Vol. 70, No. 2, 1977

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Square integrable primary representations

Calvin Cooper Moore

Vol. 70 (1977), No. 2, 413–427

If π is a unitary representation of a locally compact group G, a weight ϕ on the von Neumann algebra R(π) generated by π is called semi-invariant if it is transformed into scalar multiples of itself by the action of G. If π is primary we show such objects are essentially unique if one specifies the scaling factor (Schur’s lemma). We then study square integrable primary representations and show that a number of possible different definitions are equivalent. We show that any such representation π has a semi-invariant weight scaling by the modular function, and this object is seen to be the proper generalization of the formal degree of π. We formulate and prove generalized Schur orthogonality relations for π. Finally we specialize to the semi-finite case and identify the formal degree as an operator affiliated to R(π). For irreducible π the results reduce to those of Duflo and the author, and Phillips.

Mathematical Subject Classification 2000
Primary: 22D12
Received: 27 October 1976
Published: 1 June 1977
Calvin Cooper Moore