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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