Seminar

Let k be a field of characteristic p and let G be a split reductive group over k.  Mirkovic and Vilonen proved an equivalence between the category of (rational) representations of G and a category of perverse sheaves with coefficients in k on the affine Grassmannian for the Langlands dual complex reductive group.  They also conjectured that the perverse sheaves corresponding to dual Weyl modules satisfy a certain parity vanishing in their stalks.  The conjecture was resolved for p sufficiently large by Achar-Rider, using a case-by-case result of Juteau-M.-Williamson.  In this talk I will discuss joint work with Simon Riche, in which we strengthen the result of JMW and thus, together with the work of Achar-Rider, resolve the Mirkovic-Vilonen conjecture in the remaining characteristics.  Our proofs are uniform and use an exotic t-structure introduced and studied in characteristic zero by Bezrukavnikov, and a categorical braid group action defined by Bezrukavnikov and Riche.