Infinite Dimensional Multiplicity Free Spaces II: Limits of Commutative Nilmanifolds

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dc.creatorWolf, Joseph A.
dc.date2008-01-25
dc.date.accessioned2026-07-07T08:56:26Z
dc.date.available2026-07-07T08:56:26Z
dc.descriptionWe study direct limits $(G,K) = \varinjlim (G_n,K_n)$ of Gelfand pairs of the form $G_n = N_n\rtimes K_n$ with $N_n$ nilpotent, in other words pairs $(G_n,K_n)$ for which $G_n/K_n$ is a commutative nilmanifold. First, we extend the criterion of \cite{W3} for a direct limit representation to be multiplicity free. Then we study direct limits $G/K = \varinjlim G_n/K_n$ of commutative nilmanifolds and look to see when the regular representation of $G = \varinjlim G_n$ on an appropriate Hilbert space $\varinjlim L^2(G_n/K_n)$ is multiplicity free. One knows that the $N_n$ are commutative or 2--step nilpotent. In many cases where the derived algebras $[\gn_n,\gn_n]$ are of bounded dimension we construct $G_n$--equivariant isometric maps $ζ_n : L^2(G_n/K_n) \to L^2(G_{n+1}/K_{n+1})$ and prove that the left regular representation of $G$ on the Hilbert space $L^2(G/K) := \varinjlim \{L^2(G_n/K_n),ζ_n\}$ is a multiplicity free direct integral of irreducible unitary representations. The direct integral and its irreducible constituents are described explicitly. One constituent of our argument is an extension of the classical Peter--Weyl Theorem to parabolic direct limits of compact groups.
dc.description31 pages
dc.identifierhttps://arxiv.org/abs/0801.3866
dc.identifierhttp://arxiv.org/abs/0801.3866
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/146609
dc.subjectRepresentation Theory
dc.subjectDifferential Geometry
dc.subject22E45; 22E65; 22E25; 53C35
dc.titleInfinite Dimensional Multiplicity Free Spaces II: Limits of Commutative Nilmanifolds
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