Journal of Lie Theory Volume 7 (1997) 29–60 ◯^{C}1997 Heldermann Verlag

The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces

Roberto Camporesi

Communicated by G. Mauceri

Abstract. Let G/K be a Riemannian symmetric space of the noncompact type. For τ ∈ K̂, let E^{τ} be the homogeneous vector bundle over G/K associated with τ, and let C_{0}^{∞}(G, τ, τ) be the related convolution algebra of radial systems of sections of E^{τ}. Assuming C_{0}^{∞}(G, τ, τ) commutative, we use the theory of spherical functions of type τ on G to define a spherical transform for F ∈ C_{0}^{∞}(G, τ, τ). The corresponding inversion formula is obtained by using the Plancherel formula on G. The example of real hyperbolic spaces H^{N}(ℝ) is discussed. The Plancherel measure is written down explicitly in this case, and the vector bundles of Dirac spinors, symmetric traceless tensors, and p-forms on H^{N}(ℝ) are considered in detail.

Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy Camporesi@polito.it

Received April 18, 1996

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag