arXiv subjects
Publications and source records attributed to Geordie Williamson.
We apply Treumann's "Smith theory for sheaves" in the context of the Iwahori--Whittaker model of the Satake category. We deduce two results in the representation theory of reductive algebraic groups over fields of positive characteristic: (a) a geometric proof of the linkage principle; (b) a character formula for tilting modules in terms of the $\ell$-canonical basis, valid in all blocks and in all characteristics.
Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the combinatorial invariance conjecture for symmetric groups, a well-known conjecture formulated by Lusztig and Dyer in the 1980s.
These are lecture notes (by the first author) from a course (by the second author) given over two extended semesters at the University of Sydney. The first part provides an introduction to the Langlands correspondence from an arithmetical point of view. The second part gives enough background in geometric representation theory to understand Bezrukavnikov's equivalence, which is a categorification of Kazhdan and Lusztig's two realizations of the affine Hecke algebra.
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.
These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.
We refine an idea of Deodhar, whose goal is a counting formula for Kazhdan-Lusztig polynomials. This is a consequence of a simple observation that one can use the solution of Soergel's conjecture to make ambiguities involved in defining certain morphisms between Soergel bimodules in characteristic zero (double leaves) disappear.
The Hecke category is at the heart of several fundamental questions in modular representation theory. We emphasise the role of the "philosophy of deformations" both as a conceptual and computational tool, and suggest possible connections to Lusztig's "philosophy of generations". On the geometric side one can understand deformations in terms of localisation in equivariant cohomology. Recently Treumann and Leslie-Lonergan have added Smith theory, which provides a useful tool when considering mod p coefficients. In this context, we make contact with some remarkable work of Hazi. Using recent work of Abe on Soergel bimodules, we are able to reprove and generalise some of Hazi's results. Our aim is to convince the reader that the work of Hazi and Leslie-Lonergan can usefully be viewed as some kind of localisation to "good" reflection subgroups. These are notes for my lectures at the 2019 Current Developments in Mathematics at Harvard.
We formulate a conjecture for the second generation characters of indecomposable tilting modules for ${\rm SL}_3$. This gives many new conjectural decomposition numbers for symmetric groups. Our conjecture can be interpreted as saying that these characters are governed by a discrete dynamical system ("billiards bouncing in alcoves"). The conjecture implies that decomposition numbers for symmetric groups display (at least) exponential growth.
The Hecke category is emerging as a fundamental object in representation theory. We give a motivated introduction to this category in both its geometric (via parity sheaves) and diagrammatic (generators and relations) incarnations. We also discuss the p-canonical basis and Koszul duality for the Hecke category.
We prove the relative hard Lefschetz theorem for Soergel bimodules. It follows that the structure constants of the Kazhdan-Lusztig basis are unimodal. We explain why the relative hard Lefschetz theorem implies that the tensor category associated by Lusztig to any 2-sided cell in a Coxeter group is rigid and pivotal.
A basic question concerning indecomposable Soergel bimodules is to understand their endomorphism rings. In characteristic zero all degree-zero endomorphisms are isomorphisms (a fact proved by Elias and the second author) which implies the Kazhdan-Lusztig conjectures. More recently, many examples in positive characteristic have been discovered with larger degree zero endomorphisms. These give counter-examples to expected bounds in Lusztig's conjecture. Here we prove the existence of indecomposable Soergel bimodules in type A having non-zero endomorphisms of negative degree. This gives the existence of a non-perverse parity sheaf in type A.
We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic p in terms of p-Kazhdan-Lusztig polynomials, for p>h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if p>2h-3. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.
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 GL_n 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.
We give a monoidal presentation of Coxeter and braid 2-groups, in terms of decorated planar graphs. This presentation extends the Coxeter presentation. We deduce a simple criterion for a Coxeter group or braid group to act on a category.
These are notes for my Takagi lecture at the University of Tokyo in November, 2016. I survey what is known about simple modules for reductive algebraic groups. The emphasis is on characteristic p>0 and Lusztig's character formula. I explain ideas connecting representations and constructible sheaves (Finkelberg-Mirkovic conjecture) in the spirit of the Kazhdan-Lusztig conjecture. I also discuss a conjecture with S. Riche (a theorem for GL_n) which should eventually make computations more feasible.
Ideas from Hodge theory have found important applications in representation theory. We give a survey of joint work with Ben Elias which uncovers Hodge theoretic structure in the Hecke category ("Soergel bimodules"). We also outline similarities and differences to other combinatorial Hodge theories.
We observe that certain numbers occurring in Schubert calculus for SL_n also occur as entries in intersection forms controlling decompositions of Soergel bimodules and parity sheaves in higher rank. These numbers grow exponentially. This observation gives many counterexamples to Lusztig's conjecture on the characters of simple rational modules for SL_n over a field of positive characteristic. We explain why our examples also give counter-examples to the James conjecture on decomposition numbers for symmetric groups.
We prove the local hard Lefschetz theorem and local Hodge-Riemann bilinear relations for Soergel bimodules. Using results of Soergel and Kübel one may deduce an algebraic proof of the Jantzen conjectures. We observe that the Jantzen filtration may depend on the choice of non-dominant regular deformation direction.