Kac polynomials and Lie algebras associated to quivers and curves

TL;DR

This paper surveys the theory of Kac polynomials for quivers and curves, highlighting their representation-theoretic and geometric interpretations, and explores related infinite-dimensional Lie algebras over finite fields.

Contribution

It provides a comprehensive overview of Kac polynomials' roles in representation theory and geometry, and introduces heuristics for associated infinite-dimensional Lie algebras.

Findings

Kac polynomials encode dimensions of certain representation spaces.

They have both algebraic and geometric significance in moduli spaces.

Heuristics suggest a connection to infinite-dimensional Lie algebras.

Abstract

A survey of the theory of Kac polynomials for quivers and for curves. In particular, we describe the representation-theoretic meaning of Kac polynomials in terms of Hall algebras, and the geometric meaning of Kac polynomials in relation to the geometry of moduli spaces of representations of quivers or vector bundles on smooth projective curves. We end with some heuristics concerning a family of infinite-dimensional $\mathbb{Z}^2$-graded Lie algebras attached to curves of a fixed genus (over a finite field), whose 'Cartan datum' encodes the dimension of the spaces of absolutely cuspidal functions.