorphic and extension of L ?

REPLY [1 votes]: We don't know a priori that $L(\alpha)$ and $L(\beta)$ are isomorphic extensions of $L$. This is exactly what we are trying to prove: we have that $L(\alpha) = L$, because we assume that $\alpha$ belongs to $L$, and we are trying to prove that any other root $\beta$ of the minimal polynomial $m \in K[x]$ of $\alpha$ over $K$ also belongs to $L$, so that $m \in K[x]$ must in fact split into linear factors over $L$.
Consider, for example, $K = \mathbb{Q}$, and $L = K(\sqrt[3]{2})$, and let $\alpha = \sqrt[3]{2}$. Then, minimal polynomial of $\alpha$ over $K = \mathbb{Q}$ is $m(x) = x^3 - 2$. The splitting field $M$ of $m$ is not $L$. Let $\beta$ be one of the complex roots of $x^3 - 2$. Then, $L(\alpha) = L$, but $L(\beta) \ne L$.
What might have confused you, though, is that we do easily know that $K(\alpha)$ and $K(\beta)$ are isomorphic extensions of $K$, because they are both isomorphic to the quotient ring $K[x]/(m)$ via maps $x \mapsto \alpha$ and $x \mapsto \beta$.


TITLE: On number of solutions to $y^2 \equiv x^3 \pmod{p}$ when $p=1\pmod{6}$
QUESTION [1 upvotes]: Concerning the congruence relation $y^2 \equiv x^3 \pmod{p}$ there are $p+1$ solutions to every $p$ when $p$ denotes a prime number. (Note that we include the solution of $(x,y)=(\infty,\infty)$ when we say there are $p+1$ soutions.)
For the case of $p\in\{2,3\}$, it is trivial. For those which satisfy $p=5\pmod{6}$, the proof is relatively easy, since we can deduce it directly from the number of solutions of quadratic residue and Fermat's little theorem.
(For those who want the source, please refer the following document of wikipedia. Cubic Reciprocity)
However, I want to teach myself with the case when the prime number is of the form satisfying $p=1\pmod{6}$. The hyperlink I've attached above just shows the brief description of the proof regarding group theory.
Could somebody teach me how to prove there are $p+1$ solutions to $y^2 \equiv x^3 \pmod{p}$ when $p=1\pmod{6}$? If somebody knows more than one proof, I would be so grateful if I could see them all, only when if the somebody's time is permissible. However if the somdbody has only limited sources regarding the proof I want to be taught in the language of group theory, which is still grateful.
Also I would appreciate if there is someone who could introduce some textbooks which deal with the solutions of $y^2 \equiv x^3 \pmod{p}$, especially for the case when $p=1\pmod{6}$. The more the number of texbooks being recommended, the better it is for me.
Many thanks.

REPLY [2 votes]: $\mathbb{Z}_p^*$, which means the non-zero elements of $\mathbb{Z}_p$ as a group under multiplication, is cyclic of order $p-1$.
Reference link
Therefore, you may as well solve $2a\equiv3b \,\text{mod}(p-1)$.


TITLE: Is the modulus of curvature the speed at which tangents change direction?
QUESTION [4 upvotes]: I'm having trouble understanding an argument: the modulus of curvature it is the speed at which tangents change direction.
Let $\alpha:I \rightarrow \mathbb{R}^2$ a plane curve arc length parametrized and $k(s)$ the curvature of $\alpha$ is $s$. Consider the tangent vectors $\alpha'(s_0)$ and $\alpha'(s_0+h)$, where $s_0 \in I$ is fixed and $s_0 + h \in I$. Denote by $\phi(h)$ the angle between these two vectors, ie,
\begin{equation}
\cos\phi(h) = \langle  \alpha'(s_0) ,\alpha'(s_0+h) \rangle.
\end{equation}
(because $|\alpha'(s_0)| = |\alpha'(s_0+ h)| = 1$).
The limit $\displaystyle \lim_{h \rightarrow 0} \frac{\phi(h)}{h}$ is the speed at which tangents change direction. We have
\begin{equation}
 |\alpha'(s_0+h) - \alpha'(s_0)| = 2 \sin\frac{\phi(h)}{2}
\end{equation}
for all $h$ and so
\begin{equation}
|k(s_0)| = |\alpha''(s_0)| = \lim_{h \rightarrow 0} \frac{\phi(h)}{h}.
\end{equation}
Comments:
I'm not able to verify the last two equations:
$$|\alpha'(s_0+h) - \alpha'(s_0)|^2 = |\alpha'(s_0+h)|^2 - 2 \langle \alpha'(s_0+h) , \alpha'(s_0) \rangle + |\alpha'(s_0)|^2 = 2 - 2\cos{\phi(h)}.$$
I don't know which trigonometric identity is being used.
The last equation is also not being able to verify.
Thank you for your help.

