Semiorthogonal decompositions for representations of algebraic groups
Abstracts of Research talks (4 September)
On the semiorthogonal decompositions for twisted flag varieties
Alexey Ananyevskiy
In the talk I will outline how one can use Samokhin and van der Kallen decomposition of Db(rep P) to construct full exceptional collections on the forms of G/P twisted by a G-cocycle. Then I will discuss a generalization of the decomposition to the case of a non-simply connected split semisimple group G, and its consequences for the derived categories of inner twisted forms of G/P, categorifying Panin's computation of Quillen K-theory of projective homogeneous varieties. This is based on an ongoing work in progress with Alexander Samokhin.
The Steinberg basis and the asymptotic Hecke algebra
Stefan Dawydiak
Lusztig defined the asymptotic Hecke algebra J to capture via Kazhdan-Lusztig theory certain features of the affine Hecke algebra at “q=0.” Rigidifying the cell filtration of H, J is a direct sum indexed nilpotent orbits of G^; the best understood summand J0 corresponds to the zero orbit. For this summand, I will explain two related contexts in which the Steinberg basis appears in connection with J0, and in which further progress is frustrated by lack of understanding of the dual basis of the Steinberg basis.
Singular cohomology of BG via representation theory
Dmitry Kubrak
Let G be a split reductive group and let G(C) be the topological group of its complex points. It’s a classical problem in topology to understand the singular cohomology ring of the corresponding classifying space BG(C) known also as the ring of “characteristic classes”. While for rational coefficients a simple description was given by Borel in 1950’s, for Fp-coefficients even the dimensions of individual cohomology groups in some cases are not known. I will talk about several recently developed methods (like prismatic cohomology, even filtration and derived Satake with Fp-coefficients) that allow to approach this problem from the point of view of representation theory of either the reduction GFp as an algebraic group or its Langlands dual group. All three methods in particular produce a filtration (in the derived sense) on cohomology of BG, and it might be expected that the associated gradeds for some of these filtrations coincide: this turns out to be equivalent to the vanishing of B-cohomology of a certain explicit B-module, which I wasn’t able to prove or disprove.