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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups
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by L. Corwin, F. P. Greenleaf and G. Grélaud PDF
Trans. Amer. Math. Soc. 304 (1987), 549-583 Request permission

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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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 304 (1987), 549-583
  • MSC: Primary 22E25; Secondary 22E27
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0911085-6
  • MathSciNet review: 911085