Some Computations with Representations of Finite Chevalley Groups in Equal Characteristic

I will discuss two computations involving representations of finite Chevalley groups in equal characteristic. The first one reduces by an argument of Bao Le Hung and Tony Feng to description of a class in top homology of a certain affine Springer fiber (performed jointly with Bao Le Hung, Tony Feng and Pablo Boixeda Alvarez). The second one (joint with Michael Finkelberg, David Kazhdan and Calder Morton-Ferguson) involves a new basis for the ring O(T) of regular functions on the maximal torus T as a module over the ring O(T)^W of invariant regular functions, different from but related to the basis constructed by Steinberg in 1975.