A theory of Stiefel harmonics

Abstract

An explicit theory of special functions is developed for the homogeneous space S O ( n ) / S O ( n − m ) SO(n)/SO(n - m) generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on n × m n \times m matrix space in terms of operator valued Bessel functions of matrix argument. Underlying these results is a hitherto unnoticed relation between certain irreducible representations of S O ( n ) SO(n) and the polynomial representations of G L ( m , C ) GL(m,{\mathbf {C}}) .