The Bruhat order on a Weyl group has a representation theoretic interpretation in terms of Verma modules. The talk concerns resulting interactions between combinatorics and homological algebra. I will present several questions around the above realization of the Bruhat order and answer them based on a series of recent works, partly joint with Volodymyr Mazorchuk and Rafael Mrden.
Let G be a complex, connected, reductive, algebraic group, and χ : ℂ× → G be a fixed cocharacter that defines a grading on 𝔤, the Lie algebra of G. In my first talk I have talked about the grading, derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. In this talk I will define parabolic induction and restriction both on nilpotent cone and graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumptions together with Mautner's cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig's work on ℤ-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.
Admissible representations of real reductive Lie groups are a key player in the world of unitary representation theory. The characters of irreducible admissible representations were described by Lusztig-Vogan in the 80s in terms of a geometrically defined module over the associated Hecke algebra. In this talk, I'll describe a categorification of this module using Soergel bimodules, with a focus on examples.
This talk focuses on the Humphreys-Verma conjecture about the lifting of principal indecomposable modules for restricted Lie algebras to their ambient algebraic groups and on Donkin's tilting module conjecture. Several techniques that are needed to tackle the conjectures for small primes will be introduced and results for the exceptional group G2 will be discussed in some detail.
Kazhdan and Lusztig proved the Deligne-Langlands conjecture, a bijection between irreducible representations of unipotent principal block representations of a p-adic group with certain unipotent Langlands parameters in the Langlands dual group (plus the data of certain representations). We lift this bijection to a statement on the level of categories. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category equivalent to modules for the affine Hecke algebra (or specializing at q, unipotent principal block representations of a p-adic group). Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category.
This talk will introduce spherical elements in a finite Coxeter system. These spherical elements are a generalization of Coxeter elements, that conjecturally, for Weyl groups, index Schubert varieties in the flag variety G/B that are spherical for the action of a Levi subgroup. We will see that this conjecture extends and unifies previous sphericality results for Schubert varieties in G/B due to P. Karuppuchamy, J. Stembridge, P. Magyar–J. Weyman-A. Zelevinsky. In type A, the combinatorics of Demazure modules and their key polynomials, multiplicity freeness, and split-symmetry in algebraic combinatorics are employed to prove this conjecture for several classes of Schubert varieties.
The Schur functor and its inverses give an important connection between the representation theories of the symmetric group and the general linear group. Kleshchev and Nakano proved in 2001 that when the characteristic of the field is at least 5, the image of the Specht module under the inverse Schur functor is isomorphic to the dual Weyl module. In this talk I will address what happens in characteristics 2 and 3: in characteristic 3, the isomorphism holds, and I will give an elementary proof of this fact which covers also all characteristics other than 2; in characteristic 2, the isomorphism does not hold for all Specht modules, and I will classify those for which it does. Our approach is with Young tableaux, tabloids and Garnir relations.
In this talk I will report on the progress in the project of trying to understand categories of tilting modules as categories, meaning the morphisms (and not objects) in these categories and their relations, with the focus being on SL2 and SL3. Joint work with Paul Wedrich.
Let G be a reductive complex algebraic group with a Borel subgroup B. A spherical G-variety is an irreducible normal G-variety X where B has an open orbit. If X is affine, or if it is projective but endowed with a G-linearized ample line bundle, then the group action criteria for the sphericality is in fact equivalent to the representation theoretic statement that a certain space of functions (related to X) is multiplicity-free as a G-module. In this talk, we will discuss the following question about a class of spherical varieties: if X is a Schubert variety for G, then when do we know that X is a spherical L-variety, where L is the stabilizer of X in G.
The geometry of closures of K-orbits in the flag variety governs key properties in representation theory of real reductive groups. For example, Kazhdan-Lusztig-Vogan polynomials and characteristic cycles of Harish-Chandra modules are of current interest. We recall how small resolutions have been used to compute these invariants, describe fibres of the resolutions from last week, and describe more small resolutions for the real reductive groups Sp2n(ℝ), U(p,q), and complex groups. Along the way we consider an application to Schubert varieties.
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