Oberseminare

Olivier Dudas (Marseille)
Seminar: Lie-Theorie
Tuesday, 26.11.2024, 16:15 Uhr, room IB 1/103

Title: "Representations of GL(n,x), x an indeterminate"

Abstract:
I will begin by presenting some observations from the 1980s showing that many invariants of the representation theory of GL(n,q), the finite general linear group over a field with q elements, behave as if q were an indeterminate. After playing with a few examples (q=1, q a root of 1, negative q...), I will explain how the modern approach to the representation theory of this group should lead to a possible explanation of these phenomena.

Giulia Iezzi (RWTH Aachen)
Seminar: Lie-Theorie
Talk on Monday, 18.11.2024, 16:15 Uhr, room IA 1/109

Title: "Linear degenerations of Schubert varieties via quiver Grassmannians"

Abstract:
Quiver Grassmannians are projective varieties  parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. For instance, this method was used to study linear degenerations of flag varieties, obtaining characterizations of flatness, irreducibility and normality via rank tuples. We provide a construction for realising smooth Schubert varieties as quiver Grassmannians and desingularizing non-smooth Schubert varieties. We then exploit this construction to define linear degenerations of Schubert varieties, giving a combinatorial description of the correspondance between their isomorphism classes and the B-orbits of certain quiver representations.

Wushi Goldring (Stockholm)
Seminar: Lie-Theorie
Talk on Monday, 11.11.2024, 16:15 Uhr, via Zoom

Title: "Propagating the algebraicity of automorphic representations via functoriality"

Abstract:
My talk concerns the algebraic properties of automorphic representations. These  infinite-dimensional representations of reductive groups over number fields are defined using harmonic analysis. For every prime p, they admit p-adic analogues of Laplacian eigenvalues called Hecke eigenvalues. One of the main mysteries of the Langlands Program is that some automorphic representations have algebraic Hecke eigenvalues while others have transcendental ones. For some, the algebraicity follows from the geometry of Shimura varieties and/or locally symmetric spaces, while for others there are conjectures predicting either algebraicity or transcendence. But there are also instances where it is unclear whether to expect algebraic or transcendental eigenvalues.

I will discuss when Langlands Functoriality, another central theme of the Langlands Program, can be used to reduce the algebraicity for a representation \pi of a group G to that of some other representation \pi' of some other group G' for which algebraicity is known for geometric reasons. Via difficult dictionaries, this translates into much more elementary problems in group theory. In the negative direction, we give several group-theoretic obstructions to the existence of \pi'. In particular, this gives a conceptual explanation for why \pi' doesn't exist when \pi arises from non-holomorphic analogues of modular forms called Maass forms. In the positive direction, we exhibit new cases of algebraicity of Hecke eigenvalues for automorphic representations for which no direct link to geometry is known. For some of these, we also associate the Galois representations predicted by the Langlands correspondence.

Dmitriy Rumynin (Warwick, zZt MPIM Bonn)
Seminar: Lie-Theorie
Talk on Monday, 04.11.2024, 16:15 Uhr, room IA 1/109

Title: "Disconnected Reductive Groups"

Abstract:
A disconnected reductive group is a linear algebraic group whose connected component of the identity is a reductive group. If one is only interested in connected reductive groups, disconnected ones enter the picture as subgroups.
In this talk I will explain how to classify disconnected reductive groups up to an isomorphism. Time permitting, I will also briefly discuss the representation ring of such a group. The talk is based on joint work with Dylan Johnston and Diego Martin Duro.

Matilde Maccan (RUB)
Seminar: Lie-Theorie
Talk on Monday, 28.10.2024, 16:15 Uhr, room IA 1/109

Title: "Parabolic subgroup schemes in small characteristics"

Abstract:
Any rational projective homogeneous variety can be written as a quotient of a semi-simple algebraic group by a so-called parabolic subgroup. In this talk we complete the classification of parabolic subgroup schemes (which can be non-reduced) and formulate it in a uniform way, independent of type and characteristic. The cases we focus on are of a base field of characteristic two or three. We will then move on to a few geometric consequences.