Totally real representations and real function spaces

Calvin Cooper Moore and Joseph Albert Wolf

Vol. 38 (1971), No. 2, 537–542

DOI: 10.2140/pjm.1971.38.537

Abstract

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 = Kg⁻¹K 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

Milestones

Received: 13 October 1970

Published: 1 August 1971