2 TOMOYUKI ARAKAWA

Theorem 1.1. The embedding z(g) → V

cri(g)

induces the isomorphism

z(g)

∼

→

Z(Wcri(g,f)).

Moreover, Wcri(g,f) is free over z(g), where z(g) is regarded as a commutative ring

with the (−1)-product.

Theorem 1.1 generalizes a result of Feigin and Frenkel [FF4], who proved that

z(g)

∼

=

Wcri(g,fprin) for a principal nilpotent element fprin of g. It also generalizes a

result of Frenkel and Gaitsgory [FG], who proved the freeness of V

cri(g) over z(g).

Let G be the adjoint group of g, S the Slodowy slice at f to Ad G.f, N the

nilpotent cone of g. Set

S = S ∩ N .

It is known [P1] that the scheme S is reduced, irreducible, and normal complete

intersection of dimension dim N − dim Ad G.f.

For χ ∈ OpLg

reg

, let Wχ

cri(g,f)

be the quotient of

Wcri(g,f)

by the ideal generated

by z − χ(z) with z ∈ z(g). Then any simple quotient of

Wcri(g,f)

is a quotient of

Wχ

cri(g,f)

for some χ.

Theorem 1.2. For χ ∈ OpLg

reg

, the vertex algebra Wχ(g,f) is simple. Its asso-

ciated graded vertex Poisson algebra gr Wχ(g,f) is isomorphic to C[S∞] as vertex

Poisson algebras, where S∞ is the infinite jet scheme of S and C[S∞] is equipped

with the level 0 vertex Poisson algebra structure.

Theorem 1.2 generalizes a result of Frenkel and Gaitsgory [FG], who proved the

simplicity of the quotient of V cri(g) by the ideal generated by z − χ(z) for z ∈ z(g).

In the case that f = fprin we have Wχ(g,fprin) = C [FF4], while S is a point,

and so is S∞. Theorem 1.2 implies that this is the only case that Wcri(g,f) admits

finite-dimensional quotients.

In general little is known about the representations of Wcri(g,f). We have

shown in [A4] that at least in type A the representation theory of Wk(g,f) is

controlled by that of g at level k for any k ∈ C. Therefore the Feigin-Frenkel con-

jecture (see [AF]) implies that, at least in type A, the character of irreducible high-

est weight representations of Wcri(g,f) should be expressed in terms of Lusztig’s

periodic polynomial [Lus]. We plan to return to this in future work.

2. Associated graded vertex Poisson algebras

For a vertex algebra V , let {F

pV

} be the Li filtration [Li],

gr V =

p

F

pV/F p+1V

the associated graded vertex Poisson algebra. The vertex Poisson algebra structure

of gr V restricts to the Poisson algebra structure on Zhu’s Poisson algebra [Zhu]

RV := V/F

1V

⊂ gr V.

Moreover there is a surjective map

(RV )∞ → gr V (1)