# Introduction to Bruhat-Tits buildings

## Schedule

The course will take place on Mondays from 14.00 to 16.00. First lecture on Monday 14/10/19 and it will continue for 8 weeks.

You can contact me at marcelo.demartino 'at' maths.ox.ac.uk.

## Sinopsis

Buildings are purely combinatorial objects that, as topological spaces, are cell complexes made up out of smaller complexes, the so-called apartments. When a group G acts nicely on the building, information about the structure and the representation theory of G can be inferred from this action. The lectures will be divided into two parts:

- Part 1: We will develop a general theory of buildings in an axiomatic way and discuss Coxeter groups, Coxeter complexes and BN-pairs.
- Part 2: We will discuss some properties of spherical buildings, but the main focus will be affine buildings which are relevant to theory of p-adic groups.

## References

The main references for the course will be:

- M. Ronan, Lectures on Buildings, Academic Press, 1989.
- P. Garrett, Buildings and Classical Groups, Chapman and Hall, 1997.

Another valuable references are:

- K. Brown, Buildings, Springer-Verlag, New York, 1989.
- J. Humphreys, Reflection Groups and Coxeter Groups, Camb. Univ. Press, 1990.

Specially for the second part, aside from the above-mentioned books, the monumental treatise of Bruhat and Tits, entitled "Groupes Reductifs sur un Corps Local", parts I, II and III are, of course, very relevant. But we also single out:

- I. G. MacDonald, Spherical Functions on a Group of p-adic Type, Ramanujan Institute, 1971.
- J. Tits, Reductive groups over local fields, Proc. Symp. Pure Math, Vol. 33, part 1, pp. 29--69, 1979.

## Course Files

These are the lecture notes for the course and here are the pictures. In case you spot typos/mistakes in the
text, or has any remarks/questions, please drop me a line. Last update: 09 December 2019.

Here are the slides to the course:

## Exercises

These are the exercise sheets for the course. If you need some sort of evaluation, please
complete these exercises and send it to marcelo.demartino 'at' maths.ox.ac.uk until
Wednesday of week 8.