Vol. 38, No. 2, 1971

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Totally real representations and real function spaces

Calvin Cooper Moore and Joseph Albert Wolf

Vol. 38 (1971), No. 2, 537–542

Let G be a locally compact group. The notion of “totally real” unitary representation of G is defined and investigated in §1. In particular, if K is a compact subgroup of G, it is shown that every closed G-invariant subspace of L2(G∕K) is spanned by real-valued functions if, and only if, KgK = Kg1K for every g G. In §2 the coset space X = GlK is specialized to a Riemannian symmetric space, where the double coset condition is replaced by a simple Weyl group condition.

Mathematical Subject Classification 2000
Primary: 22D10
Secondary: 22E45, 43A65
Received: 13 October 1970
Published: 1 August 1971
Calvin Cooper Moore
Joseph Albert Wolf