Howe duality over finite fields I: The two stable ranges

Abstract: This is the first in a series of papers on type I Howe duality for finite fields, concerning the restriction of an oscillator representation of the symplectic group to a product of a symplectic and an orthogonal group. The goal of the series is describing this restriction completely. In this first paper, we construct the correspondence in the two so called stable ranges, where the rank of one of the factors is large enough with respect to the other.