Spherical functions on harmonic extensions of \(H\)-type groups
In the paper under review, the authors consider the harmonic extension \(AN\) of an \(H\)-type group \(N\) with Lie algebra \(\mu=\nu +\zeta\) with \([\nu,\nu ]=\zeta\), and characterize the positive definite spherical functions on \(AN\). The notion of \(H\)-type group was introduced by \textit{A. Kaplan} [Trans. Am. Math. Soc. 258, 147-153 (1980; Zbl 0416.35022)] as a natural generalization of the Iwasawa \(N\)-group of a rank one simple Lie group \(G\). \textit{J.-P. Anker, E. Damek} and {Ch. Yacoub} [Ann. Scuola Norm. Sup. Pisa, Cl. Sci (4) 23, 643-679 (1996; Zbl 0881.22008)] studied harmonic analysis on a solvable extension \(AN\) of an \(H\)-type group \(N\), where \(A={\mathbb R}^{+}\) acts on \(N\) by nonhomogeneous dilations and \(AN\) is the semi-direct product of \(A\) and \(N\). Analysis on the space \(AN\) has many similarities to that on a symmetric space of rank one. In particular, one can define the spherical functions \(\phi_\xi\) and they play the same role as in the study of a rank one symmetric space. An important problem in the study of rank one symmetric spaces is to determine all the positive spherical functions. First, the authors recall the basic definitions of \(H\)-type groups \(N\) and their harmonic extensions \(AN\), as well as the definition of a spherical function for \(AN\). The induced representations of \(AN\) and the Plancherel decomposition of \(L^2(N)\) under the group \(AN\) are studied and the Fourier transform of the kernel of the intertwining operator is calculated. The authors prove a necessary and sufficient condition for the spherical function \(\phi_\xi\) to be positive definite. This result was conjectured by Anker, Damek and Yacoub [op. cit.].
scientific article; zbMATH DE number 1542094
scientific article