International Journal of Mathematics and Mathematical Sciences
Volume 1 (1978), Issue 2, Pages 235-244
doi:10.1155/S0161171278000277
A Stone-Weierstrass theorem for group representations
Department of Mathematics, University of Toronto, Toronto MSS IAI, Ontario, Canada
Received 16 January 1978
Copyright © 1978 Joe Repka. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.