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
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

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)
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