Winter Meeting on Bruhat-Tits buildings
January 6th - 9th 2015
Speakers
The lecture series will be given by:
University of Münster, Germany
Linus will be giving his lectures on "Metric aspects of Euclidean buildings".
Abstract: The simplicial realizations of spherical and euclidean
buildings carry natural metrics with curvature bounded from above. In my lectures I will give a short introduction to spaces with upper curvature bounds. We will then discuss the special metric properties of these buildings, and also some characterizations of spherical and euclidean buildings among metric spaces. Finally we will discuss topological and metric rigidity questions of buildings.
University of Giessen, Germany
Bernhard will be giving his lectures on "Descent in Bruhat-Tits buildings".
Abstract: Galois descent is an extremely useful tool to reduce questions about algebraic groups over arbitrary fields to the case where the ground field is separably closed.
Bruhat and Tits used Galois-descent in order to prove the existence of an affine building associated to a reductive group (of exceptional type) over a local field. In my lectures I will first develop a purely combinatorial theory of descent for arbitrary buildings and focus on the special cases of spherical and affine buildings later. The central notion in this context is that of a (combinatorial) Tits index which is known from the classification of algebraic groups and which has a most natural geometric interpretation for 'their' buildings. During my lectures I will make the connection to algebraic groups and Bruhat-Tits theory by discussing several examples.
École polytechnique, Paris, France
Bertrand will be giving his lectures on "Integral structures in Bruhat-Tits theory".
Abstract: A given reductive group over a local field has many nice integral structures (i.e. defining equations with coefficients in the valuation ring of the ground field), parametrized by the facets of the associated Bruhat-Tits building. We will discuss some examples, their construction, their topological interpretations (in terms of compact open subgroups) and their geometric interpretation in connection with reduction modulo the maximal ideal (in terms of spherical buildings around a facet in the ambient affine building).
Institut Elie Cartan, Nancy, France
Guy will be giving his lectures on Images of line segments by retractions and application to representation theory.
Abstract: In an euclidean building there are (at least) two types of retractions onto an apartment: with center a chamber or a sector-germ. We shall describe the images, under such a retraction, of a minimal gallery or a line segment. When the center is a sector-germ, the image of a line segment is almost a LS-path. These LS-paths describe the multiplicities of the weights in a finite dimensional representation of a reductive group. We shall explain how this leads to proofs of some representation theoretic results.
The 1-hour lectures will be given by:
Ramla Abdellatif
Ecole Normale Supérieure de Lyon, France
Bruhat-Tits buildings in representation theory
Let p be a prime integer. This talk aims to explain and illustrate with some examples how useful the theory of Bruhat-Tits buildings is to study smooth representations of p-adic classical groups in any characteristic. We will especially focus on the theory of coefficient systems, developed by Schneider and Stuhler in the nineties, and on how it branches out depending on the characteristic of the coefficients field chosen for the representations. In particular, we aim to underline how representations in characteristic p are isolated in this context and to show how mysterious they remain so far, even for basic groups as GL_2(F) or SL_2(F) with F being a finite (unramified) extension of Q_p.
University of Glasgow, UK
The Davis realization of a building and lattices in Kac-Moody groups
We will describe the Davis geometric realization of a building, including its metric and topological properties. We then explain how in joint work with Inna Capdeboscq, we have used this realization to construct cocompact lattices in some complete Kac-Moody groups.