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Transactions of the American Mathematical Society

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



Lie group representations and harmonic polynomials of a matrix variable

Author: Tuong Ton-That
Journal: Trans. Amer. Math. Soc. 216 (1976), 1-46
MSC: Primary 22E45
MathSciNet review: 0399366
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Abstract: The first part of this paper deals with problems concerning the symmetric algebra of complex-valued polynomial functions on the complex vector space of n by k matrices. In this context, a generalization of the so-called ``classical separation of variables theorem'' for the symmetric algebra is obtained.

The second part is devoted to the study of certain linear representations, on the above linear space (the symmetric algebra) and its subspaces, of the complex general linear group of order k and of its subgroups, namely, the unitary group, and the real and complex special orthogonal groups. The results of the first part lead to generalizations of several well-known theorems in the theory of group representations.

The above representation, of the real special orthogonal group, which arises from the right action of this group on the underlying vector space (of the symmetric algebra) of matrices, possesses interesting properties when restricted to the Stiefel manifold. The latter is defined as the orbit (under the action of the real special orthogonal group) of the n by k matrix formed by the first n row vectors of the canonical basis of the k-dimensional real Euclidean space. Thus the last part of this paper is involved with questions in harmonic analysis on this Stiefel manifold. In particular, an interesting orthogonal decomposition of the complex Hilbert space consisting of all square-integrable functions on the Stiefel manifold is also obtained.

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Keywords: Lie group representations, harmonic polynomial functions of a matrix variable, irreducible holomorphic representations of the special orthogonal groups, Borel-Weil-Bott theory, symmetric algebras of polynomial functions, rings of differential operators, G-harmonic polynomials, "generalized separation of variables'' theorem, theory of polynomial invariants, generalized spherical harmonics, Stiefel manifolds, Fourier transforms, generalized Hankel transforms
Article copyright: © Copyright 1976 American Mathematical Society