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.

Reductive groups, the loop Grassmannian, and the Springer resolution

“…Their work has significant consequences for the representation theory of quantum groups: they lead to alternative proofs of Lusztig's character formula for simple modules (see [9, §1.2]) and of Soergel's character formula for tilting modules (using [60]). We believe that the results of the present paper will likewise have consequences for the representation theory of G. In particular, we expect to use them to establish the character formulas for simple and tilting G-modules conjectured by the second author and G. Williamson in [50]. See §1.7 below for details.…”

“…First, in [50], the second author and G. Williamson conjecture that the multiplicities of standard/costandard modules in indecomposable tilting modules in Rep ∅ (G) can be expressed in terms of the values at 1 of some ℓ-Kazhdan-Lusztig polynomials (in the sense of [35]), which compute the dimensions of the stalks of some indecomposable parity complexes on the affine flag variety Fl of Ġ∨ . This conjecture is proved in the case G = GL n (k) in [50], but the methods used in this proof do not make sense for a general reductive group.…”

The Center of Small Quantum Groups I: The Principal Block in Type A

“…The second reason is that the principal block of the big quantum group at a root of unity categorifies the antispherical module of the affine Hecke algebra: this is exactly the induced module of the affine Hecke algebra from the sign character of the finite Hecke subalgebra. A recent work of Riche and Williamson [RW15] has given a categorical explanation of this phenomenon in type A for algebraic groups over finite characteristic fields via categorification. Their method also applies to (big) quantum groups at roots of unity.…”

“…We will continue working on obtaining further evidence and, hopefully, a proof for the formulated conjectures on the structure of the center of small quantum groups. Furthermore, it would be interesting to find a connection between centers of small quantum groups and categorified small quantum groups, as initiated in [KQ15,EQ16], and to understand how the Hecke categories in [RW15] are related to the center of small quantum group, which we also plan to pursue in the future. nik, A. Johan de Jong, Dennis Gaitsgory, Jiuzu Hong, Steve Jackson, Mikhail Khovanov, Peng Shan, Geordie Williamson and Chenyang Xu for helpful discussions.…”

La théorie de Hodge des bimodules de Soergel (d'après Soergel et Elias-Williamson)

“…Depuis ces travaux fondateurs, les bimodules de Soergel (sous différentes formes) se sont révélés être des outils extrèmement utiles en théorie des représentations (voir [S2,S6,WW,Do,BY,RW] pour quelques exemples), car ils permettent souvent de faire un lien avec la géométrie d'une variété de drapeaux appropriée. Mais dans tous les cas la preuve de certaines de leurs propriétés sort du cadre algébrique de leur définition, et repose sur la géométrie.…”