Howe duality over finite fields III: Full computation and the Gurevich-Howe conjectures

In this third paper in a series on type I Howe duality for finite fields, we give a complete description of the restriction of the oscillator representation over a finite field to products of dual pairs of symplectic and orthogonal groups in all cases that occur. We also provide a dictionary with the notation of S.-Y. Pan, who identified which tensor products of irreducible representations occur with non-zero multiplicity. As an application, we give a recursive construction of all irreducible complex representations of finite symplectic and orthogonal groups and a recursive formula for the characters of unipotent cuspidal representations. We also give a proof of the Gurevich-Howe rank and exhaustion conjectures for type C groups.