Vol. 42, No. 2, 1972

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Representation theory of almost connected groups

Ronald Leslie Lipsman

Vol. 42 (1972), No. 2, 453–467

Let G be a locally compact group and G0 its connected component of the identity. If G∕G0 is compact, then G is a projective limit of Lie groups. In fact, there exist arbitrarily small compact normal subgroups H G such that G∕H is a Lie group. Suppose H is such a compact, co-Lie subgroup of G. Then any unitary representation of G∕H can be lifted to G in a natural way. Conversely, given a unitary representation π of G, one may ask whether it really lives on a Lie factor—that is, does there always exist a compact normal subgroup H G such that G∕H is a Lie group and π(h),h H, is the identity operator ? In this paper it is shown that this is indeed the case whenever π is irreducible (or more generally whenever π is factorial). The dual space Ĝ (= equivalence classes of irreducible unitary representations) is then realized as an inductive limit of the dual spaces of Lie groups. This inductive limit is first cast in a topological setting (using the dual topology on Ĝ); and then, when G is also unimodular and type I, one obtains a measure-theoretic interpretation of the inductive limit (using the Plancherel measure). One application of these results is the fact that an almost connected group whose solvable radical is actually nilpotent must be a type I group.

Mathematical Subject Classification 2000
Primary: 22D10
Received: 21 May 1971
Published: 1 August 1972
Ronald Leslie Lipsman