, Volume 160, Issue 3, pp 271-312
Date: 03 Jun 2009

Flat orbits, minimal ideals and spectral synthesis

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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.

Communicated by K.H. Gröchenig.
Jean Ludwig and Carine Molitor-Braun are supported by the research grants R1F104C09 and AHCM07 of the University of Luxembourg.
Some of the results on minimal ideals have been proven by D. Alexander in his Ph.D. thesis [1]. These results will be mentioned as such in the text.