Fourier transforms on a semisimple symmetric space

Abstract

Let G=H be a semisimple symmetric space, that is, G is a connected semisimple real Lie group with an involution ?, and H is an open subgroup of the group of xed points for ? in G. The main purpose of this paper is to study an explicit Fourier transform on G=H. In terms of general representation theory the (`abstract') Fourier transform of a compactly supported smooth function f 2 C 1 c (G=H) is given by (see [6]) (1) ^ f(?)() =?(f) = ZG=H f(x)?(x) dx; for (?; H?) a unitary irreducible representation of G and 2 (H ? an H-invariant distribution vector for ?. Here dx is the invariant measure on G=H. Thus ^ f f?)() is a smooth vector for H?, depending linearly on . Our goal is to obtain an explicit version of the restriction of this Fourier transform to representations (?; H?) in the (minimal) unitary principal series (??;; H?;) for G=H, under the assumption that the center of G is nite. In the sequel [10] to this paper it is proved that a function f 2 C 1 c (G=H) is uniquely determined by the restriction of ^ f to this series (a priori it is known that f is determined by ^ f )