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.
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