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

kλ is an injective B-module, since induction

takes injective modules to injective modules. Since B

∼

=

T U, we have indT

B

kλ

∼

=

k[U]⊗kλ. By (4.2), resT B indT

B

kλ

∼

=

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

kλ are ≥ λ.

The B-module V can be embedded in an injective module

I0

by taking a direct

sum of copies of indT

B

kλ 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