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)
1
Proceedings of Symposia in Pure Mathematics
Volume 86, 2012
c 2012 American Mathematical Society
1
http://dx.doi.org/10.1090/pspum/086/1409
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