Wavelet transforms associated to a principal series representation of semisimple Lie groups. II
[For Part I of this paper see the preceding review.] 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 \(ds\). In Part I we supposed that there exists a \(\psi\in {\mathcal S}' (\overline N)\) satisfying the following admissible condition: for all \(\omega\in V_T'\) \[ \text{(i)} \quad \sigma_\omega (\psi)\sigma_\omega (\psi)^* = n_\psi(\omega)I, \qquad \text{(ii)} \quad 0< \int_Sn_\psi \bigl(Ad(s)\omega\bigr) ds=c_{S,\psi} < \infty, \] where \(c_{S,\psi}\) is independent of \(\omega\). Then for all such \(\psi\) we can deduce the inversion formula \[ f(x)=(c_\psi^S)^{-1} \int \int_{\overline N \times S} \bigl\langle f,\pi_{-i\rho} (\overline ns) \psi\bigr\rangle \pi_{-i\rho} (\overline ns) \psi(x)d \overline nds \quad \text{for all } f\in{\mathcal S} (\overline N), \] where \(\langle \cdot, \cdot \rangle\) is the inner product of \(L^2 (\overline N)\). A number of well-known examples of wavelet transforms arises from this scheme through the explicit form of \(\psi\). However, in the case of \(G=SL(n+2, \mathbb{R})\) \((n\geq 1)\) and \(\overline N \cong H_n\), the \((2n + 1)\)-dimensional Heisenberg group, the above formula does not cover the three examples constructed by \textit{C. Kalisa} and \textit{B. Torrésani} [Ann. Inst. Henri Poincaré, Phys. Théor. 59, 201-236 (1993)]. Therefore, in order to obtain a widespread application we need to generalize this formula. In this paper we suppose that \(S\) is an arbitrary measurable set with map \(\ell:S \to G\) and then we consider a distribution vector \(\psi\) in \({\mathcal S}' (\overline N)\) which depends on \(s\in S\).
scientific article; zbMATH DE number 897676