On Virtual Representations of Finite Chevalley Groups in Defining Characteristic

Let G be a finite Chevalley group, i.e., the group of F_q points of a reductive group over F_q. Virtual representations of G in defining characteristic can be constructed in two ways, either by Brauer-Nesbitt reduction of complex representations, or by restricting an algebraic representation. G. Lusztig conjectured the shape of formulas connecting the two procedures; I will discuss a realization of his proposal related to decomposition of the class of diagonal for G/B coming from summands in the push-forward of the structure sheaf under Frobenius. 

Time permitting I will discuss a different, unrelated at present, way to describe such virtual representations linking it to homology of an affine Springer fiber. This found application in the work of Tony Feng and Viet Bao Le Hung on Breuil-Mezard conjectures. 

Based on joint works with Finkelberg, Kazhdan and Morton-Ferguson and with Boixeda Alvarez, McBreen and Yun respectively.