Character varieties with Zariski closures of $$\mathrm{GL}_n$$ GL n -conjugacy classes at punctures

Abstract

In the paper [20], we gave a conjectural formula for the mixed Hodge polynomials of character varieties with generic semisimple conjugacy classes at punctures and we prove a formula for the \(E\)-polynomial. We also proved that these character varieties are irreducible [21]. In this paper, we extend the results of [20, 21] to character varieties with Zariski closures of arbitrary generic conjugacy classes at punctures working with intersection cohomology. We also study Weyl group action on the intersection cohomology of the partial resolutions of these character varieties and give a conjectural formula for the two-variable polynomials that encode the trace of the elements of the Weyl group on the subquotients of the weight filtration. Finally, we compute the generating function of the stack count of character varieties with Zariski closure of unipotent regular conjugacy class at punctures.