Abstract

In the modular (char p) representation theory of algebraic reductive groups, the Frobenius twist is a great self-symmetry of the category of representations. Geometrically this self-symmetry is related to the embedding of the affine grassmannian, which is the based loop space of the reductive group, into itself as based loops that repeat themselves for p times. I'll explain an interpretation of the Donkin's tensor product conjecture as a consequence of this geometry and point out some potential ways to turn this into a proof. I'll also explain how to prove the quantum group version of the Donkin's tensor product theorem using this geometry.