Related papers: Conjugating Representations in $PGL(k, \mathbb{C})…

Let $S$ be a closed, orientable surface of genus at least 2. The cotangent bundle of the "hyperbolic'' Teichm\"uller space of $S$ can be identified with the space $\CP$ of complex projective structures on $S$ through measured laminations,…

We study which representations $\rho$ of the fundamental group of a compact oriented surface $X$ admit Higgs data that depend holomorphically on the Riemann surface $\Sigma\,=\, (X,\, J)$ via non-abelian Hodge correspondence. For…

We show that a compact representation of a semisimple Lie group has an orthogonal decomposition into finite length representations. This generalises and simplifies a number of more special spectral theorems in the literature. We apply it to…

We study geometric representations of GL(n,R) for a ring R. The structure of the associated Hecke algebras is analyzed and shown to be cellular. Multiplicities of the irreducible constituents of these representations are linked to the…

I prove the existence of slices for an action of a reductive complex Lie group on a K\"ahler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the…

We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group $G$, the derived and the stable categories…

The purpose of this paper is to extend the Donaldson-Corlette theorem to the case of vector bundles over cell complexes. We define the notion of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham…

We describe the centralizer of irreducible representations from a finitely generated group $\Gamma$ to $PSL(p,\mathbb{C})$ where $p$ is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of…

Let $\Gamma$ be a discrete subgroup of $\textrm{PGL}(d,\R)$ and fix some euclidean norm $\|\ \|$ on $\R^d.$ Let $N_\Gamma(t)$ be the number of elements in $\Gamma$ whose operator norm is $\leq t.$ In this article we prove an asymptotic for…

Let G(K) be the group of K-rational points of a connected adjoint simple algebraic group defined over a non-archimedean local field K. In this paper we classify the unipotent representations of G(K) in terms of the geometry of the Langlands…

This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain…

We identify type-preserving representations $\phi: \pi_1(\Sigma)\to \mathrm{PSL}(2,\mathbb{R})$ of the fundamental group of every punctured surface $\Sigma = \Sigma_{g,p}$ that are not Fuchsian yet send all non-peripheral simple closed…

In the present paper we are dealing with reflection equation algebras ${\cal L}(R)$ corresponding to even skew-invertible Hecke symmetries. Our main result consists in computing the characters of the spectral values of the generating matrix…

Let $\Sigma_{g,n}$ be a compact oriented surface with genus $g\geq 2$ bordered by $n$ circles. Due to Witten, the twisted Reidemeister torsion coincides with a power of the Atiyah-Bott-Goldman-Narasimhan symplectic form on the space of…

We give explicit computations of the $\Gamma$-Euler characteristic of several families of orbit space definable translation groupoids. These include the translation groupoids associated to finite-dimensional linear representations of the…

We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective…

Using the Morse-theoretic techniques introduced by Hitchin, we prove that the moduli space of $\Sp(2p,2q)$-Higgs bundles over a compact Riemann surface of genus $g\geq 2$ is connected. In particular, this implies that the moduli space of…

We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on…

Let G be a complex semisimple Lie group and H a complex closed connected subgroup. Let g and h be their Lie algebras. We prove that the regular representation of G in $L^2(G/H)$ is tempered if and only if the orthogonal of h in g contains…

We develop Fenchel-Nielsen coordinates for representations of surface groups into Sp(2n,R) with maximal Toledo invariant. Analogous to classical Fenchel-Nielsen coordinates on the Teichm\"uller space they consist of a parametrization of…