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.

The Zilber—Pink conjecture describes the points on an algebraic variety which have 'special' properties. In this talk, I will discuss some new results which can be proved about this, focusing on the examples of subvarieties of a torus, an abelian variety, or a product of modular curves. The method of proof is a generalisation of the Buium—Coleman proof of the Manin—Mumford conjecture. Parts of this are joint work with Sudip Pandit (KCL) and with Arnab Saha (IIT Gandhinagar).

Weighted flag varieties are generalizations of flag varieties and weighted projective spaces.  Although they are not usually homogeneous varieties, they are orbifolds and admit a torus action with isolated fixed points, and like ordinary flag varieties, their equivariant cohomology admits a Schubert basis.  This talk will be an introduction to weighted flag varieties, and will also discuss positivity.  Abe and Matsumura proved that the equivariant cohomology of weighted Grassmannians has a positivity property analogous to that for ordinary (non-weighted) flag varieties.  We prove a strengthened version of this result for arbitrary weighted flag varieties, along the way providing a geometric interpretation of the weighted roots of Abe and Matsumura.  This is joint work with Scott Larson.