Mixed Hodge polynomials of character varieties. With an appendix by Nicholas M. Katz.
Let \(g\geq 0\) and \(n>0\) be integers. Let \(\zeta_n\in \mathbb C\) be a primitive \(n\)-th root of unity. Let \(I_n\) be the identity \(n\times n\)-matrix. For two square matrices \(A\) and \(B\) denote their commutator \(ABA^{-1}B^{-1}\) by \([A, B]\) and define \(\mathcal M_n\) to be the following affine GIT quotient by the conjugation action of \(\mathrm{GL}(n, \mathbb C)\): \[ \{ (A_1, B_1,\dots, A_g, B_g) \mid A_i, B_i\in \mathrm{GL}(n, \mathbb C),\quad [A_1, B_1]\cdot\dots\cdot[A_g, B_g]=\zeta_n I_n \}// \mathrm{GL}(n, \mathbb C). \] \(\mathcal M_n\) is a twisted character variety of a genus \(g\) closed Riemann surface. The paper under review deals with such varieties. The \(E\)-polynomials of \(\mathcal{M}_{n}\) are calculated by counting points over finite fields using the character table of the finite group of Lie-type \(\mathrm{GL}(n,\mathbb{F}_{q})\) and a theorem proved in the appendix of the paper by N. Katz. Several geometric results are deduced from this calculation, for example, the value of the topological Euler characteristic of the associated \(\mathrm{PGL}(n,\mathbb C)\)-character variety. The calculation also leads to several conjectures about the cohomology of \(\mathcal{M}_{n}\): an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. These conjectures are proven for \(n=2\). The paper consists of 6 sections. 1. A motivation with relevant references is presented in the introduction. An outline of the paper is given, the main results of the paper are formulated. 2. In Section~2 various facts needed later are collected. 3. Section~3 deals with the calculation of the \(E\)-polynomial of \(\mathcal M_n\). 4. In Section~4 a conjecture on the mixed Hodge polynomial of \(\mathcal M_n\) is formulated, some consequences are derived. 5. The proof of all the conjectures from Section~4 is given in Section~5 in the case \(n=2\). 6. Section~6 is an appendix written by Nicholas M. Katz. It presents some important results about \(E\)-polynomials, zeta-equivalence, and polynomial-count varieties. Those results are used to compute the \(E\)-polynomials of \(\mathcal M_n\) in Section~3.
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