16 PRAMOD N. ACHAR
are both (G × Gm)-equivariant. Our convention on the Gm-action means that the
graded rings k[u] and k[N ] are concentrated in even, nonpositive degrees. We do
not endow U with a Gm-action, so the map (4.1) only gives rise to an isomorphism
(4.2) k[U]

U(k[u])
of ungraded T -representations.
Any graded B-representation V gives rise to a locally free (G×Gm)-equivariant
sheaf on B, denoted S (V ). Given a weight λ Λ, consider the object
A(λ) =
Rπ∗p∗S
(kλ).
This is called the Andersen–Jantzen sheaf of weight λ. (It is known [KLT, Theo-
rem 2] that A(λ) is actually a coherent sheaf, rather than a complex of sheaves, for
λ dominant, but we will not use this fact.) For any λ Λ, let
D λ, D
λ

DbCohG×Gm
(N )
be the full triangulated subcategories generated by all Tate twists of Andersen–
Jantzen sheaves A(μ) n with μ λ or μ λ, respectively.
Consider now the object in
DbRep(G)
given by RΓ(A(λ))

=
RΓ(p∗p∗S
(kλ)).
By the projection formula, we have
p∗p∗S
(kλ)

= S (k[u] kλ), so
(4.3) RΓ(A(λ))

= RindB(k[u]
G
kλ).
The following lemma on representations of a Borel subgroup is certainly well-
known, but we include a proof for completeness.
Lemma 4.1. Let μ Λ, and let U and V be rational B-representations such
that all weights of U are μ but no weights of V are μ. Then RHom(U, V ) = 0.
Proof. Let λ Λ. Note that indT
B
is an injective B-module, since induction
takes injective modules to injective modules. Since B

=
T U, we have indT
B


=
k[U]⊗kλ. By (4.2), resT B indT
B


=
resT B U(k[u])⊗kλ. The weights of k[u] are sums
of positive roots, so we see that all weights of indT
B
are λ.
The B-module V can be embedded in an injective module
I0
by taking a direct
sum of copies of indT
B
as λ varies over weights of V . It follows from the preceding
paragraph that every weight of I is some weight of V . More generally, we can
extend this an injective resolution
(In)n≥0
of V in which every weight of every term
is some weight of V .
Since no weight of V is μ, it follows that no weight of
In
is μ either, so
Hom(U,
In)
= 0 for all n. Thus, RHom(U, V ) = 0.
4.5. Perverse coherent sheaves. Recall that G acts on N with finitely
many orbits, and that each orbit has even dimension. For an orbit C, let ηC
be its generic point, and let iC : {ηC } C be the inclusion map. An object
F
DbCohG(N
) is said to be a perverse coherent sheaf if the following two condi-
tions hold:
Hi(iC
F) = 0 for all i
1
2
codim C,
Hi(iC !
F) = 0 for all i
1
2
codim C.
The second condition is equivalent to requiring that
Hi(iC
DF) = 0 for all i
1
2
codim C,
16
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