A note on square-integrable representations

Abstract

In this short note, irreducible square-integrable representations of left Hilbert algebras are studied in detail. Orthogonality relations are formulated and proved which contain as a special case the orthogonality relations for square-integrable representations of unimodular locally compact groups. A self-adjoint (generally unbounded) operator is defined on the representation space, the inverse of whose square has some claim to the title “formal dimension operator” since in the case of unimodular groups the inverse of the square of this operator is just the formal dimension times the identity operator. Although the methods used are quite different, this note was inspired by some similar results of M. Duflo and C. Moore for nonunimodular groups.