14 PRAMOD N. ACHAR
for all d ≥ 1, so it follows that
Σu p ) = 0 as well. It remains to show
Σs n ) = 0.
Consider the exact sequence
Qs n ) →
Σs n ) →
The ﬁrst term is already known to vanish because Qs n ∈
and the last term
vanishes by Corollary 3.9 because P has Ξ-standard ﬁltration. Thus, we have
Σs n ) = 0, as desired.
3.4. Main result. Given a set Ξ ⊂ S × Z, let ΞD ⊂ D denote the full tri-
angulated subcategory generated by
In other words,
is full subcategory
consisting of objects X all of whose cohomology objects
is the heart of a bounded t-structure on
so we have a realization func-
tor real :
Composition with the inclusion functor gives us a natural
Theorem 3.15. For any convex set Ξ ⊂ S × Z, the natural functor Db(ΞA) →
D is fully faithful. In particular, the realization functor
is an equivalence of categories.
Proof. We ﬁrst treat the case where Ξ is ﬁnite. For a ﬁxed object X ∈ ΞA,
is a universal δ-functor, and Proposition 3.14 tells us that the
is eﬀaceable. Since their 0th parts agree, there is a
canonical isomorphism of functors
Therefore, the natural
→ D is full and faithful.
Now, suppose Ξ is inﬁnite. For any ﬁnite convex subset Ψ ⊂ Ξ, consider the
chain of maps
We claim that g and h are both isomorphisms for all i. This is clearly the case
for i = 0 and i = 1. Suppose, in fact, that it is known for i = 0, 1,...,d − 1.
By Lemma 2.1, both g and h are injective for i = d. We know by the previous
paragraph that the composition h ◦ g is an isomorphism for all i, so it follows that
g and h are isomorphisms as well.
We have shown that ExtΞ(X,
Y ) when X, Y ∈ ΨA. But given
any two objects X, Y ∈ ΞA, we know from Lemma 3.12 that there exists a ﬁnite con-
vex subset Ψ ⊂ Ξ such that X, Y ∈ ΨA. It follows that ExtΞ(X,
for all X, Y ∈ ΞA, so Db(ΞA) → D is full and faithful.
4. Notation for semisimple groups
4.1. Representations and varieties. As noted in Section 2, we will work
over a ﬁxed algebraically closed ﬁeld k. For an algebraic group H over k, let Rep(H)
denote the category of rational representations of H, and Repf (H) ⊂ Rep(H) the
subcategory of ﬁnite-dimensional representations. If H ⊂ K, we have the usual
induction and restriction functors indH
: Rep(H) → Rep(K) and resH
: Rep(K) →
Rep(H). We also use the derived functor RindH