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.
