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 = 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 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
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