# On the number of points of nilpotent quiver varieties over finite fields

## TL;DR

This paper provides explicit formulas for counting points over finite fields of Lusztig nilpotent varieties associated with quivers, connecting these counts to Kac's A-polynomials and their variants, and explores their stratification related to crystal graphs.

## Contribution

It introduces nilpotent versions of Kac's A-polynomials and derives closed formulas for point counts of Lusztig nilpotent varieties with various quiver configurations.

## Key findings

- Closed formulas for point counts over finite fields.
- Introduction of nilpotent Kac A-polynomials and their explicit formulas.
- Computation of points on stratifications related to crystal graphs.

## Abstract

We give a closed expression for the number of points over finite fields (or the motive) of the Lusztig nilpotent variety associated to any quiver, in terms of Kac's A-polynomials. When the quiver has 1-loops or oriented cycles, there are several possible variants of the Lusztig nilpotent variety, and we provide formulas for the point count of each. This involves nilpotent versions of the Kac A-polynomial, which we introduce and for which we give a closed formula similar to Hua's formula for the usual Kac A-polynomial. Finally we compute 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.