MA 357: Topics in Representation Theory
Credits: 3:0
Pre-requisites :
- working knowledge of local fields
- Linear Algebraic groups (as covered in MA 379)
- generalities of root systems and Coxeter groups
This course will be an introduction to Bruhat-Tits theory. Given a connected, reductive group $G$ over a non-archimedean local field $F$, the theory constructs a contractible topological space $B(G)$, called the Bruhat-Tits building of $G(F)$. This space has the structure of a poly-simplicial complex and the topological group $G(F)$ acts on the building via automorphisms that preserve this poly-simplicial structure. To each point $x$ in $B(G)$, one can associate various subgroups of $G(F)$, the most obvious one being the stabilizer of the point $x$. The building serves the purpose of organizing the various compact open subgroups of $G(F)$ and these subgroups play a tremendous role in the study of representations of $p$-adic groups.
Organization: The first part of the course will be on affine root systems, Tits’ systems, and the Tits building. Then, we will construct the Bruhat-Tits building and various associated objects for two examples: The group $SL(2)$ and the quasi-split group $SU(3)$. Finally, after a review of the theory of reductive groups over general fields, we will embark on the construction of the building of a connected, reductive group over a non-archimedean local field, first by doing it for quasi-split groups, and then “descending this construction” to the general case.
Suggested books and references:
- F. Bruhat and J. Tits, Groupes reductifs sur un corps local., Publ. Math. IHES 41 (1972).
- F. Bruhat and J. Tits, Groupes reductifs sur un corps local II., Publ. Math. IHES 60 (1984).
- E. Landvogt, A compactification of the Bruhat-Tits building., Lecture Notes in Math., 1619 Springer-Verlag, Berlin, viii+152 pp. (1996).
- T. Kaletha and G. Prasad, Bruhat-Tits theory - a new approach, New Math. Monogr. 44, Cambridge University Press, Cambridge, xxx+718 pp. (2023).