2 PRAMOD N. ACHAR
Conjecture 1.3. We have
This conjecture is known to hold in characteristic 0 by [B4]. The proof given
there involves relating
) 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
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
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