REPLY [1 votes]: The first equation uses cosine double-angle formula:
$$
\cos\phi=1-2\sin^2{\phi\over2}.
$$
The second equation features a standard trick to compute the limit:
$$
|\alpha''(s_0)|
=\lim_{h\to0}{|\alpha'(s_0+h) - \alpha'(s_0)|\over h}
= \lim_{h\to0}\left({2\over h} \sin\frac{\phi(h)}{2}\right)
\\
= \lim_{h\to0}\left({\phi(h)\over h}\cdot {\sin(\phi(h)/2)\over\phi(h)/2}\right)
= \lim_{h\to0}\left({\phi(h)\over h}\right),
$$
because $\displaystyle{\sin\theta\over\theta}\to1$ for $\theta\to0$.


TITLE: Principal component analysis of complex-valued time series
QUESTION [0 upvotes]: My problem is the following: I have many, say around 800 complex-valued time-series, but I have a hunch that with a proper unitary transformation they can actually be transformed into another 800 time series where any significant change happens in only two of them, and the rest of the series should be close to constant zero. (The series are the coefficients of basis vectors from a quantum mechanics simulation, and based on the physics of the problem I have this feeling.) Can you give me hints on how to find this unitary transformation? I was thinking about principal component analysis and singular value decomposition, but I am not sure that they are satisfactory in case of complex numbers.

REPLY [1 votes]: Suppose $m$ is the number of time series you have (800 in your example), and $n$ is the size of each time series. You can construct a complex data matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ containing your time series. Then you can apply SVD to diagonalize and obtain:
$$
\mathbf{A} = \mathbf{U}\mathbf{\Sigma}\mathbf{V}^H
$$
where $\mathbf{U}\in\mathbb{C}^{m\times m}$ and $\mathbf{V}\in\mathbb{C}^{n\times n}$ are unitary matrices and $\mathbf{\Sigma}\in\mathbb{C}^{m\times n}$ is a matrix containing the singular values in its diagonal (there is also the reduced-form representation of SVD where $\mathbf{\Sigma}$ is square!).
Then you can represent your data using the eigenvectors with largest singular values.


TITLE: The tangent space of the graph submanifold
QUESTION [1 upvotes]: I have gotten stuck at an exercise that is concerned with the submanifold graph.

Here is what I have:
Define $q:=f(p)$. Because $M,N$ are smooth manifolds we find open sets $U,V$ in $M,N$ respectively such that $p \in M, q \in N$ and
charts
$$
\varphi: U \to \mathbb{R}^m \\
\psi: V \to \mathbb{R}^n.
$$
Define
$$
\delta: U \times V \to \mathbb{R}^m \times \mathbb{R}^n, \
(x,y) \mapsto (\varphi(x), \psi(y)-(\psi \circ f \circ \varphi^{-1})(y))
$$.
This is injective and maps $U \times V$ to $\delta(U \times V)$ and is hence invertible. Moreover, because $f$ is smooth it follows that $\psi \circ f \circ \varphi^{-1}$ is smooth and hence continous. Because $\varphi, \psi$ are continous, $\delta$ is also continous. So $\delta$ is a chart on $M \times N$. Let $\tilde{U}:=\varphi(U)$. By construction $\delta(\Gamma_f \cap (U \times V))=\tilde{U} \times \{0\}$ which is an open set in $\mathbb{R}^m \times \mathbb{R}^n$. Thus $\Gamma_f$ is a submanifold.
But now I am unsure about the tangent space. The tangent space consists of linear maps
$$
X: C^{\infty}(\Gamma_f) \to \mathbb{R}, \
g \mapsto Xg
$$
such that there is a smooth curve $\gamma: (-\varepsilon, \varepsilon) \to \Gamma_f$ with $\gamma(0)=p$ and $Xg=(g \circ \gamma)'(0)$.
The coordinate charts $\varphi, \psi$ are of the form $\varphi=(x_1,...,x_m), \psi=(y_1,...,y_n)$. Take $g \in C^{\infty}(\Gamma_f)$ and define $X_i,X_j$ via
$$
X_i g:=\partial_i (g \circ \delta^{-1}) (\delta((p,q))), \
i \in \{1,...,m\} \tag{1} \\
$$
$$
X_j g:=\partial_j (g \circ \delta^{-1}) (\delta((p,q))), \
j \in \{1,...,n\} \tag{2}.
$$
where $\delta$ denotes the chart from above. The functions of the form (1) and (2) form a basis of the tangent space. So, if I can find an explicit formula for the vectors I know what the tangent space looks like. However I am a bit at a loss about computing the vectors, because $\delta^{-1}$ is made up of homeomorphisms wich aren't necessarily differentiable by themselves. Is there perhaps another way to find a description of the tangent space?

