Oberseminar Algebra und Geometrie

Petra Schwer (Otto-von-Guericke-Universität Magdeburg)

Abstract:
Groups like GL_n, SL_n or SP_n  play an important role in many areas of mathematics. It has been known for a long time that some of their properties (when studied over the reals or complex numbers) are best understood via the associated symmetric spaces. Jaques Tits later introduced buildings as a tool to study the respective groups over other field and developed, together with Bruhat, a theory that also captures reductive groups evaluated over non-archimedian local fields with discrete valuations, like the p-adic numbers.

In this talk I will explain how some of the subgroup structures of such a reductive group over a non-Archimedian local field can be explained via Coxeter combinatorics and the geometry of an (affine) Bruhat-Tits building, its apartments and retractions. The building for example simultaneously encodes the (affine) flag variety and (affine) Grassmannian associated to the group. But it also permits to explain more complicated structures such as representation theoretic data or other associated varieties in purely combinatorial terms.