On the number of points of the Lusztig nilpotent variety over a finite field

TL;DR

This paper provides a closed-form formula for counting points over finite fields on Lusztig's nilpotent variety related to quivers without edge loops, connecting it to Kac's A-polynomial, and discusses extensions to quivers with loops.

Contribution

It offers a new explicit formula for point counts on Lusztig's nilpotent variety in terms of Kac's A-polynomial, and proposes a conjecture for cases with edge loops.

Findings

Derived a closed expression for point counts over finite fields.

Connected point counts to Kac's A-polynomial.

Proposed a conjecture for quivers with edge loops.

Abstract

In this note we give a closed expression for the number of points over finite fields of the Lusztig nilpotent variety associated to any quiver without edge loops, in terms of Kac's A-polynomial. We conjecture a similar result for quivers in which edge loops are allowed. Finally, we give a formula for the number of points over a finite field of the various stratas of the Lusztig nilpotent variety involved in the geometric realization of the crystal graph.