Contemporary Mathematics

Volume 565, 2012

http://dx.doi.org/10.1090/conm/565/11184

W-algebras at the critical level

Tomoyuki Arakawa

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

Wcri

associated (g, f) at the

critical level coincides with the Feigin-Frenkel center of g, (2) the centerless

quotient Wχ(g, f) of

Wcri(g,f)

corresponding to an

Lg-oper

χ on the disc is

simple, and (3) the simple quotient Wχ(g,f) is a quantization of the jet scheme

of the intersection of the Slodowy slice at f with the nilpotent cone of g.

1. Introduction

Let g be a complex simple Lie algebra, f a nilpotent element of g, U(g,f) the

finite W -algebra [P1] associated with (g,f). In [P2] it was shown that the center

of U(g,f) coincides with the center Z(g) of the universal enveloping algebra U(g)

of g (Premet attributes the proof to Ginzburg).

Let

Wk(g,f)

be the (aﬃne) W -algebra [FF3, KRW, KW] associated with

(g,f) at level k ∈ C. One may [A3, DSK] regard Wk(g,f) as a one-parameter

chiralization of U(g,f). Hence it is natural to ask whether the analogous iden-

tity holds for the center Z(Wk(g,f)) of Wk(g,f), which is a commutative vertex

subalgebra of Wk(g,f).

Let V k(g) be the universal aﬃne vertex algebra associated with g at level k,

Z(V k(g)) the center of V k(g). The embedding Z(V k(g)) → V k(g) induces the

vertex algebra homomorphism

Z(V

k(g))

→

Z(Wk(g,f))

for any k ∈ C. However, both Z(V

k(g))

and

Z(Wk(g,f))

are trivial unless k is the

critical level

cri :=

−h∨,

where

h∨

is the dual Coxeter number of g. Therefore the question one should

ask is that whether the center

Z(Wcri(g,f))

of the W -algebra at the critical level

coincides with the Feigin-Frenkel center [FF4, F1] z(g) := Z(V

cri(g)),

which can

be naturally considered as the space of functions on the space of OpLg

reg

of

Lg-opers

on the disc. Here

Lg

is the Langlands dual Lie algebra of g.

2000 Mathematics Subject Classification. Primary 14B69, 17B68, 17B67.

This work is partially supported by the JSPS Grant-in-Aid for Scientific Research (B) No.

20340007 and the JSPS Grant-in-Aid for challenging Exploratory Research No. 23654006.

c 2012 American Mathematical Society

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