Microlocal characterization of Lusztig sheaves for affine quivers and $g$-loops quivers

TL;DR

This paper characterizes Lusztig sheaves for affine and g-loops quivers using nilpotency of their singular support, confirming a conjecture for affine quivers and extending results to quivers with loops.

Contribution

It proves Lusztig's conjecture for affine quivers and extends the characterization to g-loops quivers, introducing new nilpotent varieties for quivers with loops.

Findings

Lusztig sheaves are characterized by nilpotent singular support for affine quivers.

Confirmed Lusztig's conjecture for affine quivers.

Proved conjecture for g-loops quivers ($g eq 1$).

Abstract

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures for similar results for quivers with loops, for which we have to use the appropriate notion of nilpotent variety, due to Bozec, Schiffmann and Vasserot. We prove our conjecture for $g$-loops quivers ($g\geq 2$).