Summer School 2023


The registration takes place on Monday from 9:00 to 9:30 in front of lecture hall 8 (building G, level 10, see here).







9:30 Bode Pépin (9:00) Kohlhaase Kohlhaase Bode
11:00 Strauch Strauch (10:30) Bode Strauch Ardakov
14:00 Huyghe Kohlhaase (11:45) Huyghe Linden
15:30 Pépin Sorensen Junger
16:45 Junger (17:00) Reception

Konstantin Ardakov: Localisation of Orlik-Strauch representations

This is joint work in progress with Tobias Schmidt. About ten years ago, Orlik and Strauch introduced a functor from a certain parabolic category O to the category of admissible locally analytic representations of a semisimple p-adic Lie group. This functor is exact, faithful and sends simple g-modules to topologically irreducibile representations. We will describe the coadmissible equivariant D-modules on the flag variety that arise as localisations of the representations in the image of this functor. Using the geometric induction equivalence, we sketch a strategy to obtain an alternative proof of the topological irreducibility of the Orlik-Strauch representations.

Andreas Bode: Mini-Course on Localisation and p-adic representations

D-modules, modules over the sheaf of differential operators on a variety, provide a powerful geometric tool in the representation theory of Lie algebras and algebraic groups. In the algebraic setting (over C, say), this approach has been well-developed since the early 1980s. Recently, a theory of completed D-modules on p-adic rigid analytic varieties has started to take shape, mainly due to Ardakov and Wadsley, which encapsulates in the same way admissible locally analytic representations. We first give an overview over the ‘classical’ theory: the BGG category O and the Kazhdan-Lusztig conjectures, the theory of algebraic D-modules on smooth varieties, and Beilinson-Bernstein Localisation, which expresses representations as the global sections of D-modules on the associated flag variety. We then turn to the p-adic analytic theory, introducing the sheaf D-cap and coadmissible D-cap-modules, as well as their G-equivariant analogues, and discuss Ardakov’s version of Localisation, relating modules over the distribution algebra D(G, K) to G-equivariant D-cap-modules on the rigid analytic flag variety. If time permits, we will also discuss recent advances in the development of D-cap-module theoretic operations and questions of holonomicity.  

Christine Huyghe: Introduction to rigid analytic geometry and D-modules on Zariski-Riemann spaces

Talk 1: Introduction to rigid analytic geometry
In the first lecture, we will explain basics of rigid analytic geometry, by focusing on two points of view: Tate's construction of rigid analytic spaces and the point of view of Zariski-Riemann spaces.

Talk 2: D-modules on Zariski-Riemann spaces
In the second lecture, we will explain how to construct a ring of differential operators on some Zariski-Riemann spaces that are generic fibers of smooth formal schemes. This construction provides a variant of the localization theorem that Andreas Bode will explain in his last lecture.

Damien Junger: Equivariant line bundle on the Drinfeld tower and p-adic local Langlands correspondance

(Slides Part 1)
(Slides Part 2)
In a fundamental paper, Drinfeld has constructed a tower of coverings for the Drinfeld upper half-plane which is the open of the projective line on Q_p obtained by removing all the Q_p-rational points. It is known that the supercuspidal part of the geometric étale \ell-adique cohomology with compact support of these spaces provides geometric realizations of the classical local Langlands and Jacquet-Langlands correspondances.

On the other hand, some aspects of the p-adic local Langlands correspondence should be visible in the cohomology of line bundles on the coverings of the tower. In this two part serie talk we will describe two results and strategies that illustrate this principle (which generalizes some conjectures of Breuil-Strauch). In the first one, we will explain the study of the structure sheaf and the differential forms (which are informally "of weight 0 and -2") on the coverings done in a paper of Dospinescu-Le Bras and see how the contragredient of their global sections naturally contain any locally analytic representations of GL_2(Q_p) that corresponds to de Rham Galois representation of Hodge-Tate weights (0,1).

In the second talk, we will concentrate on integral structures of equivariant line bundles on the upper half-plane and give their complete classification following a recent paper of the speaker. In particular, we will exhibit a subclass of such objects (of "weight -1") for which taking the contragredient of the global sections on the special fiber gives a one-to-one correspondence with supersingular mod p representations of GL_2(Q_p). These line bundle also provide Banach space representations which should corresponds to de Rham Galois representation of Hodge-Tate weights (0,0).

Jan Kohlhaase: Schneider-Stuhler theory in natural characteristic

Let G be the group of rational points of a connected reductive group over a non-archimedean local field F of residue characteristic p. The Bruhat-Tits building X of G is a topological G-space which plays a fundamental role in the study of smooth G-representations. Over the complex numbers, P. Schneider and U. Stuhler gave a comprehensive analysis of the relation between smooth G-representations, G-equivariant coefficient systems and G-equivariant sheaves on X. In my course, I will introduce these various concepts and explain what is known about the relation between smooth representations, coefficient systems and equivariant sheaves in characteristic p.

Georg Linden: Equivariant Vector Bundles on the Drinfeld Upper Half Space

The Drinfeld upper half space is defined as the complement of all K-rational hyperplanes in the projective space over a local non-archimedean field K. The global sections of homogeneous vector bundles on the projective space restricted to the Drinfeld upper half space naturally give rise to locally analytic GL_d(K)-representations. We describe the structure of these representations via an adaptation of the modified parabolic induction functors F^G_P due to Orlik and Strauch. This generalizes work of Orlik (in turn based on work of Schneider-Teitelbaum and Pohlkamp) to the effect that it becomes applicable to local fields of positive characteristic as well.

Cédric Pépin: The Iwahori-Hecke algebra of a p-adic group

This algebra provides an explicit way to study certain smooth representations of a reductive group over a non-archimedean local field. We will present its basic structural properties in the case of complex coefficients, and state its relation to the geometric representation theory of the Langlands dual group.

Claus Sorensen: An overview of the Breuil-Schneider conjecture

This talk will be a swift survey of the Breuil-Schneider conjecture for the uninitiated. I'll first motivate and state the conjecture precisely. Then I'll go over the current status and progress over the last 15 years, with emphasis on the automorphic aspects.

Matthias Strauch: Mini-Course on Locally Analytic Representations

Talk 1: Topological vector spaces over non-archimedean fields, locally analytic functions and distributions
Locally convex vector spaces, initial topologies, final topologies, Banach spaces, Frechet spaces, compact inductive limits, continuous dual spaces, reflexivity, completed tensor products [all folllowing NFA], locally analytic functions, distributions [following S-T, JAMS].

Talk 2: Locally analytic representations and distribution algebras
In this talk we will introduce locally analytic representations (following the work of Schneider and Teitelbaum) and discuss several examples. The second subject of this talk is the distribution algebra D(G), its Frechet-Stein structure, and the category of admissible locally analytic representations.

Talk 3: From Lie algebra representations to locally analytic representations
In this talk we will introduce analogues of the Bernstein-Gelfand-Gelfand category O for semisimple (more generally, reductive) Lie algebras. Then we discuss the functor F^G_P from (a suitable version of) this category to the category of locally analytic representations (as defined in joined work with Sascha Orlik), and give an overview of some of its applications. For example, we plan to explain its use in recent work of Abe-Herzig on the irreducibility of continuous principal series representations.

Last modified: 04.08.2023

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