Buildings and the Structure of p-adic Groups

D2.314 at Paderborn University and online, Wednesdays from 11:00-13:00 and Fridays from 10:00-11:00.
Please contact Carsten Peterson (clhpeterson1870@gmail.com) to be added to the appropriate mailing list (and especially for the zoom link).
Buildings are combinatorial and geometric objects which generalize many of the essential features of flag manifolds and Riemannian symmetric spaces. The two most important classes are the spherical buildings and the affine (also called Euclidean) buildings, which in “most” cases are constructed from reductive algebraic groups. Spherical buildings arise from studying the combinatorics of parabolic subgroups in reductive algebraic groups over arbitrary fields and, in the real case, arise as the boundaries of non-compact Riemannian symmetric spaces. On the other hand, affine buildings, which will be the main focus of this course, arise from studying the combinatorics of compact open (more specifically, parahoric) subgroups of reductive algebraic groups over non-archimedean local fields (such as the p- adic numbers). From a geometric, algebraic, and analytic perspective, affine buildings are remarkably similar to symmetric spaces but in many ways “easier” to work with because of their discrete structure. Furthermore they are an indispensable tool for studying the structure and representation theory of reductive groups over non-archimedean local fields which plays a big role in, e.g., the theory of automorphic forms.

Rough selection of topics to be covered: (affine) root systems, (affine) Weyl/Coxeter groups, Coxeter complexes, structure of non-archimedean local fields, basic structure theory of algebraic groups (Chevalley groups), (B,N)-pairs and their associated buildings, matrix decompositions (Bruhat, Cartan, Iwasawa, Iwahori) and their geometric interpretations, spherical building at infinity of an affine building, Hecke algebras associated to buildings, Satake isomorphism.

Material will be taken from a number of sources including:

Mark Ronan. Lectures on buildings. Vol. 7. Perspectives in Mathematics. Academic Press, Inc., Boston, MA, 1989, pp. xiv+201

Tasho Kaletha and Gopal Prasad. Bruhat-Tits theory—a new approach. Vol. 44. New Mathematical Monographs. Cambridge University Press, Cambridge, 2023, pp. xxx+718

Kenneth S. Brown. Buildings. Springer-Verlag, New York, 1989, pp. viii+215

Peter Abramenko and Kenneth S. Brown. Buildings. Vol. 248. Graduate Texts in Mathematics. Theory and applications. Springer, New York, 2008, pp. xxii+747 1

Paul Garrett. Buildings and classical groups. Chapman & Hall, London, 1997, pp. xii+373 • James Parkinson. // “Buildings and Hecke Algebras”. PhD thesis. University of Sydney, 2005. url: https://www.maths.usyd.edu.au/u/jamesp/0.pdf

Joseph Rabinoff. The Bruhat-Tits building of a p-adic Chevalley group and an application to representation theory. 2007. url: https://services.math.duke.edu/~jdr/ papers/building.pdf

William Casselman. Geometry of the tree. 2019. url: https://personal.math.ubc. ca/~cass/research/pdf/Tree.pdf