Weil representation

Summary

The Weil representation is a unitary representation of the metaplectic group, a two-fold cover of the symplectic group over a local field. Over the real numbers it is the oscillator representation. It comes from the Stone–von Neumann theorem. Twisting the Schrödinger representation of the Heisenberg group by a symplectic map gives an equivalent representation. The intertwining operators then form the Weil representation. It is the source of theta functions and of the theta correspondence in number theory. In physics it describes the symmetries of the quantum harmonic oscillator.