, Volume 2, Issue 3, pp 213-248

Harmonic analysis on solvable extensions of H-type groups

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Abstract

To each groupN of Heisenberg type one can associate a generalized Siegel domain, which for specialN is a symmetric space. This domain can be viewed as a solvable extensionS =NA ofN endowed with a natural left-invariant Riemannian metric. We prove that the functions onS that depend only on the distance from the identity form a commutative convolution algebra. This makesS an example of a harmonic manifold, not necessarily symmetric. In order to study this convolution algebra, we introduce the notion of “averaging projector” and of the corresponding spherical functions in a more general context. We finally determine the spherical functions for the groupsS and their Martin boundary.

This work has been partially supported by the Italian Consiglio Nazionale delle Ricerche. We profited from conversation with various colleagues, including P. Biler, J. Faraut, W. Hebisch, A. Korányi, A. Lunardi, A. Hulanicki, M. Pavone, and T. Pytlik.
Communicated by Guido Weiss