Related papers: Root data in character varieties

We study properties of irreducible and completely reducible representations of finitely generated groups Gamma into reductive algebraic groups G in in the context of the geometric invariant theory of the G-action on Hom(Gamma,G) by…

Let $k$ be a field, let $H \subset G$ be (possibly disconnected) reductive groups over $k$, and let $\Gamma$ be a finitely generated group. Vinberg and Martin have shown that the induced morphism of character varieties \[…

Given a finitely-generated group G, and a finite group \Gamma, Philip Hall defined \delta_\Gamma to be the number of factor groups of G that are isomorphic to \Gamma. We show how to compute the Hall invariants by cohomological and…

Let $\Gamma$ be a finitely generated group, $G_\mathbb{C}$ be a classical complex group and $G_\mathbb{R}$ a real form of $G_\mathbb{C}$. We propose a definition of the $G_\mathbb{R}$-character variety of $\Gamma$ as a subset…

Let $\Gamma$ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety $\mathrm{Hom}(\Gamma,G)$ and the character variety $\mathrm{Hom}(\Gamma,G)//G$ each carry a natural topology,…

Suppose $k$ is a field, $G$ is a connected reductive algebraic $k$-group, $T$ is a maximal $k$-torus in $G$, and $\Gamma$ is a finite group that acts on $(G,T)$. From the above, one obtains a root datum $\Psi$ on which…

Let $M_C(G)$ be the moduli space of semistable principal $G-$bundles over a smooth curve $C$. We show that a flat degeneration of this space $M_{C_{\Gamma}}(G)$ associated to a singular stable curve $C_{\Gamma}$ contains the free group…

In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good GIT quotients regardless of…

Let $\Gamma$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $\Gamma$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $\Gamma$.…

Let G be a connected reductive affine algebraic group. In this short note we define the "variety of G-characters" of a finitely generated group F and show that the quotient of the G-character variety of F by the action of the trace…

Let $G$ be a semi-simple algebraic group over a perfect field $k$. A lot of progress has been made recently in computing the Chow motives of projective $G$-homogenous varieties. When $k$ has positive characteristic, a broader class of…

Let $\mathcal{X}_{\Gamma}G:=\mathrm{Hom}(\Gamma,G)/\!/G$ be the $G$-character variety of $\Gamma$, where $G$ is a complex reductive group and $\Gamma$ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and…

We find finite, reasonably small, generator sets of the coordinate rings of G-character varieties of finitely generated groups for all classical groups G. This result together with the method of Grobner basis gives an algorithm for…

Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation…

An algebraic zip datum is a tuple $\CZ := (G,P,Q,\phi)$ consisting of a reductive group $G$ together with parabolic subgroups $P$ and $Q$ and an isogeny $\phi\colon P/R_uP\to Q/R_uQ$. We study the action of the group $E := \{(p,q)\in…

For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $\Gamma$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety)…

Suppose that $\tilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\tilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the connected part of the…

Let $G$ be a reductive algebraic group---possibly non-connected---over a field $k$ and let $H$ be a subgroup of $G$. If $G= GL_n$ then there is a degeneration process for obtaining from $H$ a completely reducible subgroup $H'$ of $G$; one…

Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed by a set…

Let k be an algebraically closed field of characteristic zero, F its algebraically closed extension, and G be the group of k-automorphisms of F endowed with a natural topology. One of the purposes of this paper is to show that any…