Monatshefte für Mathematik

, Volume 160, Issue 3, pp 271–312

Flat orbits, minimal ideals and spectral synthesis

Article

DOI: 10.1007/s00605-009-0122-2

Cite this article as:
Ludwig, J. & Molitor-Braun, C. Monatsh Math (2010) 160: 271. doi:10.1007/s00605-009-0122-2

Abstract

Let G = exp \({\mathfrak{g}}\) be a connected, simply connected, nilpotent Lie group and let ω be a continuous symmetric weight on G with polynomial growth. In the weighted group algebra \({L^{1}_{\omega}(G)}\) we determine the minimal ideal of given hull \({\{\pi_{l'} \in \hat{G} | l' \in l + \mathfrak{n}^{\perp}\}}\), where \({\mathfrak{n}}\) is an ideal contained in \({\mathfrak{g}(l)}\), and we characterize all the L(G/N)-invariant ideals (where \({N = {\rm exp}\, \mathfrak{n}}\)) of the same hull. They are parameterized by a set of G-invariant, translation invariant spaces of complex polynomials on N dominated by ω and are realized as kernels of specially built induced representations. The result is particularly simple if the co-adjoint orbit of l is flat.

Keywords

Nilpotent Lie group Irreducible representation Co-adjoint orbit Flat orbit Minimal ideal Spectral synthesis Weighted group algebra 

Mathematics Subject Classification (2000)

22E30 22E27 43A20 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité Paul Verlaine-MetzMetz Cedex 1France
  2. 2.Unité de Recherche en MathématiquesUniversité du LuxembourgLuxembourgLuxembourg

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