18 PRAMOD N. ACHAR

The last assertion of the lemma is just the general fact that

Homi(X,

Y ) always

vanishes for i 0 if X and Y are in the heart of some t-structure.

Lemma 5.3. Let λ, μ ∈ Λ be two weights in the same W -orbit. If μ ≤ λ, then

A(μ) ∈ D

λ

∗ A(λ)−2 ,

where is the length of the shortest element w ∈ W such that wλ = μ.

Proof. The statement is trivial if μ = λ, so assume that μ λ. It is easily

seen by induction on that it suﬃces to prove this in the case where μ = sλ for

some simple reflection s, say corresponding to the simple root α. Let n = α∨,λ .

Since sλ λ, we have n 0.

Let Pα ⊂ G be the minimal parabolic subgroup corresponding to α, and let

pα : G/B → G/Pα be the projection map. Let ρ =

1

2

∑

α, where the sum runs over

all positive roots. Recall that G is assumed to be simply connected, so ρ lies in the

weight lattice for G. Let Q = kρ−α ⊗ resBα

P

indBα

P

kλ−ρ. Since α∨,λ − ρ = n − 1,

the weights of indBα

P

kλ−ρ are λ − ρ, λ − ρ − α, . . . , λ − ρ − (n − 1)α. Thus, the

weights of Q are

λ − α, λ − 2α,...,λ − nα = sλ.

A standard fact relating induction, restriction, and tensor products tells us that

(5.1) RindBα

P

Q

∼

=

RindBα

P

kρ−α

L

⊗ RindBα

P

kλ−ρ = 0,

where the last equality follows from the fact that

α∨,ρ

− α = −1.

From the weights of Q, we see that there is a short exact sequence of B-modules

0 → ksλ → Q → K1 → 0,

where the weights of K1 are λ. Applying Rπ∗ ◦ p∗ ◦ S , we see that

(5.2) A(sλ) ∈ D

λ

∗

Rπ∗p∗S

(Q).

Similarly, there is a short exact sequence

0 → K2 → Q ⊗ kα → kλ → 0

where K2 has weights that are λ. We deduce that

(5.3)

Rπ∗p∗S

(Q ⊗ kα) ∈ D

λ

∗ A(λ).

In view of (5.2) and (5.3), we see that the lemma will follow once we prove that

(5.4)

Rπ∗p∗S

(Q ⊗ kα)−2

∼

=

Rπ∗p∗S

(Q).

Let uα be the Lie algebra of the unipotent radical of Pα, and consider its

coordinate ring k[uα]. It is the quotient of the graded ring k[u] by the ideal generated

by α ∈

u∗

= (k[u])−2. In other words, we have a short exact sequence 0 →

k[u] ⊗ kα−2 → k[u] → k[uα] → 0 of (B × Gm)-equivariant k[u]-modules, or

equivalently of objects in

CohB×Gm

(u). A construction analogous to that of S

then gives us a short exact sequence

0 →

p∗S

(kα)−2 → O

˜

N

→ i∗O

˜

N

α

→ 0

in

CohG×Gm

(

˜

N ). Here

˜

N

α

= G ×B uα, and i :

˜

N

α

→

˜

N is the inclusion map.

Tensoring with p∗S (Q), we see that (5.4) would follow if we knew that

(5.5) Rπ∗(i∗O

˜

N

α

⊗

p∗S

(Q)) = 0.

18