Algebraic representations and constructible sheaves

G. Williamson

Abstract: I survey what is known about simple modules for reductive algebraic groups. The emphasis is on characteristic $p>0$ and Lusztig's character formula. I explain ideas connecting representations and constructible sheaves (Finkelberg--Mirković conjecture) in the spirit of the Kazhdan--Lusztig conjecture. I also discuss a conjecture with S. Riche (a theorem for $\mathrm{GL}_n$) which should eventually make computations more feasible.