18 PRAMOD N. ACHAR
The last assertion of the lemma is just the general fact that
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
∗ 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 =
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
kλ−ρ. Since α∨,λ − ρ = n − 1,
the weights of indBα
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
kλ−ρ = 0,
where the last equality follows from the fact that
− 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
Similarly, there is a short exact sequence
0 → K2 → Q ⊗ kα → kλ → 0
where K2 has weights that are λ. We deduce that
(Q ⊗ kα) ∈ D
In view of (5.2) and (5.3), we see that the lemma will follow once we prove that
(Q ⊗ kα)−2
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 ∈
= (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
(u). A construction analogous to that of S
then gives us a short exact sequence
(kα)−2 → O
N ). Here
= G ×B uα, and i :
N is the inclusion map.
Tensoring with p∗S (Q), we see that (5.4) would follow if we knew that
(Q)) = 0.