Counting absolutely cuspidals for quivers

TL;DR

This paper proves that the space of cuspidal functions for any quiver has a dimension polynomial in q, introduces a variant counting absolutely cuspidal functions, and provides formulas for totally negative quivers.

Contribution

It establishes the polynomial nature of cuspidal function dimensions, defines a new counting polynomial for absolutely cuspidal functions, and derives explicit formulas for totally negative quivers.

Findings

Dimension of cuspidal functions is given by a polynomial in q.

A new polynomial counting absolutely cuspidal functions has integral coefficients.

Explicit formulas are provided for totally negative quivers.

Abstract

For an arbitrary quiver Q and dimension vector d we prove that the dimension of the space of cuspidal functions on the moduli stack of representations of Q of dimension d over a finite field F_q is given by a polynomial in q with rational coefficients. We define a variant of this polynomial counting absolutely cuspidal functions, prove that it has integral coefficients and conjecture that it has positive coefficients. In the case of totally negative quivers (such as the g-loop quiver for g >1) we provide a closed formula for these polynomials in terms of Kac polynomials.