Arithmetic harmonic analysis on character and quiver varieties II

TL;DR

This paper explores deep connections between the topology of character varieties, Macdonald polynomials, quiver representations, Hilbert schemes, modular forms, and tensor product multiplicities, revealing new insights across algebraic geometry and representation theory.

Contribution

It introduces novel links between diverse mathematical structures, advancing understanding of character varieties and their related algebraic and geometric objects.

Findings

Established new relationships between character varieties and Macdonald polynomials.

Connected quiver representations with Hilbert schemes and modular forms.

Provided insights into tensor product multiplicities in finite general linear groups.

Abstract

We study connections between the topology of generic character varieties of fundamental groups of punctured Riemann surfaces, Macdonald polynomials, quiver representations, Hilbert schemes on surfaces, modular forms and multiplicities in tensor products of irreducible characters of finite general linear groups.