Unitary representations of groups of automorphisms of riemann symmetric spaces of null curvature

Abstract

Kalgebra ~ , and G is the corresponding compact form. We shall not discuss this here. In the present article we shall study unitary representations of the group ~ : we shall calculate their character (section 3 below), the Plancherel measure on the group (section 4) and on a symmetric space (section 5), we shall study zonal harmonics for r epresentations of class 1 (section 6), and we shall consider a problem of integral geometry in Euclidean space which arises from representations of the group ~ (section 7). The formulas for the representations of the groups 5 , G, G K have much in common. The questions listed above for the groups G K are studied in the classical works of G. Weyl; for the groups G many of them have not yet been completely answered(Planche rel's formula integral geometry on symmetric spaces of negative curvature).