Representations of 𝑆𝑂(𝑘,𝐶) on harmonic polynomials on a null cone

Debarre, Olivier; Ton-That, Tuong

doi:10.1090/S0002-9939-1991-1033957-3

Representations of $\textrm {SO}(k,\textbf {C})$ on harmonic polynomials on a null cone
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by Olivier Debarre and Tuong Ton-That PDF

Proc. Amer. Math. Soc. 112 (1991), 31-44 Request permission

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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The theory of linear representations of complex and real Lie groups