Character varieties with Zariski closures of \(\mathrm{GL}_n\)-conjugacy classes at punctures. (English) Zbl 1400.14122
To define the character varieties in the title of this paper, one needs to fix the following data: a compact Riemann surface \(\Sigma\) of genus \(g\); a set \(S = \{a_1,\ldots,a_k\} \subset \Sigma\); a base point of \(\Sigma\setminus S\) and loops \(\gamma_i\in \pi_1(\Sigma\setminus S)\) for each puncture \(a_i\); a \(k\)-tuple \(\mathcal{C} = (C_1,\ldots,C_k)\) of conjugacy classes in GL\(_n(\mathbb{C})\). Then one can consider the variety \(\mathcal{U}_{\overline{\mathbb{C}}}\) of homomorphisms \(\rho: \pi_1(\Sigma\setminus S) \to\) GL\(_n(\mathbb{C})\) such that \(\rho(\gamma_i) \in \overline{C_i}\) for each \(i\) (it is in this sense that one puts the closure of the conjugacy class \(C_i\) at the puncture \(a_i\)). Then the character variety is the GIT-quotient \(\mathcal{M}_{\overline{\mathcal{C}}}:=\mathcal{U}_{\overline{\mathcal{C}}}//\)GL\(_n(\mathbb{C})\).
The author proves some basic properties of these character varieties under the assumption that \(\mathcal{C}\) is generic in a suitable sense; in particular it is shown that the quotient map \(\mathcal{U}_{\overline{\mathcal{C}}} \to \mathcal{M}_{\overline{\mathcal{C}}}\) is a principal PGL\(_n\)-bundle in the étale topology (so that the PGL\(_n\)-orbits in \(\mathcal{U}_{\overline{\mathcal{C}}}\) are all closed of the same dimension), and there is a dimension formula for \(\mathcal{M}_{\overline{\mathcal{C}}}\). Much of the work can be done by reducing to the case where the tuple \(\mathcal{C}\) consists of semisimple conjugacy classes, which has been previously studied by the same author T. Hausel, E. Letellier and F. Rodriguez-Villegas [T. Hausel et al., Adv. Math. 234, 85–128 (2013; Zbl 1273.14101)].
Once the basic properties have been worked out, the paper proceeds to consider the mixed Hodge polynomial \(IH_c(\mathcal{M}_{\overline{\mathcal{C}}};x,y,t)\) for the intersection cohomology of the character variety, with a conjecture that this polynomial depends only on \(xy\) and \(t\). One of the main results of the paper shows that the conjecture is true if \(t\) is specialized to \(-1\). There are several other results concerning the intersection cohomology of character varieties, many of which work over the algebraic closure of a finite field as well as over \(\mathbb{C}\).
The author proves some basic properties of these character varieties under the assumption that \(\mathcal{C}\) is generic in a suitable sense; in particular it is shown that the quotient map \(\mathcal{U}_{\overline{\mathcal{C}}} \to \mathcal{M}_{\overline{\mathcal{C}}}\) is a principal PGL\(_n\)-bundle in the étale topology (so that the PGL\(_n\)-orbits in \(\mathcal{U}_{\overline{\mathcal{C}}}\) are all closed of the same dimension), and there is a dimension formula for \(\mathcal{M}_{\overline{\mathcal{C}}}\). Much of the work can be done by reducing to the case where the tuple \(\mathcal{C}\) consists of semisimple conjugacy classes, which has been previously studied by the same author T. Hausel, E. Letellier and F. Rodriguez-Villegas [T. Hausel et al., Adv. Math. 234, 85–128 (2013; Zbl 1273.14101)].
Once the basic properties have been worked out, the paper proceeds to consider the mixed Hodge polynomial \(IH_c(\mathcal{M}_{\overline{\mathcal{C}}};x,y,t)\) for the intersection cohomology of the character variety, with a conjecture that this polynomial depends only on \(xy\) and \(t\). One of the main results of the paper shows that the conjecture is true if \(t\) is specialized to \(-1\). There are several other results concerning the intersection cohomology of character varieties, many of which work over the algebraic closure of a finite field as well as over \(\mathbb{C}\).
Reviewer: Michael Bate (York)
Keywords:
character varieties; Hodge polynomial; intersection cohomology; character theory of finite general linear groupsCitations:
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