Seminar Darstellungstheorie

Das Seminar Darstellungstheorie findet normalerweise dienstags, 14.15 - 15.15 Uhr in G.13.18 statt.

Beteiligte Dozenten: Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth.

Andreas Bode: Auslander regularity of p-adic Banach algebras via almost mathematics

Auslander regularity is a cohomological regularity condition for non-commutative rings. We introduce the notion of almost Auslander regular algebras and use the machinery of almost mathematics to prove the Auslander regularity of several Banach algebras appearing naturally in p-adic representation theory: the completed enveloping algebra of a Lie algebra, completed Weyl algebras, and the norm-completion of the distribution algebra of a compact p-adic Lie group. This generalizes earlier results by Ardakov--Wadsley and Schmidt in the case of a discretely valued base field to arbitrary nonarchimedean fields of mixed characteristic.

Damian Sercombe: Unipotent normal subgroups of algebraic groups

Let G be an affine algebraic group over a field k. We show there exists a unipotent normal subgroup of G which contains all other such subgroups; we call it the restricted unipotent radical Rad_u(G) of G. We investigate some properties of Rad_u(G), and study those G for which Rad_u(G) is trivial. In particular, we relate these notions to their well-known analogues for smooth connected affine k-groups.

Jonathan Gruber: Centers and centralizers in (double) affine Hecke algebras

The affine Hecke algebra and its center are important objects of study in combinatorial, geometric and categorical representation theory. In this talk, I will discuss a new commutative subalgebra of the affine Hecke algebra of type A, which arises from a centralizer construction in the double affine Hecke algebra. This subalgebra contains the center, and it admits a canonical basis akin to the Kazhdan-Lusztig basis of the affine Hecke algebra. I will explain how the canonical basis can be used as a tool to compute composition multiplicities in Gaitsgory's central sheaves on affine flag varieties.

Raoul Hallopeau: Holonomicity for coadmissible D-modules over formal curves

In order to study differential equations on a smooth variety, one can introduce a sheaf D of differential operators and look at the associated D-modules that encode many properties of the initial differential equations. More generally, D-modules have other applications beyond differential equations, like representation theory and the Riemann-Hilbert correspondence.

Some interesting and relevant D-modules to look at, generalizing connections, are called holonomic D-modules. They are well-known for complex varieties. In this situation, holonomic D-modules form a well-behaved category, stable by classical operations and with good finiteness properties. They are for example of finite length and have a finite dimensional cohomohology. Such a notion was introduced by Berthelot for arithmetic D-modules with a Frobenius. One motivation for that was to compute the rigid cohomology in characteristic p. More recently, both Huyghe-Schmidt-Strauch and Ardakov-Wadsley have defined categories of coadmissible D-modules over smooth rigid analytic spaces. We do not currently have a good notion of holonomicity in this setting. The aim of this talk is to explain how one can introduce holonomic coadmissible D-modules in the dimension one case.

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