Locally analytic representations of p-adic groups

Friday 10 September 2021, hybrid, local organiser: Rachel Camina

Locally analytic p-adic representations of p-adic groups are of
great interest not just to representation theorists, but also to
other areas: they arise naturally in number theory (where they
form key objects in the mainly conjectural p-adic local Langlands
correspondence) and in arithmetic geometry, with many
constructions drawing inspiration from geometric representation theory.

This meeting, which can be viewed as a continuation of the
introductory lectures of the LMS Autumn Algebra School 2020, aims
to bring together young researchers and experts in the field to
present current trends in research in this highly dynamic area.

The timetable is as follows, where all times are given in British Summer Time:

1:00-2:00pm: Simon Wadsley
2:15-2:45pm: Adam Jones
2:45-3:15pm: Tea and Coffee break
3:15-3:45pm: Nicolas Dupré
4:00-5:00pm: Gabriel Dospinescu

To register for the event and to receive the talk links, please
email Andreas Bode (andreas.bode@uni-due.de).

The FCG Research Group is supported by an LMS Joint Research
Groups in the UK Scheme 3 grant. Limited funding is available for
PhD students, allocated on a first come first served basis.

Abstracts

Simon Wadsley: Introduction to locally analytic representations of p-adic groups

This talk will aim to introduce locally analytic representation theory of p-adic groups to those who are familiar with basic representation theory, say of finite groups, over the complex numbers but with no previous knowledge of the p-adic setting.

Adam Jones: Representations of p-adic Lie groups via the Iwasawa algebra

When studying the representation theory of a p-adic Lie group GG, a common approach is to associate an algebra AA to GG, whose module structure describes a class of representations of interest, e.g. Hecke algebras if considering smooth representations, or the distribution algebra when considering locally analytic representations. Taking A=Λ(G)A=Λ(G) to be the Iwasawa algebra of GG, we can similarly describe the class of continuous, pseudocompact representations of GG. Focusing on the case where G is compact, I will outline some techniques in representation theory that can be employed to completely describe the prime ideal structure of Λ(G)Λ(G), and some results that have followed from this approach.

Nicolas Dupré: p-adic quantum groups

In a paper from 2007, Soibelman suggested that it should be possible to develop a quantum analogue to Schneider and Teitelbaum's theory of admissible locally analytic representations. While the representation theory of p-adic groups had been widely studied, notably for its connection to the Langlands programme, the idea of developing quantum analogues of p-adic groups and their representation theory was entirely new. To a certain extent, this idea still has not yet been explored much. In this talk, we will attempt to begin correcting this by introducing certain p-adic analytic constructions of quantum groups and explain how to construct quantum D-modules with them. Indeed, in the last decade or so, techniques using D-modules have been introduced in the study of locally analytic representations of p-adic groups as well as continuous representations, notably using analogues of the Beilinson-Bernstein localisation theorem. We will finish by explaining how to obtain a quantum version of one of these analytic localisation theorems.

Gabriel Dospinescu: A few remarks on a locally analytic version of Scholze's functor

A few years ago Scholze defined a functor from smooth torsion representations of GL_n(Q_p) to representations of the product of the units in a suitable division algebra and the absolute Galois group of Q_p, using the cohomology of the Drinfeld tower. We will explain a variation on this construction, taking as input locally analytic representations, and how this is related to recent work of Lue Pan. This is joint work (very much in progress) with Juan Esteban Rodriguez Camargo.