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We prove some combinatorial conjectures extending those proposed in [13, 14]. The proof uses a vertex operator due to Nekrasov, Okounkov, and the first author [4] to obtain a gluing formula for the relevant generating series, essentially reducing the computation to the case of complex projective space with three punctures.
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The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.
In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that these varieties are symplectic singularities and classify which admit symplectic resolutions. The classification reduces to the semi-simple case, where we show that a resolution exists if and only if either $g=1$ and $G$ is a product of special linear groups of any rank and copies of the group $PGL_2$, or if $g=2$ and $G = (SL_2)^m$ for some $m$.
We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma-Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P=W conjecture for a suitable wild Hitchin system.
We explore induced mappings between character varieties by mappings between surfaces. It is shown that these mappings are generally Poisson. We also explicitly calculate the Poisson bi-vector in a new case.
We calculate the E-polynomial for a class of the (complex) character varieties $mathcal{M}_n^{tau}$ associated to a genus $g$ Riemann surface $Sigma$ equipped with an orientation reversing involution $tau$. Our formula expresses the generating function $sum_{n=1}^{infty} E(mathcal{M}_n^{tau}) T^n$ as the plethystic logarithm of a product of sums indexed by Young diagrams. The proof uses point counting over finite fields, emulating Hausel and Rodriguez-Villegas.
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