Quivers with loops and Lagrangian subvarieties

TL;DR

This paper generalizes Lusztig Lagrangian varieties to arbitrary quivers with loops, introducing new combinatorial structures and a convolution algebra framework for their irreducible components.

Contribution

It extends Lusztig's construction to quivers with loops, develops a new combinatorial structure, and constructs a related convolution algebra of functions.

Findings

Established Lagrangian property for generalized varieties

Discovered a new combinatorial structure beyond usual crystals

Constructed a convolution algebra of constructible functions

Abstract

In this article we define a generalization of Lusztig Lagrangian varieties in the case of arbitrary quivers, possibly carrying loops. As opposed to the Lagrangian varieties constructed by Lusztig, which consisted in nilpotent representations, we have to consider here slightly more general representations. That this is necessary is already clear from the Jordan quiver case. Our proof of the Lagrangian character is based on induction, but with non trivial first steps, consisting in the study of quivers with one vertex but possible loops. From our proof emerges a new combinatorial structure on the set of irreducible components, which is more general than the usual crystals, in that there are now more operators associated to a vertex with loops. We finally consider a convolution algebra of constructible functions on our varieties, and construct a family of constructible functions naturally attached to the irreducible components.