In 1980 Victor Kac proved that the number of indecomposable representations of a quiver over a finite field is a polynomial.The desire to understand this polynomial has been a key driving force in geometric representation theory, cf. Schiffmann’s 2018 ICM. Quiver varieties are by construction type A objects, i.e. attached to the group GL_n. In this talk we explore what the reductive analogue of Kac polynomials can be.
We restrict to certain quivers that are called star shaped quivers. Here a theorem of Hausel, Letellier and Villegas relates Kac’s polynomial to the number of points of moduli of representations of connections on vector bundles on surfaces. Replacing vector bundles by G-bundles allows us to explore the G-analogues of Kac’s polynomials. This talk is based on the arXiv preprint: https://arxiv.org/abs/2409.04735