GROUPS, ARITHMETIC AND ALGEBRAIC GEOMETRY SEMINAR - Quiver varieties with multiplicities

Representations of quivers with multiplicities are representations of quivers over rings of truncated power series. Such representations were studied by Geiss, Leclerc and Schröer in a series of works, where connections with symmetrisable Kac-Moody algebras were established. These results generalise earlier constructions of Lusztig, Kashiwara and Saito from symmetric to symmetrisable Kac-Moody algebras.

I will report on joint work with Victoria Hoskins and Joshua Jackson, where we construct analogues of Nakajima quiver varieties for quivers with multiplicities. Our construction relies on Geometric Invariant Theory results for non-reductive groups due to Hamilton, Hoskins and Jackson. We define stability conditions for quiver representations with multiplicities, generalising the stability conditions introduced by King in the nineties. Time permitting, I will discuss a conjectural construction of nilpotent quiver varieties with multiplicities and applications to geometric realisations of crystals for irreducible highest-weight representations of symmetrisable Kac-Moody algebras.