REPLY [2 votes]: You're overcomplicating it with charts. The only thing you need to know is that tangent vectors at a point are initial velocities of curves starting at that point. The claim is that $$T_{(p,f(p))}\Gamma_f = \{ (v,{\rm d}f_p(v)) \mid v \in T_pM\}.$$
First, let $(v,w) \in T_{(p,f(p))}\Gamma_f$. Then there is $\gamma\colon (-\epsilon,\epsilon)\to \Gamma_f$ such that $\gamma(0) = (p,f(p))$ and $\gamma'(0) = (v,w)$. Writing $\gamma(t) = (\alpha(t),\beta(t))$, the fact that $\gamma(t)\in \Gamma_f$ says that $\beta(t) = f(\alpha(t))$. So $$(v,w) = \gamma'(0) = (\alpha'(0),(f\circ \alpha)'(0)) = (\alpha'(0), {\rm d}f_{\alpha(0)}(\alpha'(0))),$$and thus $$v = \alpha'(0) \implies w = {\rm d}f_{\alpha(0)}(\alpha'(0)) = {\rm d}f_p(v).$$Conversely, for any $v\in T_pM$, let's show that $(v,{\rm d}f_p(v))\in T_{(p,f(p))}\Gamma_f$. Let $\alpha\colon (-\epsilon,\epsilon) \to M$ be such that $\alpha(0)=p$ and $\alpha'(0)=f(p)$. Now, define $\gamma\colon (-\epsilon,\epsilon)\to \Gamma_f$ by $\gamma(t) = (\alpha(t),f(\alpha(t)))$. Then $\gamma(0) = (p,f(p))$ says that $\gamma'(0)\in T_{(p,f(p))}\Gamma_f$. But what is $\gamma'(0)$? Compute $$\gamma'(0) = (\alpha'(0),(f\circ\alpha)'(0)) = (\alpha'(0),{\rm d}f_{\alpha(0)}(\alpha'(0))) = (v, {\rm d}f_p(v)),$$as required.


TITLE: Norm of right shift on finite dimensional space
QUESTION [0 upvotes]: Let $X$ be a Banach space with a (normalized) Schauder basis $(e_{j})_{j=1}^{\infty}$. Then, for all $x\in X$, there exists a unique sequence of scalars $(\lambda_{j})_{j=1}^{\infty}$ such that
\begin{equation} \limsup_{N\to\infty}\left\Vert x-\sum_{j=1}^{N}\lambda_{j}e_{j}\right\Vert =0 \end{equation}
and the projection operators $P_{N}(x)=P_{N}\left(\sum_{j=1}^{\infty}\lambda_{j}e_{j}\right)=\sum_{j=1}^{N}\lambda_{j}e_{j}$ are known to be linear and (as $X$ is Banach) continuous. Consider the linear function $T:P_{N}(X)\to P_{N+M}(X)$ defined by:
\begin{equation} T(x)=T\left(\sum_{j=1}^{N}\lambda_{j}e_{j}\right)=(0,\ldots,0,\lambda_{1}e_{1+M},\ldots,\lambda_{N}e_{N+M})  \end{equation}
This function is automatically continuous because $\dim(P_{N}(X))<\infty$ and clearly,
\begin{equation} \|T(e_{1})\|=\|e_{1+M}\|=1 \implies 1\leq\sup\{\|T(x)\| \mid x\in P_{N}(X) \text{ and }\|x\|=1\}=\|T\|_{\text{op}}.\end{equation}
My intuition says that, in fact, the operator norm of $T$ should be equal to $1$ because this function would be an isometry if $X=\ell_{p}$ for any $p\in[1,\infty]$. Is this true? If so, could someone give me a hint as to how to prove it or suggest a reference? I have tried supposing $\|x\|<\|T(x)\|$ for some unit norm $x\in P_{N}(X)$ to derive a contradiction, but playing around with the inequality $0<\|T(x)\|-\|x\|$ hasn't yielded anything. I have also tried to use the fact that all norms are equivalent on a finite-dimensional space and this hasn't seemed to help either. Any suggestions would be greatly appreciated!

REPLY [2 votes]: You meant $T(\sum\limits_{j=1}^N \lambda_j e_j)=\sum\limits_{j=1}^N \lambda_j e_{j+N}$, right? (You wrote the output in coordinates). However, this need not be an isometry, for example let $X=\mathbb{C}^3$, $e_1=(1, 0, 0)$, $e_2=(0,1,0)$ and $e_3=(0, \frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$. For $N=2$, $M=1$ we have $||T(e_1+e_2)||=||e_2+e_3||\neq ||e_1+e_2||$ so $T$ is not an isometry. One case in which $T$ would be an isometry is if $X$ were a Hilbert space and $\{e_j\}_j$ were an orthonormal sequence.


