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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A theory of Stiefel harmonics

Author: Stephen S. Gelbart
Journal: Trans. Amer. Math. Soc. 192 (1974), 29-50
MSC: Primary 43A85; Secondary 22E45, 33A75
MathSciNet review: 0425519
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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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Keywords: Generalized spherical harmonics, Stiefel manifold, representations of $ SO(n)$, holomorphic representations of $ GL(m,{\mathbf{C}})$, Fourier transforms on matrix space, generalized Hankel transforms, Bessel functions of matrix argument
Article copyright: © Copyright 1974 American Mathematical Society

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