Abstract. In this paper we address the problem of describing in explicit algebraic terms the collective structure of the coadjoint orbits of a connected, simply connected exponential solvable Lie group G. We construct a partition p of the dual g* of the Lie algebra 9 of G into finitely many Ad*(G)-invariant algebraic sets with the following properties. For each ii e p, there is a subset X of ii which is a cross-section for the coadjoint orbits in ii and such that the natural mapping ii/Ad*(G) -► X is bicontinuous. Each Z is the image of an analytic Ad*(G)-invariant function P on ii and is an algebraic subset of 0* . The partition p has a total ordering such that for each ii € p, U{£2' : ii' < ii} is Zariski open. For each ii there is a cone W c fl* , such that ii is naturally a fiber bundle over £ with fiber W and projection P. There is a covering of I by finitely many Zariski open subsets O such that in each O , there is an explicit local trivialization 0: P-1 (O) -> W x O . Finally, we show that if ii is the minimal element of p (containing the generic orbits), then its cross-section X is a differentiable submanifold of a* . It follows that there is a dense open subset U of G~ such that U has the structure of a differentiable manifold and G~ ~ U has Plancherel measure zero.
Let N be the Heisenberg group. We consider left-invariant multiplicity free subspaces of L 2 (N ). We prove a necessary and sufficient density condition in order that such subsspaces possess the interpolation property with respect to a class of discrete subsets of N that includes the integer lattice. We exhibit a concrete example of a subspace that has interpolation for the integer lattice, and we also prove a necessary and sufficient condition for shift invariant subspaces to possess a singly-generated orthonormal basis of translates. (2000): 42C15, 92A20, 43A80.
Mathematics Subject Classification
Abstract. Given a semidirect product G = N ⋊ H where N is nilpotent, connected, simply connected and normal in G and where H is a vector group for which ad(h) is completely reducible and R-split, let. In this paper we give an explicit construction of admissible vectors in the case where G is not unimodular and the stabilizers in H of its action on b N are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of L 2 (G) into G-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.
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