A Stone-Weierstrass theorem for group representations

Abstract

It is well known that if G is a compact group and π a faithful (unitary) representation, then each irreducible representation of G occurs in the tensor product of some number of copies of π and its contragredient. We generalize this result to a separable type I locally compact group G as follows: let π be a faithful unitary representation whose matrix coefficient functions vanish at infinity and satisfy an appropriate integrabillty condition. Then, up to isomorphism, the regular representation of G is contained in the direct sum of all tensor products of finitely many copies of π and its contragredient.We apply this result to a symplectic group and the Weil representation associated to a quadratic form. As the tensor products of such a representation are also Weil representations (associated to different forms), we see that any discrete series representation can be realized as a subrepresentation of a Weil representation.