Proceedings of Symposia in Pure Mathematics

Perverse coherent sheaves on the nilpotent cone

in good characteristic

Pramod N. Achar

Abstract. In characteristic zero, Bezrukavnikov has shown that the cate-

gory of perverse coherent sheaves on the nilpotent cone of a simply connected

semisimple algebraic group is quasi-hereditary, and that it is derived-equivalent

to the category of (ordinary) coherent sheaves. We prove that graded versions

of these results also hold in good positive characteristic.

1. Introduction

Let G be a simply connected semisimple algebraic group over an algebraically

closed field k of good characteristic. Let N denote the nilpotent variety in the Lie

algebra of G. There is a “scaling” action of Gm on N that commutes with the

G-action. Following [B1], we may consider the category of (G × Gm)-equivariant

perverse coherent sheaves on N , denoted

PCohG×Gm

(N ). This category has some

features in common with ordinary perverse sheaves, but it lives inside the derived

category of (equivariant) coherent sheaves. In this note, we prove the following two

homological facts about

PCohG×Gm

(N ).

Theorem 1.1. The category

PCohG×Gm

(N ) is quasi-hereditary.

Theorem 1.2. We have

DbPCohG×Gm

(N )

∼

=

DbCohG×Gm

(N ).

Theorem 1.1 means that the category contains a class of distinguished objects,

called “standard” and “costandard” objects, that lead to a kind of Kazhdan–Lusztig

theory. This result was proved in characteristic 0 in [B2]. (See also [A].) In fact,

the proof given there “almost” works in positive characteristic as well; it is quite

close to the proof given here. The same arguments also establish the corresponding

result for

PCohG(N

), where the Gm-action is forgotten.

On the other hand, our proof of Theorem 1.2 makes use of the Gm-action in a

crucial way (it means that various Ext-groups carry a grading which we exploit),

so it cannot easily be forgotten. The proof is quite elementary: it relies only on

general notions from homological algebra, and it is similar in spirit to the methods

of [BGS]. Unfortunately, for the moment, these methods seem to be inadequate

to prove the following natural analogue of Theorem 1.2.

2010 Mathematics Subject Classification. Primary 20G05; secondary 14F05, 17B08.

The author received support from NSF grant DMS-1001594.

c

0000 (copyright holder)

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Proceedings of Symposia in Pure Mathematics

Volume 86, 2012

c 2012 American Mathematical Society

1

http://dx.doi.org/10.1090/pspum/086/1409