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Invariant theory of the dual pairs $ (\mathrm{SO}^*(2n), \mathrm{Sp}(2k, \mathbf{C}))$ and $ (\mathrm{Sp}(2n, \mathfrak{R}), \mathrm{O}(N))$

Authors: Eric Y. Leung and Tuong Ton-That
Journal: Proc. Amer. Math. Soc. 120 (1994), 53-65
MSC: Primary 22E46; Secondary 17B99, 22E60
MathSciNet review: 1165060
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Abstract: Let $ G \equiv {\text{Sp}}(2k,{\mathbf{C}})$ or $ {\text{O}}(N)$ and $ G' \equiv {\text{S}}{{\text{O}}^{\ast}}(2n)$ or $ {\text{Sp}}(2n,\Re )$. The adjoint representation of $ G'$ on its Lie algebra $ \mathcal{G}'$ gives rise to the coadjoint representation of $ G'$ on the symmetric algebra of all polynomial functions on $ \mathcal{G}'$. The polynomials that are fixed by the restriction of the coadjoint representation to a block diagonal subgroup $ K'$ of $ G'$ form a subalgebra called the algebra of $ K'$-invariants. Using the theory of invariants of Procesi for the "dual pair" $ (G',G)$, a finite set of generators of this algebra is explicitly determined.

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Keywords: Invariant polynomials, Casimir invariants, dual groups
Article copyright: © Copyright 1994 American Mathematical Society