Matrix-valued commuting differential operators and their joint eigenfunctions (Various Issues relating to Representation Theory and Non-commutative Harmonic Analysis)

Abstract

We give an example of vector-valued analogue of the theory of the Heckman-Opdam hypergeometric function associated with a root system. We construct matrix-valued commuting differential operators associated with root system of type A_{2} and their joint eigenfunctions. In group case, the differential operators are radial parts of invariant differential operators on a certain homogeneous vector bundle over a Riemannian symmetric space and the radial part of a matrix coefficient of a principal series representation gives a joint eigenfunction that is analytic at the origin. Allowing the root multiplicity to be an arbitrary complex number, we give matrix-valued commuting differential operators and connection coefficients (c-functions) for their joint eigenfunctions given by power series.