2 PRAMOD N. ACHAR

Conjecture 1.3. We have

DbPCohG(N

)

∼

=

DbCohG(N

).

This conjecture is known to hold in characteristic 0 by [B4]. The proof given

there involves relating

CohG(N

) to perverse sheaves on the aﬃne flag variety Fl for

the Langlands dual group. It is likely (and perhaps already known to experts) that

a similar approach using mixed perverse sheaves would allow one to bring in the

Gm-action, leading to a characteristic-0 proof of Theorem 1.2 that is quite diﬀerent

from the one given here.

The reason for the restriction to characteristic 0 in [B4] is that the arguments

there require the base ﬁeld k for G to coincide with the ﬁeld of coeﬃcients of

sheaves on Fl. The sheaves in [B4], like nearly all constructible sheaves used in

representation theory in the past thirty-ﬁve years, have their coeﬃcients in Q . But

so-called modular perverse sheaves—perverse sheaves with coeﬃcients in a ﬁeld of

positive characteristic—have recently begun to appear in a number of important

applications [F, Ju, JMW, S]. It would be very interesting to develop a sheaf-

theoretic approach to Theorem 1.2 or Conjecture 1.3 in positive characteristic using

modular perverse sheaves.

In the present paper, the assumptions that G is simply connected and that k

is of good characteristic are needed in order to invoke certain results from [BK]

and [Ja]. In characteristic 0, it can be deduced from Theorems 1.1 and 1.2 that

corresponding results hold for arbitrary connected reductive groups. In positive

characteristic, however, isogenous groups need not have isomorphic nilpotent cones

(see, e.g., [Ja, Remark 2.7]). In the latter case, the main theorems hold for groups

with simply connected derived group, but not for arbitrary reductive groups.

The paper is organized as follows. Sections 2 and 3 lay the homological-algebra

foundations for the main results, starting with notation and deﬁnitions. The key

result of that part of the paper is Theorem 3.15, which states that any quasi-

exceptional set satisfying certain axioms gives rise to a derived equivalence. In

Section 4, we return to the setting of algebraic groups. Section 5 contains a number

of technical lemmas on the so-called Andersen–Jantzen sheaves. The main theorems

are proved in Section 6.

Acknowledgments. While this project was underway, I beneﬁtted from nu-

merous conversations with A. Henderson, S. Riche, and D. Treumann. I would

also like to express my gratitude to the organizers of the Southeastern Lie Theory

Workshop series for having given me the opportunity to participate in the May

2010 meeting.

2. Preliminaries on abelian and triangulated categories

2.1. Generalities. Fix an algebraically closed ﬁeld k. Throughout the paper,

all abelian and triangulated categories will be k-linear and skeletally small (that

is, the class of isomorphism classes of objects is assumed to be a set). Later, all

schemes and algebraic groups will be deﬁned over k as well. For an abelian category

A, we write Irr(A) for its set of isomorphism classes of simple objects. We say that

A is a ﬁnite-length category if it is noetherian and artinian.

Now, let T be a triangulated category. For objects X, Y ∈ T, we write

Homi(X,

Y ) = Hom(X, Y [i]).

A full subcategory A ⊂ T is said to be admissible if it stable under extensions and

direct summands, and if it satisﬁes the condition of [BBD, §1.2.5]. (Thus, our

2