Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

Abstract

Let G be a connected reductive group over an algebraically closed field F of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of G, and Iwahori-constructible F-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirković-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence./ / Tilt(E G×Gm ( N )) commutes. Hence to conclude we only have to prove that the diagram

On monoidal Koszul duality for the Hecke category

“…This category was used in [AMRW19] to formulate and prove a positive characteristic monoidal Koszul duality for Kac-Moody groups. The latter result, combined with a recent string of advances in modular geometric representation theory (Achar-Rider [AR16b], Mautner-Riche [MR18], Achar-Riche [AR18]), yields a character formula for tilting modules of connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, confirming (the combinatorial consequence of) the Riche-Williamson conjecture [RW18], for p greater than the Coxeter number. This article will not discuss this or other applications of Koszul duality to representation theory (aside from brief remarks in §4.3).…”

An Iwahori-Whittaker model for the Satake category

“…Of course we can assume that E = E λ for some λ ∈ X ∨ + . Recall that the forgetful functor For I − sends indecomposable parity objects to indecomposable parity objects (see [MR,Lemma 2.4]). In view of the classification of such objects in the I − -equivariant and I − -constructible derived categories, this means that any I − -constructible parity complex on Gr belongs to the essential image of For I − .…”

A non-perverse Soergel bimodule in type A

“…Moreover, a recent conjecture of Lusztig and the second author implies that parity sheaves can be arbitrarily far from being perverse on the affine flag manifold of SL 3 [LW18]. However, in [JMW16,MR15] it is proved that parity sheaves on the affine Grassmannian are perverse as long as p is a good prime. (Recall that a prime p is good for a fixed root system if it does not divide any coefficient of the highest root when expressed in the simple roots.)…”