Let k be a field and G a connected split semisimple algebraic group defined over k. The group of rational points G(k) (e.g., SL_n(C), SL_n(Q), SL_n(F_p)) is called a Chevalley group. Chevalley groups admit explicit generators and relations, which has many interesting applications including the constructions of finite simple groups (of Lie type) and maximal compact subgroups of G(k) when k is a local field (by Iwahori and Matsumoto). The latter construction is a precursor of Bruhat-Tits theory. This course is an introduction to this area of mathematics and the main reference is Steinberg’s lectures on Chevalley groups.

Theory of complex semisimple Lie algebra.

1.     Representation theory, Fulton and Harris

2.     Lectures on Chevalley groups, Steinberg

3.     Linear algebraic groups, Borel

4.     Reductive groups over local fields, Tits.

5.     The Bruhat-Tits building of a p-adic Chevalley group and an application to representation theory