Howe Duality Over Finite Fields

In this talk, I will completely describe the decomposition of an oscillator representation under the tensor embedding of a product of a symplectic and orthogonal group in the case of finite fields via a correspondence proposed by Roger Howe. Applications include a proof of the rank conjecture and a very stable case of the character sum conjecture of Shamgar Gurevich and Roger Howe. My approach is based on studying the endomorphism algebras from the point of view of interpolation of tensor categories.