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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups

Authors: L. Corwin, F. P. Greenleaf and G. Grélaud
Journal: Trans. Amer. Math. Soc. 304 (1987), 549-583
MSC: Primary 22E25; Secondary 22E27
MathSciNet review: 911085
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Abstract: Let $ K$ be a Lie subgroup of the connected, simply connected nilpotent Lie group $ G$, and let $ \mathfrak{k}$, $ \mathfrak{g}$ be the corresponding Lie algebras. Suppose that $ \sigma $ is an irreducible unitary representation of $ K$. We give an explicit direct integral decomposition of $ {\operatorname{Ind} _{k \to G}}\sigma $ into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between $ G^\wedge$ and the coadjoint orbits in $ {\mathfrak{g}^{\ast}}$ (and similarly for $ K^\wedge,\,{\mathfrak{k}^{\ast}}$). Let $ P:{\mathfrak{k}^{\ast}} \to {\mathfrak{g}^{\ast}}$ be the canonical projection, let $ {\mathcal{O}_\sigma } \subset {\mathfrak{k}^{\ast}}$ be the orbit corresponding to $ \sigma $, and, for $ \pi \in G^\wedge$, let $ {\mathcal{O}_\pi } \subset {\mathfrak{g}^{\ast}}$ be the corresponding orbit. The main result of the paper says essentially that $ \pi \in G^\wedge$ appears in the direct integral iff $ {P^{ - 1}}({\mathcal{O}_\sigma })$ meets $ {\mathcal{O}_\pi }$; the multiplicity of $ \pi $ is the number of $ {\operatorname{Ad} ^{\ast}}(K)$-orbits in $ {\mathcal{O}_\pi } \cap {P^{ - 1}}({\mathcal{O}_\sigma })$. There is also a natural description of the measure class in the integral.

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