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
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
cri(g)

=
C[N∞]. (10)
Furthermore, it was proved in [FG] that the vertex algebra
cri(g)
is simple (thus
in particular
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)
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