A Stone-Weierstrass theorem for group representations

Joe Repka

International Journal of Mathematics and Mathematical Sciences, 1978, vol. 1, 1-10

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.

Date: 1978
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:174034

DOI: 10.1155/S0161171278000277

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