Admissibility for a Class of Quasiregular Representations

Currey, Bradley N.

Canadian journal of mathematics, Tome 59 (2007) no. 5, pp. 917-942

Résumé

Given a semidirect product $G\,=\,N\,\rtimes \,H$ where $N$ is nilpotent, connected, simply connected and normal in $G$ and where $H$ is a vector group for which $ad(\mathfrak{h})$ is completely reducible and $\mathbf{R}$ -split, let $\tau $ denote the quasiregular representation of $G$ in ${{L}^{2}}(N)$ . An element $\psi \,\in \,{{L}^{2}}(N)$ is said to be admissible if the wavelet transform $f\,\mapsto \,\left\langle f,\,\tau (\cdot )\psi\right\rangle $ defines an isometry from ${{L}^{2}}(N)$ into ${{L}^{2}}(G)$ . 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 $\hat{N}$ 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.

Détail

DOI : 10.4153/CJM-2007-039-0

Mots-clés : 22E27, 22E30

Bibliographie

[1] [1] Arnal, D.,BenAmmar, M., Currey, B., and Dali, B., Explicit construction of canonical coordinates for completely solvable Lie groups. J. Lie Theory 15(2005), 521–560. Google Scholar

[2] [2] Currey, B., A continuous trace composition sequence for C*(G) where G is an exponential solvable Lie group . Math. Nachr. 159(1992), 189–212. Google Scholar

[3] [3] Currey, B., An explicit Plancherel formula for completely solvable Lie groups. MichiganMath. J. 38(1991), no. 1, 75–87. Google Scholar

[4] [4] Currey, B., Explicit orbital parameters and the Plancherel measure for exponential Lie groups. Pacific J. Math. (to appear). Google Scholar

[5] [5] Currey, B., Smooth decomposition of finite multiplicity monomial representations for a class of completely solvable homogeneous spaces. Pacific J. Math. 170(1995), no. 2, 429–460. Google Scholar

[6] [6] Currey, B., The structure of the space of coadjoint orbits of an exponential solvable Lie group. Trans. Amer. Math. Soc. 332(1992), no. 1, 241–269. Google Scholar

[7] [7] Currey, B. and Penney, R., The structure of the space of co-adjoint orbits of a completely solvable Lie group. Mich. Math. J. 36(1989), no. 2, 309–320. Google Scholar

[8] [8] Duflo, M. and Moore, C., On the regular representation of a nonunimodular locally compact group. J. Functional Analysis 21(1976), no. 2, 209–243. Google Scholar

[9] [9] Duflo, M. and Raïs, M., Sur l’analyse harmonique sur les groupes de Lie résolubles. Ann. Sci. École Norm. Sup. 9(1976), no. 1, 107–144. Google Scholar

[10] [10] Fabec, R., and Ólafsson, G., The continuous wavelet transform and symmetric spaces. Acta Applicandae Math. (to appear) Google Scholar

[11] [11] Führ, H., Admissible vectors for the regular representation. Proc. Amer. Math. Soc. 130(2002), no. 10, 2959–2970.(electronic). Google Scholar

[12] [12] Ishi, H., Wavelet transforms for semidirect product groups. J. Fourier Anal. Appl. 12(2006), no. 1, 37–52. Google Scholar

[13] [13] Laugesen, R. S., Weaver, N., Weiss, G., and Wilson, E. N., A characterization of the higher dimensional groups associated with continuous wavelets. J. Geom. Anal. 12(2002), no. 1, 89–102. Google Scholar

[14] [14] Lipsman, R., Harmonic analysis on exponential solvable homogeneous spaces: the algebraic or symmetric cases. Pacific. J. Math. 140(1989), no. 1, 117–147. Google Scholar

[15] [15] Lipsman, R., Induced representations of completely solvable Lie groups. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17(1990), no. 1, 127–164. Google Scholar

[16] [16] Lipsman, R., The multiplicity function on exponential and completely solvable homogeneous spaces. Geom. Dedicata 39(1991), no. 2, 155–161. Google Scholar

[17] [17] Lipsman, R. and Wolf, J., Canonical semi-invariants and the Plancherel formula for parabolic groups. Trans. Amer. Math. Soc. 269(1982), no. 1, 111–131. Google Scholar

[18] [18] Liu, H. and Peng, L., Admissible wavelets associated with the Heisenberg group. Pacific J. Math. 180(1997), no. 1, 101–123. Google Scholar

[19] [19] Pedersen, N. V., On the infinitesimal kernel of irreducible representations of nilpotent Lie groups. Bull. Soc. Math. France 112(1984), no. 4, 423–467. Google Scholar

[20] [20] Pukánszky, L., On the characters and the Plancherel formula of nilpotent Lie groups. J. Functional Analysis 1(1967), 255–280. Google Scholar

[21] [21] Pukánszky, L., On the unitary representations of exponential Lie groups. J. Functional Analysis 2(1968), 73–113. Google Scholar

[22] [22] Weiss, G. and Wilson, E. N., The mathematial theory of wavelets. In: Twentieth Century Harmonic Analysis – A Celebration, NATO Sci. Ser. II Math. Phys. Chem. 33, Kluwer Acad. Publ., Dordrecht, 2001, pp. 329–366. Google Scholar