On the Exotic t-Structure in Positive Characteristic

Abstract

Abstract. In this paper we study Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of a connected reductive algebraic group defined over a field of positive characteristic with simply-connected derived subgroup. In particular, we show that the heart of the exotic t-structure is a graded highest weight category, and we study the tilting objects in this heart. Our main tool is the "geometric braid group action" studied by Bezrukavnikov and the second author.

Kostant section, universal centralizer, and a modular derived Satake equivalence

“…Now the fact that the right-hand side vanishes follows from the similar claim on N (which itself follows from [MR1,Corollary 4.3.7]) by the same arguments as in [MR2,Proposition 5.5]. …”

“…Acknowledgements. This work is part of a joint project with Carl Mautner, see [MR1,MR2], and was motivated by discussions with him. We also thank Zhiwei Yun for useful conversations on regular elements and for confirming a sign mistake in [YZ], and Sergey Lysenko for his help with some references.…”

On the Humphreys Conjecture on Support Varieties of Tilting Modules

“…This t-structure will be called the exotic t-structure, and its heart will be denoted E G×Gm ( N ) k . By [MR1,Corollary 3.11], the objects ∆ k (λ) and ∇ k (λ) belong to E G×Gm ( N ) k . (Again, in [MR1] it is assumed that k is algebraically closed, but the general case follows.)…”

“…By [MR1,Corollary 3.11], the objects ∆ k (λ) and ∇ k (λ) belong to E G×Gm ( N ) k . (Again, in [MR1] it is assumed that k is algebraically closed, but the general case follows.) It follows that E G×Gm ( N ) k is a graded highest weight category in the sense of Definition 2.1 (with weight poset (X, ≤ )), and that the realization functor…”

“…proving that sw > w, and hence that (1) holds. By the Iwahori-Matsumoto formula for the length in W (see, for instance, [MR1,(2.2)]), we have…”