W-ALGEBRAS AT THE CRITICAL LEVEL 3
of vertex Poisson algebras [Li, A5]. Here XV = Spec RV , (RV )∞ = C[(XV )∞],
where X∞ denotes the infinite jet scheme of a scheme X of finite type, and (RV )∞
is equipped with the level zero vertex Poisson algebra structure [A5, 2.3].
Let
Dch(Cr)
be the βγ-system of rank r, that is, the vertex algebra generated
by fields a1(z),...,ar(z), a1(z),...,ar(z),

satisfying the following OPE’s:
ai(z)aj(z)∗

δij
z w
, ai(z)aj(z) ai
∗(z)aj ∗(z)
0.
It is straightforward to see that RDch(Cr)

=
C[T ∗Cr] as Poisson algebras and that
(1) gives the isomorphism
(RDch(Cr))∞

gr
Dch(Cr).
(2)
Let g, f be as in Introduction, rk g the rank of g, ( | ) the normalized invariant
bilinear form of g. Let s = {e, h, f} be an sl2-triple in g, and let gj = {x g; [h, x] =
2jx} so that
g =
j∈
1
2
Z
gj. (3)
Fix a triangular decomposition g = n− h n such that h h g0 and n g≥0 :=
j≥0
gj. We will identify g with
g∗
via ( | ).
The Slodowy slice to Ad G.f at f is by definition the affine subspace
S = f +
ge
of g, where
ge
is the centralizer of e in g. It is known [GG] that the Kirillov-Kostant
Poisson structure of
g∗
= g restricts to S.
Let be an ad h-stable Lagrangian subspace of g1/2 with respect to the sym-
plectic form g1/2 × g1/2 C, (x, y) (f|[x, y]). Set
m =
j≥1
gj,
and let M be the unipotent subgroup of G whose Lie algebra is m, m⊥ = {x
g; (x|y) = 0 for all y m}. Then [GG] we have the isomorphism of affine varieties
M × S

f +
m⊥,
(g, x) Ad(g)(x). (4)
This induces the following isomorphism of jet schemes:
M∞ × S∞

(f +
m⊥)∞.
(5)
Denote by I and I∞ the defining ideals of f + m⊥ and (f + m⊥)∞ in g and g∞,
respectively. By (4) and (5) we have
C[S]

=
(C[g]/I)M
, C[S∞]

=
(C[g∞]/I∞)M∞
.
Let
g = g[t,
t−1]
CK CD
be the affine Kac-Moody algebra associated with g, where K is the central element
and D is the degree operator. Set g = g[t,
t−1]
CK, the derived algebra of g.
The universal affine vertex algebra V
k(g)
associated with g at level k C is the
induced g-module U(g) ⊗U(g[t]
CK)
Ck, equipped with the natural vertex algebra
structure (see e.g. [Kac, FBZ]). Here Ck is the one-dimensional representation of
g[t] CK on which g[t] acts trivially and K acts as a multiplication by k. The Li
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