The Journal of Geometric Analysis

, Volume 2, Issue 3, pp 213–248

Harmonic analysis on solvable extensions of H-type groups

  • Ewa Damek
  • Fulvio Ricci
Article

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.

Math Subject Classification

43A20 43A90 53C25 

Key Words and Phrases

Harmonic manifolds Heisenberg type groups Martin boundary solvable Lie groups spherical functions 

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Copyright information

© CRC Press, Inc 1992

Authors and Affiliations

  • Ewa Damek
    • 1
  • Fulvio Ricci
    • 2
  1. 1.Instytut Matematyczny Universytetu WroclawskiegoWroclawPoland
  2. 2.Dipartimento di MatematicaPolitecnico di TorinoTorinoItaly

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