Radial pairs (Various Issues relating to Representation Theory and Non-commutative Harmonic Analysis)

Abstract

Let G=KAN be a real reductive Lie group and H a graded Hecke algebra defined by the restricted root system of G. The representation theory of G and that of H resemble very much. For example, the H-counterpart of the Helgason-Fourier transform for C^{∞}(G/K) is the Opdam-Cherednik transform for C^{∞}(A). In this report I explain the notion of radial pairs which I introduced in [O2] for the sake of precise description of the resemblance mentioned above. We also discuss some functors between the category of G- and H-modules.