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Journal of Lie Theory 19 (2009), No. 3, 557--612
Copyright Heldermann Verlag 2009

Decomposition and Multiplicities for Quasiregular Representations of Algebraic Solvable Lie Groups

Bradley N. Currey
Dept. of Mathematics and Computer Science, Saint Louis University, St. Louis, MO 63103, U.S.A.

We obtain an explicit irreducible decomposition for the quasiregular representation τ of a connected algebraic solvable Lie group induced from a co-normal Levi factor. In the case where the multiplicity function is unbounded, we show that τ is a finite direct sum of subrepresentations τε where for each ε, τε is either infinite or has finite but unbounded multiplicity. We obtain a criterion by which the cases of bounded multiplicity, finite unbounded multiplicity, and infinite multiplicity are distinguished.

Keywords: Quasiregular representation, coadjoint orbit, Plancherel formula, multiplicity function.

MSC: 22E45, 22E25; 43A25

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