Wavelet transforms associated to a principal series representation of semisimple Lie groups. I
Let \(G\) be a locally compact Lie group and \(\pi\) a continuous representation of \(G\) on a Hilbert space \({\mathcal H}\). Let \({\mathcal H}_\infty\) denote the space of \(C^\infty\)-vectors in \({\mathcal H}\), endowed with a natural Sobolev-type topology, and \({\mathcal H}_{- \infty}\) the dual of \({\mathcal H}_\infty\) endowed with the strong topology. We denote the corresponding representation on \({\mathcal H}_{-\infty}\) by the same letter \(\pi\). Let \(S\) be a subset of \(G\) and \(ds\) a measure on \(S\). A vector \(\psi\in {\mathcal H}_{-\infty}\) is said to be \(S\)-strongly admissible for \(\pi\) if, as a functional on \({\mathcal H}_\infty\), \[ f=c^{-1}_{S,\psi} \int_S \bigl\langle f,\pi(s)\psi \bigr\rangle_{\mathcal H} \pi(s) \psi ds \quad \text{for all } f\in{\mathcal H}_\infty. \] We call \(\langle f, \pi(s)\psi \rangle\) the wavelet transform of \(f\) associated to \((G,\pi,S, \psi)\) in the sense that, by specializing \((G,\pi,S, \psi)\), the above formula yields a group theoretical interpretation of various well-known wavelet transforms. Let \(G\) be a noncompact connected semisimple Lie group with finite center and \(P=MAN\) a parabolic subgroup of \(G\). Let \(\pi_\lambda = \text{Ind}^G_P(1 \otimes e^\lambda \otimes 1)\) \((\lambda \in a^*_C)\) denote a principal series representation of \(G\) and \((\pi_\lambda, L^2 (\overline N, e^{-2{\mathfrak I} \lambda(H (\overline n))}d \overline n)\) \((\overline N = \theta(N))\) the noncompact picture of \(\pi_\lambda\). Let \(\sigma_\omega\) denote an irreducible unitary representation of \(\overline N\) corresponding to \(\omega\in\overline n^*_C\) and \((S,ds)\) a subset of \(MA\) with measure \(ds\). We suppose that there exists a \(\psi\in {\mathcal S}' (\overline N)\) satisfying the following admissible condition: for all \(\omega \in V_T'\) \[ \text{(i)} \quad \sigma_\omega (\psi) \sigma_\omega (\psi)^* = n_\psi(\omega)I, \qquad \text{(ii)} \quad 0<\int_S n_\psi \bigl(Ad(s) \omega\bigr) ds=c_{S,\psi} < \infty, \] where \(c_{S,\psi}\) is independent of \(\omega\). Then for all such \(\psi\) we can deduce the inversion formula \[ f(x) = (c^S_\psi)^{-1} \int\int_{\overline N \times S} <f,\pi_{-i\rho} (\overline ns) \psi>\pi_{-i\rho} (\overline ns) \psi(x)d \overline nds \quad \text{for all } f\in{\mathcal S} (\overline N), \] where \(\langle \cdot,\cdot\rangle\) is the inner product of \(L^2 (\overline N)\). This is a generalization of the Calderón identity and the Grossmann-Morlet transform. A transform associated to the analytic continuation of the holomorphic discrete series and its limit will be treated in a forthcoming paper.
scientific article; zbMATH DE number 897675