MA 357: Topics in Representation Theory

Credits: 3:0

Pre-requisites :

  1. working knowledge of local fields
  2. Linear Algebraic groups (as covered in MA 379)
  3. 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:

  1. F. Bruhat and J. Tits, Groupes reductifs sur un corps local., Publ. Math. IHES 41 (1972).
  2. F. Bruhat and J. Tits, Groupes reductifs sur un corps local II., Publ. Math. IHES 60 (1984).
  3. E. Landvogt, A compactification of the Bruhat-Tits building., Lecture Notes in Math., 1619 Springer-Verlag, Berlin, viii+152 pp. (1996).
  4. T. Kaletha and G. Prasad, Bruhat-Tits theory - a new approach, New Math. Monogr. 44, Cambridge University Press, Cambridge, xxx+718 pp. (2023).

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Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 14 Jul 2024