Flat orbits, minimal ideals and spectral synthesis

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.