The oscillator representation of a metaplectic group (a two-fold cover of the symplectic group) over a local field turns out to define a remarkable correspondence between representations of certain pairs of subgroups. This correspondence is germane to broad areas of mathematics, including invariant theory, automorphic forms, and mathematical physics. The analogous representation for the symplectic group over a finite field fails to automatically define such a bijection. However, it turns out that there is a semigroup that may be built naturally from the symplectic group and that allows one to define a bijection of representations. We call this the oscillator semigroup. This chapter describes the construction of the oscillator semigroup and establishes the existence of the bijection.