  These are notes for my Takagi lecture at the University of Tokyo in November,
2016. 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-Mirkovic conjecture) in the spirit of the Kazhdan-Lusztig
conjecture. I also discuss a conjecture with S. Riche (a theorem for GL_n)
which should eventually make computations more feasible.
