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

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.