Bradley Currey

Construction of canonical coordinates for exponential Lie groups

Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Holder basis. Thus we obtain a stratification of g * into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ C Ω for coadjoint orbits in Ω, so that each pair (Ω, Σ) behaves predictably under the associated restriction maps on g * . The cross-section mapping σ: Ω → Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ∈ Ω. For each Ω, algebras e 0 (Ω) and e 1 (Ω) of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let 2d > 0 be the dimension of coadjoint orbits in Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q 1 , q 2 , ..., q d } and {p 1 ,p 2 ,...,p d } such that on each coadjoint orbit O in Ω, the canonical 2-form is given by Σdp k ^ dq k . The functions {q 1 ,q 2 , ..., q d } belong to e 0 (Ω), and the functions {p 1 ,p 2 , ... ,p d } belong to e 1 (Ω). The associated geometric polarization on each orbit O coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p 1 , ... , p d , q 1 , ... , q d ) (restricted to O). Finally, the linear evaluation functions l ↦ l(X) are shown to be quantizable as well.

Admissibility for a Class of Quasiregular Representations

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 τ denote the quasiregular representation of G in L2(N). An elementψ ∈ L2(N) is said to be admissible if the wavelet transform f 7→ 〈 f , τ (·)ψ〉 defines an isometry from L2(N) into L2(G). 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 L2(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. Received by the editors November 3, 2004; revised March 16, 2005. AMS subject classification: 22E27, 22E30. c ©Canadian Mathematical Society 2007. 917

A density condition for interpolation on the Heisenberg group

(N). We prove a necessary and sufficient density conditionin order that such subsspaces possess the interpolation property with respect toa class of discrete subsets of N that includes the integer lattice. We exhibit aconcrete example of a subspace that has interpolation for the integer lattice, andwe also prove a necessary and sufficient condition for shift invariant subspaces topossess a singly-generated orthonormal basis of translates.Mathematics Subject Classification (2000): 42C15, 92A20, 43A80.Keywords and phrases: The Heisenberg group, Heisenberg frame, Gabor frame, multiplicity freesubspaces, sampling spaces, the interpolation property

THE ORTHONORMAL DILATION PROPERTY FOR ABSTRACT PARSEVAL WAVELET FRAMES

In this work we introduce a class of discrete groups containing subgroups of abstract translations and dilations, respectively. A variety of wavelet systems can appear as

On the dual of an exponential solvable Lie group

\pi(\G)\psi

Characterization of Shift-Invariant Spaces on a Class of Nilpotent Lie Groups with Applications

, where

EXPLICIT ORBITAL PARAMETERS AND THE PLANCHEREL MEASURE FOR EXPONENTIAL LIE GROUPS

\pi

The structure of the space of coadjoint orbits of an exponential solvable Lie group

is a unitary representation of a wavelet group and

Gabor fields and wavelet sets for the Heisenberg group

\G

Characterization of regularity for a connected Abelian action

is the abstract pseudo-lattice