22 PRAMOD N. ACHAR

This map is at least an isomorphism over C, since q is an isomorphism over C.

Thus, its cone K has support contained in Z. Applying i∗, we have a distinguished

triangle

IC(C, V) → Rπ∗F → i∗K → .

Since Rπ∗F

∈ D and i∗K ∈ DZ

b

CohG×Gm

(N ) ⊂ D, we conclude that IC(C, V) ∈ D,

as desired.

6. Proofs of the main results

The results of Section 5 ﬁt the framework of Sections 2–3 and allow us to

quickly deduce the main results. For λ ∈

Λ+,

let δλ denote the length of the

shortest element w ∈ W such that wλ = w0λ. We then put

(6.1)

∇λ

= A(λ)−δλ ,

Δλ = A(w0λ) δλ

Proposition 6.1. The objects

∇λ

constitute an abelianesque dualizable graded

quasi-exceptional set in

DbCohG×Gm

(N ), and the Δλ form the dual set.

Likewise, the objects U(Δλ) constitute an abelianesque dualizable ungraded

quasi-exceptional set in

DbCohG(N

), and the U(Δλ) form the dual set.

Proof. Referring to Deﬁnition 2.4, we see that conditions (1)–(2) are proved

in Proposition 5.6. To see that condition (4) holds, note that every graded ﬁnite-

dimensional B-representation arises by extensions among 1-dimensional representa-

tions kλ n . By Corollary 5.8, the objects p∗S (kλ n ) generate

DbCohG×Gm

(

˜

N ),

and then by Lemma 5.9, the objects A(λ) n , where λ ∈ Λ and n ∈ Z, generate

DbCohG×Gm

(N ). The fact that it suﬃces to take the A(λ) n with λ dominant

follows from Lemma 5.3 with an induction argument with respect to . Thus, the

{A(λ) δλ} with λ ∈

Λ+

form a graded quasi-exceptional set.

In fact, the aforementioned induction argument also shows that each D

λ

is

generated by the A(μ) n with μ ∈

Λ+,

μ λ. So this category coincides with

the one that would have been denoted

(λ)D

in Section 2. By Lemma 5.3, there

is a morphism A(w0λ) → A(λ)−2δλ whose cone lies in D

λ

. Combining this

observation with Proposition 5.6(4), we see that the {A(w0λ) δλ} forms a dual

set. The fact that it is abelianesque is contained in Lemma 5.2.

For the ungraded version, we omit part (3) of Deﬁnition 2.4. Since Propo-

sition 5.6(3) was the only result of Section 5 without an ungraded analogue (see

the remarks at the beginning of Section 5), the ungraded version of the present

proposition also holds.

Theorem 6.2. The categories

PCohG×Gm

(N ) and

PCohG(N

) are quasi-heredi-

tary, with standard and costandard objects as in (6.1).

Proof. By Theorem 2.10, the objects Δλ and

∇λ

determine a t-structure on

each of

DbCohG×Gm

(N ) and

DbCohG(N

) whose heart A is quasi-hereditary and

in which those objects are standard and costandard, respectively. But it is easily

seen from Lemma 5.2 and the deﬁnition given in Theorem 2.10 that every perverse

coherent sheaf lies in A. The heart of one bounded t-structure cannot be properly

contained in the heart of another bounded t-structure, so it must be that A coincides

with

PCohG×Gm

(

˜

N ) or

PCohG(N

).

22