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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Representations of $ {\rm SO}(k,{\bf C})$ on harmonic polynomials on a null cone

Authors: Olivier Debarre and Tuong Ton-That
Journal: Proc. Amer. Math. Soc. 112 (1991), 31-44
MSC: Primary 22E45
MathSciNet review: 1033957
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Abstract: The linear action of the group $ SO(k,{\mathbf{C}})$ on the vector space $ {{\mathbf{C}}^{n \times k}}$ extends to an action on the algebra of polynomials on $ {{\mathbf{C}}^{n \times k}}$. The polynomials that are fixed under this action are called $ SO(k,{\mathbf{C}})$-invariant. The $ SO(k,{\mathbf{C}})$-harmonic polynomials are common solutions of the $ SO(k,{\mathbf{C}})$-invariant differential operators. The ideal of all $ SO(k,{\mathbf{C}})$-invariants without constant terms, the null cone of this ideal, and the orbits of $ SO(k,{\mathbf{C}})$ on this null cone are studied in great detail. All irreducible holomorphic representations of $ SO(k,{\mathbf{C}})$ are concretely realized on the space of $ SO(k,{\mathbf{C}})$-harmonic polynomials.

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Keywords: $ SO(k,{\mathbf{C}})$-invariant polynomials, $ SO(k,{\mathbf{C}})$-harmonic polynomials, $ SO(k,{\mathbf{C}})$-invariant ideal, null cone, $ SO(k,{\mathbf{C}})$-orbits, holomorphic representations of $ SO(k,{\mathbf{C}})$
Article copyright: © Copyright 1991 American Mathematical Society