Tilting modules and the p-canonical basis

Abstract

In this paper we propose a new approach to tilting modules for reductive algebraic groups in positive characteristic. We conjecture that translation functors give an action of the (diagrammatic) Hecke category of the affine Weyl group on the principal block. Our conjecture implies character formulas for the simple and tilting modules in terms of the p-canonical basis, as well as a description of the principal block as the anti-spherical quotient of the Hecke category. We prove our conjecture for GLnpkq using the theory of 2-Kac-Moody actions. Finally, we prove that the diagrammatic Hecke category of a general crystallographic Coxeter group may be described in terms of parity complexes on the flag variety of the corresponding Kac-Moody group.

Hecke category actions via Smith–Treumann theory

“…The proof is almost identical to that given in [RW18, §§ 10.3–10.6] for [RW18, Theorem 10.3.2], so for efficiency we merely annotate the meaningful points of difference in specific sections of that paper.…”

“…In the following, our schemes will be defined over and we will work in the ‘étale context’ described in [RW18, § 9.3(2)], referring to (possibly equivariant) derived categories of étale sheaves over a coefficient ring . Such equivariant derived categories were introduced in [BL06] for the topological setting; the necessary adjustments for étale sheaves are provided in [Wei13].…”

“…We follow the proof of [RW18, Lemma 10.2.1]. Write , and choose closed finite unions of -orbits through which , factor and upon which , are supported.…”

“…We define to be or according to whether or and hence reflections Now, the data , together with the assignment of the simple reflections in to the , define the loop realisation of . In fact, this realisation is a quotient of the ‘traditional’ realisation associated to the affinisation of the Lie algebra of , discussed in [RW18, Remark 10.7.2(2)].…”

“…The starting point of our approach is a realisation of the Hecke category via parity complexes on the neutral component of the affine flag variety of , first proved in [RW18]. Here we need a mild modification to incorporate loop rotation -equivariance.…”