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 = 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 = for
some simple reflection s, say corresponding to the simple root α. Let n = α∨,λ .
Since λ, we have n 0.
Let G be the minimal parabolic subgroup corresponding to α, and let
: 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 = 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 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 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
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