A geometric construction of affine gl_n and its semicanonical basis

Abstract

The semicanonical basis is a basis of the negative or positive half of the universal enveloping algebra of a Kac-Moody algebra of symmetric type constructed by Lusztig. These bases are realized via certain constructible functions on Lusztig’s nilpotent quiver varieties and possess favorable properties. Bozec recently extended this framework from quivers without loops to those with possible loops. In affine type A, these constructions encompass both affine sl_n and affine gl_1. In this talk, I will present a construction that extends the work of Lusztig and Bozec to affine gl_n, using a semi-nilpotent variant of Lusztig’s nilpotent quiver varieties. If time allowed, I’ll discuss a proof of Hennecart’s conjecture in the case of the Jordan quiver.