Mittagsseminar (2023)

Tuesdays 10-12h
(Talk starts at 10:15 and ends at 11:45)

Seminarraumzentrum (SRZ) 216/217
Orléans-Ring 12
(With Occasional Room Changes)

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Spaces of p-adic automorphic forms and categorical p-adic local Langlands

A categorical approach to the (p-adic) Langlands program searches for a functor from categories of representations of groups such as G = GL_n(F), F a finite extension of Q_p, to coherent sheaves on stacks of L-parameters. These functors are also supposed to feature in a description of the étale cohomology of Shimura varieties in terms of coherent cohomology of spaces of Galois representations. In this talk we will focus on locally analytic representations of G and we are looking for the description (in terms of coherent sheaves and Galois representations) of the cohomology of Shimura varieties with coefficients in certain overconvergent p-adic coefficient systems (closely related to overconvergent p-adic automorphic forms). I will explain what this description looks like in the case of the modular curve and discuss geometric properties of the spaces of Galois representations involved.

Equivariant motivic characteristic classes of Schubert cells

We explain in the context of complete flag varieties X = G/B the inductive calculation of equivariant motivic characteristic classes of Schubert cells via suitable Demazure-Lusztig operators, fitting with convolution actions of corresponding Hecke-algebras and Weyl groups. Applications include solutions of:
(1) a conjecture of Bump, Nakasuji and Naruse. Here a specialization of the equivariant K-theory of G/B was identified, as a Hecke module, to the Iwahori-fixed part of the the principal series representation of a p-adic Langlands dual group. Under this identification, certain classes related to motivic Chern classes of Schubert cells, were sent to the standard basis elements of the principal series representation, and the fixed point classes were sent to the Casselman basis elements;
(2) a positivity conjecture of Aluffi-Mihalcea for the non-equivariant MacPherson Chern classes of Schubert cells;
(3) a positivity conjecture about the Euler characteristic of generic triple intersections of Schubert cells.
This is joint work with P. Aluffi, L. Mihalcea and C. Su, and for (3) with C. Simpson and B. Wang.

Taming perfectoid fields

Tilting perfectoid fields allows to transfer results between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. We present a model-theoretic approach to tilting via ultraproducts, which allows to transfer many first-order properties between a perfectoid field and its tilt (and conversely). In particular, our method yields a simple proof of the Fontaine-Wintenberger Theorem which states that the absolute Galois group of a perfectoid field and its tilt are canonically isomorphic. A key ingredient in our approach is an Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof).

Square roots of symplectic L-functions and Reidemeister torsion

We give a purely topological formula for the square class of the central value of the L-function of a symplectic representation on a curve. We also formulate a topological analogue of the statement, in which the central value of the L-function is replaced by Reidemeister torsion of 3-manifolds. This is related to the theory of epsilon factors in number theory and Meyer’s signature formula in topology among other topics. We will present some of these ideas and sketch aspects of the proof. This is joint work with Akshay Venkatesh.

Motivic homotopy theory of the classifying stack of finite groups of Lie type

Let G be a split reductive group over F_p with associated finite group of Lie type G^F. Let T be a split maximal torus contained inside a Borel B of G. We relate the (rational) Tate motives of BG^F with the T-equivariant Tate motives of the flag variety G/B. We explain how this could lead to representation theoretic applications, using recent advances in motivic geometric representation theory of the flag variety. On the way, we show that for a split reductive group G over a field k acting on a smooth k-scheme X, we get an isomorphism Aⁿ_G(X, m)_Q ≅ Aⁿ_T(X, m)_Q^W extending the classical result of Edidin-Graham to higher equivariant Chow groups.

Pentecost

Pentecost is a Christian holiday which takes place on the 50th day (the seventh Sunday) after Easter Sunday. It commemorates the descent of the Holy Spirit upon the Apostles and other followers of Jesus Christ while they were in Jerusalem celebrating the Feast of Weeks, as described in the Acts of the Apostles (Acts 2:1-31).

The affine Grassmannian as a presheaf quotient

The affine Grassmannian of a reductive group G is usually defined as the étale sheafification of the quotient of the loop group LG by the positive loop subgroup. I will discuss various triviality results for G-torsors which imply that this sheafification is often not necessary.