TITLE: Rotation systems. Problem interpreting an equation
QUESTION [1 upvotes]: In this equation:
$$
\mathbf a_i\overset{\rm def}{=}\left(\frac{d^2\mathbf r}{dt^2}\right)_i=\left(\frac{d\mathbf v}{dt}\right)_i=\left[\left(\frac{d}{dt}\right)_r+\boldsymbol\Omega\times\right]\left[\left(\frac{d\mathbf r}{dt}\right)_r+\boldsymbol\Omega\times\mathbf r\right]
$$
(from Wikipedia), why is
$$\left(\frac{d}{dt}\right)_r \boldsymbol{\Omega} \times \mathbf{r}=\frac{d\boldsymbol{\Omega}}{dt}\times\bf{r}+\bf{\Omega}\times \bf{V_r}$$
In particular, I have qualms with the term
$$\frac{d\bf{\Omega}}{dt}\times\bf{r}$$
Why are we deriving the angular velocity? Why is it a derivative not of the rotational type (Namely $(\frac{d}{dt})_r$ )
Other sources do not point out that term I have problems with. In any case, I want to know how you evaluate that derivative.

REPLY [1 votes]: Other sources do not point out that term I have problems with.

Other sources explicitly assume a constant angular velocity and thus ignore that component. The wikipedia article you cited is correct.

In any case, I want to know how you evaluate that derivative.

Given any vector quantity $\mathbf q$ that is the same (other than component representation) in the inertial and rotating frame, the time derivative of that vector from the perspective of an inertial versus rotating observer is
$$\left(\frac{d\mathbf q}{dt}\right)_I = \left(\frac{d\mathbf q}{dt}\right)_R + \boldsymbol\Omega \times \mathbf q$$
In dynamics, this is sometimes called the transport theorem (but there are a number of other things called the transport theorem).
Applying the transport theorem to the angular velocity vector yields
$$
\left(\frac{d\boldsymbol\Omega}{dt}\right)_I =
  \left(\frac{d \boldsymbol\Omega}{dt}\right)_R
  + \boldsymbol\Omega \times \boldsymbol\Omega
= \left(\frac{d \boldsymbol\Omega}{dt}\right)_R
$$
In other words, angular acceleration is fundamentally the same vector in the inertial and rotating frame.
Applying the transport theorem instead to angular momentum yields
$$\left(\frac{d\mathbf L}{dt}\right)_I = \left(\frac{d\mathbf L}{dt}\right)_R + \boldsymbol\Omega \times \mathbf L$$
The rotational analog of Newton's second law provides an alternative representation of the left-hand side of the above: $\frac {d\mathbf L}{dt} = \boldsymbol \tau_{\text{ext}}$ where the derivative is calculated from the perspective of an inertial frame and $\boldsymbol \tau_{\text{ext}}$ is the external torque on the system. If the system is a rigid body, the angular momentum is given $\mathbf L = \mathrm I \boldsymbol\Omega$ where $\mathrm I$ is the object's inertia tensor. Since the inertia tensor of a rigid body is constant in a frame rotating with the body, the time derivative of the angular momentum vector from the perspective of an observer rotating with the object simplifies to $\left(\frac{d\mathbf L}{dt}\right)_R = \mathrm I\left(\frac{d \boldsymbol\Omega}{dt}\right)_R$. Putting all of the above together yields
$$
\boldsymbol\tau_{\text{ext}} =
  \mathbf I \frac{d \boldsymbol\Omega}{dt}
  + \mathbf \Omega \times (\mathrm I \, \boldsymbol\Omega)$$
or
$$
\frac {d \boldsymbol\Omega}{dt} =
  {\mathbf I}^{-1}
  \left(
    \boldsymbol\tau_{\text{ext}}
    - \boldsymbol \Omega \times (\mathrm I \, \boldsymbol \Omega)
  \right)
$$
This yields a way to calculate $\frac {d\boldsymbol \Omega}{dt}$ at any point in time for a rigid body. Whether this is integrable via elementary methods is a different question. In most cases, it isn't. It's rather challenging to find a non-trivial rotational system that has an analytic solution. One typically has to revert to numerical methods to determine the rotational behavior of an object.

REPLY [0 votes]: If you just follow your nose, then...
$$\left(\frac{d}{dt}\right)_{rotating} {\bf{\Omega}} =\frac{d\bf{\Omega}}{dt}+\bf{\Omega}\times {\bf{\Omega}}$$
Do you know what the second term is equal to? Hopefully this clears up the problem you have.