W-algebras at the critical level

TL;DR

This paper explores the structure of W-algebras at the critical level, establishing their centers, simplicity, and connection to geometric objects like jet schemes and Slodowy slices.

Contribution

It proves the center of W-algebras at the critical level matches the Feigin-Frenkel center and characterizes their simple quotients as quantizations of specific jet schemes.

Findings

Center of W^{cri}(g,f) equals Feigin-Frenkel center

Simple quotient W_{χ}(g,f) is a quantization of a jet scheme

W_{χ}(g,f) is a simple algebra

Abstract

Let g be a complex simple Lie algebra, f a nilpotent element of g. We show that (1) the center of the W-algebra $W^{cri}(g,f)$ associated with (g,f) at the critical level coincides with the Feigin-Frenkel center of the affine Lie algebra associated with g, (2) the centerless quotient $W_{\chi}(g,f)$ of $W^{cri}(g,f)$ corresponding to an oper $\chi$ on the disc is simple, (3) the simple quotient $W_{\chi}(g,f)$ is a quantization of the jet scheme of the intersection of the Slodowy slice at f with the nilpotent cone of g.