Complete Irreducibility and X-Spherical Representations

✍ Scribed by H. Stetkaer

• Book ID: 102592665
• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 458 KB
• Volume: 113
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1993.1056

✦ Synopsis

Let (G) be a locally compact group, (K) a compact subgroup, and (\chi: K \rightarrow{|=|=1}) a continuous homomorphism. Let (\pi) be a continuous irreducible representation of (G) on a complete locally convex space (V) such that the subspace ({v \in V \mid \pi(k) v=) (\chi(k) v, \forall k \in K) }as dimension 1 . Then (\pi) is completely irreducible. Furthermore (\pi) is Naimark related to a representation on a reflexive Fréchet space which is a closed subspace of (\mathscr{G}(G)) of the form (\overline{\operatorname{span}}{L(g) \phi \mid g \in G}) where (\phi) is a (\bar{\gamma})-spherical function. A corollary is that irreducible eigenspace representations are completely irreducible. 1993 Academic Press, Inc.