14 PRAMOD N. ACHAR

for all d ≥ 1, so it follows that

Homd(P,

Σu p ) = 0 as well. It remains to show

that

Homd(P,

Σs n ) = 0.

Consider the exact sequence

Homd−1(P,

Qs n ) →

Homd(P,

Σs n ) →

Homd(P, ∇s

n ).

The ﬁrst term is already known to vanish because Qs n ∈

ΨA,

and the last term

vanishes by Corollary 3.9 because P has Ξ-standard ﬁltration. Thus, we have

Homd(P,

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

ΞA.

In other words,

ΞD

is full subcategory

consisting of objects X all of whose cohomology objects

Hi(X)

lie in

ΞA.

Note

that

ΞA

is the heart of a bounded t-structure on

ΞD,

so we have a realization func-

tor real :

Db(ΞA)

→

ΞD.

Composition with the inclusion functor gives us a natural

functor

Db(ΞA)

→ D.

Theorem 3.15. For any convex set Ξ ⊂ S × Z, the natural functor Db(ΞA) →

D is fully faithful. In particular, the realization functor

real :

Db(A)

→ D

is an equivalence of categories.

Proof. We ﬁrst treat the case where Ξ is ﬁnite. For a ﬁxed object X ∈ ΞA,

{Extd(·,X)}d≥0

is a universal δ-functor, and Proposition 3.14 tells us that the

δ-functor

{Homd(·,X)}d≥0

is eﬀaceable. Since their 0th parts agree, there is a

canonical isomorphism of functors

Extd(·,X)

∼

→

Homd(·,X).

Therefore, the natural

functor Db(ΞA)

→ D is full and faithful.

Now, suppose Ξ is inﬁnite. For any ﬁnite convex subset Ψ ⊂ Ξ, consider the

chain of maps

ExtΨ(X,

i

Y )

g

→ ExtΞ(X,

i

Y )

h

→

Homi(X,

Y ).

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,

d

Y )

Homd(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,

d

Y )

Homd(X,

Y )

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

K

: Rep(H) → Rep(K) and resH

K

: Rep(K) →

Rep(H). We also use the derived functor RindH

K

:

DbRep(H)

→

DbRep(K).

(Of

course, resH

K

is exact.)

14