Pure Maths Colloquium: Jay Taylor

Finite reductive groups hold a special place amongst all finite groups, partly due to their role in the classification of finite simple groups but also for their natural interactions with geometric objects. Examples of finite reductive groups are the matrix groups \(\mathrm{GL}_n(q)\), \(\mathrm{SL}_n(q)\), \(\mathrm{SU}_n(q)\), \(\mathrm{Sp}_{2n}(q)\), …, defined over a finite field of \(q\) elements as well as more exotic examples like \(G_2(q)\) and \(E_8(q)\).

The project of trying to classify and compute the irreducible representations of these groups started in earnest in 1907 with the independent work of Jordan and Schur who computed the ordinary character table of \(\mathrm{SL}_2(q)\). Much has happened in the following 115 years with outstanding progress being made due to the enormous contributions of Lusztig.

In this talk I’ll survey some of the key historical results on representations of finite reductive groups, indicate some of the remaining challenges, and present some more recent work that contributes to the classification problem in the modular setting.