A duality for symmetric spaces with applications to group representations. III. Tangent space analysis

Abstract

1. IN~ODUCTI~N Let X = G/K be a symmetric space of the noncompact type, G being a connected semisimple Lie group with finite center and K a maximal compact subgroup. Let X0 be the tangent space to X at the fixed pont o of K and let G, denote the group of af%ne transformations of X,, generated by the translations and the natural action of K on X,, . The group GO which is the semi-direct product K x S X,, is often called the Car-tan motion group. While the irreducible unitary representations of GO are described by Mackey’s general semi-direct product theorem [13, p. 1311, and their characters given by Gindikm [4], our purpose here is to develop GO-invariant analysis on the space X0 . More specifically, let D(G,,/K) denote the algebra of G,,-invariant differential operators on Go/K = X,,. We are interested in the joint eigenfunctions of D(G,,/K) and in the representations of GO on the joint eigenspaces, ([8(d)] Ch. IV). As tools for this study we prove Paley-Wiener type theorems for the spherical transform and for the generalized Bessel transform on X0. The first of these comes from Theorem 2.1 which is an analog for entire functions of exponential type of Chevalley’s restriction theorem for polynomials. An important property of the generalized Bessel transform comes from Theorem 4.6 which shows that the generalized Bessel function is divisible by the Jg-matrix. Another useful tool is Theorem 4.8 which represents the generalized Bessel function as a certain derivative of the zonal spherical function. These results are used in Section 6 to derive an integral formula for the K-finite joint eigenfunctions of D(G,/K). This in turn is used to give (in Theorem 6.6) an explicit irreducibility criterion for the associated eigenspace representation of G,, . This is the principal application of our results. The results for the flat space X, = GO/K are to a large extent analogous to those proved in [8(g)] for the curved space X = G/K, most of the proofs are