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