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

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 OrlikStrauch 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 padic Lie group. This functor is exact, faithful and sends simple gmodules to topologically irreducibile representations. We will describe the coadmissible equivariant Dmodules 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 OrlikStrauch representations.
Andreas Bode: MiniCourse on Localisation and padic representations
Dmodules, 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 welldeveloped since the early 1980s. Recently, a theory of completed Dmodules on padic 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 KazhdanLusztig conjectures, the theory of algebraic Dmodules on smooth varieties, and BeilinsonBernstein Localisation, which expresses representations as the global sections of Dmodules on the associated flag variety. We then turn to the padic analytic theory, introducing the sheaf Dcap and coadmissible Dcapmodules, as well as their Gequivariant analogues, and discuss Ardakov’s version of Localisation, relating modules over the distribution algebra D(G, K) to Gequivariant Dcapmodules on the rigid analytic flag variety. If time permits, we will also discuss recent advances in the development of Dcapmodule theoretic operations and questions of holonomicity.
Christine Huyghe: Introduction to rigid analytic geometry and Dmodules on ZariskiRiemann 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 ZariskiRiemann spaces.
Talk 2: Dmodules on ZariskiRiemann spaces
In the second lecture, we will explain how to construct a ring of differential operators on some ZariskiRiemann 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 padic local Langlands correspondance
(Slides Part 1)
(Slides Part 2)
In a fundamental paper, Drinfeld has constructed a tower of coverings for the Drinfeld upper halfplane which is the open of the projective line on Q_p obtained by removing all the Q_prational points. It is known that the supercuspidal part of the geometric étale \elladique cohomology with compact support of these spaces provides geometric realizations of the classical local Langlands and JacquetLanglands correspondances.
On the other hand, some aspects of the padic 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 BreuilStrauch). 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 DospinescuLe 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 HodgeTate weights (0,1).
In the second talk, we will concentrate on integral structures of equivariant line bundles on the upper halfplane 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 onetoone 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 HodgeTate weights (0,0).
Jan Kohlhaase: SchneiderStuhler theory in natural characteristic
Let G be the group of rational points of a connected reductive group over a nonarchimedean local field F of residue characteristic p. The BruhatTits building X of G is a topological Gspace which plays a fundamental role in the study of smooth Grepresentations. Over the complex numbers, P. Schneider and U. Stuhler gave a comprehensive analysis of the relation between smooth Grepresentations, Gequivariant coefficient systems and Gequivariant 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 Krational hyperplanes in the projective space over a local nonarchimedean 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 SchneiderTeitelbaum and Pohlkamp) to the effect that it becomes applicable to local fields of positive characteristic as well.
Cédric Pépin: The IwahoriHecke algebra of a padic group
This algebra provides an explicit way to study certain smooth representations of a reductive group over a nonarchimedean 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 BreuilSchneider conjecture
(Slides)
This talk will be a swift survey of the BreuilSchneider 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: MiniCourse on Locally Analytic Representations
Talk 1: Topological vector spaces over nonarchimedean 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 ST, 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 FrechetStein 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 BernsteinGelfandGelfand 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 AbeHerzig on the irreducibility of continuous principal series representations.
Last modified: 04.08.2023