Showing papers by "Simon Riche published in 2019"
TL;DR: In this paper, it was shown that for any simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide when p is sufficiently large.
Abstract: Let G be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number. The "support variety" of a G-module M is a certain closed subvariety of the nilpotent cone of G, defined in terms of cohomology for the first Frobenius kernel of G. In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for G=SLn in earlier work of the second author.
In this paper, we show that for any G, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when p is sufficiently large. We also prove variants of these statements involving "relative support varieties."
16 citations
01 Dec 2019
TL;DR: In this article, it was shown that the Satake category of G can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian.
Abstract: In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q_l, or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.
8 citations
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TL;DR: For quantum groups at the root of unity, there is a web of theorems connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan-Lusztig cells in the affine Weyl group.
Abstract: For quantum groups at a root of unity, there is a web of theorems (due to Bezrukavnikov and Ostrik) connecting the following topics: (i) tilting modules; (ii) vector bundles on nilpotent orbits; and (iii) Kazhdan-Lusztig cells in the affine Weyl group. In this paper, we propose a (partly conjectural) analogous picture for reductive algebraic groups over fields of positive characteristic, inspired by a conjecture of Humphreys.
7 citations
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TL;DR: In this paper, a character formula expressing the classes of simple representations in the principal block of a simply-connected algebraic group G in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than 2h-1, where h is the Coxeter number of G.
Abstract: In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group G in terms of baby Verma modules, under the assumption that the characteristic of the base field is bigger than 2h-1, where h is the Coxeter number of G. This provides a replacement for Lusztig's conjecture, valid under a reasonable assumption on the characteristic.
7 citations
TL;DR: In this article, the authors studied three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic 0 and those in characteristic p (for groups defined over discrete valuation rings).
Abstract: We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic 0 and those in characteristic p (for groups defined over discrete valuation rings). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups.
3 citations
TL;DR: Soergel bimodules were introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras as mentioned in this paper.
Abstract: Soergel bimodules are certain bimodules over polynomial algebras, associated with Coxeter groups, and introduced by Soergel in the 1990's while studying the category O of complex semisimple Lie algebras. Even though their definition is algebraic and rather elementary, some of their crucial properties were known until recently only in the case of crystallographic Coxeter groups, where these bimodules can be interpreted in terms of equivariant cohomology of Schubert varieties. In recent work Elias and Williamson have proved these properties in full generality by showing that these bimodules possess "Hodge type" properties. These results imply positivity of Kazhdan-Lusztig polynomials in full generality, and provide an algebraic proof of the Kazhdan-Lusztig conjecture.
2 citations