A theory of Stiefel harmonics

Gelbart, Stephen S.

doi:10.1090/S0002-9947-1974-0425519-8

A theory of Stiefel harmonics
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by Stephen S. Gelbart

Trans. Amer. Math. Soc. 192 (1974), 29-50

DOI: https://doi.org/10.1090/S0002-9947-1974-0425519-8

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Abstract:

An explicit theory of special functions is developed for the homogeneous space $SO(n)/SO(n - m)$ generalizing the classical theory of spherical harmonics. This theory is applied to describe the decomposition of the Fourier operator on $n \times m$ 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 $SO(n)$ and the polynomial representations of $GL(m,{\mathbf {C}})$.

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