4 TOMOYUKI ARAKAWA

filtration of V

k(g)

is essentially the same as the standard filtration of

U(g[t−1]t−1)

under the isomorphism

U(g[t−1]t−1)

∼

= V

k(g),

see [A5]. We have

RV

k(g)

∼

=

C[g∗]

(6)

and (1) gives the isomorphism

C[g∞]

∗

∼ → gr V

k(g)

(7)

of vertex Poisson algebras. Let z(g) be the Feigin-Frenkel center Z(V

cri(g))

as in

Introduction. It is known [FF4, F1, F2] that the Li filtration of V

cri(g)

restricts

to the Li filtration of z(g). Moreover we have

Rz(g)

∼

=

C[g∗]G,

(8)

and (1) gives the isomorphism

(Rz(g))∞ ∼ → gr z(g). (9)

(Hence the vertex Poisson algebra structure of z(g) is trivial1.) The isomorphisms

(8) and (9) imply [BD1, EF] that

gr z(g)

∼

=

C[(g∗//G)∞]

=

C[g∞]G∞ ∗

.

For χ ∈ OpLg

reg

, let Vχ

cri(g)

be the quotient of V

cri(g)

by the ideal generated by

z − χ(z) for z ∈ z(g). Because z(g) acts freely on V cri(g) [FG], it follows from (7)

and (9) that

RVχ

cri(g)

∼

=

C[N ], gr Vχ

cri(g)

∼

=

C[N∞]. (10)

Furthermore, it was proved in [FG] that the vertex algebra Vχ

cri(g)

is simple (thus

in particular Vχ

cri(g)

is simple as a g-module).

Let

χ0 ∈

OpLgreg

be the unique element such that {z − χ0(z); z ∈ z(g)} is the argumentation ideal

z(g)∗ of z(g). We set

Vres(g) = Vχ0

cri

(g),

and call it the restricted aﬃne vertex algebra associated with g. As a g-module,

Vres(g) is isomorphic to the irreducible highest weight representation with highest

weight −h∨Λ0.

For k ∈ C and a V

k(g)-module

M, one can define the complex (C(M),d) of

the BRST cohomology of the generalized quantized Drinfeld-Sokolov reduction as-

sociated with (g,f) ([KRW]). We have C(M) = M ⊗ Dch(Cm) ⊗

∞

2

+•

, where

m =

1

2

dim g

1

2

, and

∞

2

+•

is the Clifford vertex superalgebra of rank dim n. The

complex (C(M),d) can be identified with Feigin’s complex which defines the semi-

infinite cohomology H

∞

2

+•(g

0

[t,

t−1],M

⊗

Dch(Cm)),

where g

0

[t,

t−1]-module

structure of

Dch(Cm)

is described in [A2,

§3]2.

Let

Hf2

∞ +•

(M) :=

H•(C(M),d).

1The

vertex Poisson algebra structure considered in this article is different from the one in

[F2]

2In

[A2]

Dch(Cm)

is denoted by

Fne(χ)