Invariant theory of the dual pairs (𝑆𝑂*(2𝑛),𝑆𝑝(2𝑘,𝐂)) and (𝐒𝐩(2𝐧,ℜ),𝔒(𝔑))

Abstract

Let G ≡ Sp ( 2 k , C ) G \equiv {\text {Sp}}(2k,{\mathbf {C}}) or O ( N ) {\text {O}}(N) and G ′ ≡ S O ∗ ( 2 n ) G’ \equiv {\text {S}}{{\text {O}}^{\ast }}(2n) or Sp ( 2 n , ℜ ) {\text {Sp}}(2n,\Re ) . The adjoint representation of G ′ G’ on its Lie algebra G ′ \mathcal {G}’ gives rise to the coadjoint representation of G ′ G’ on the symmetric algebra of all polynomial functions on G ′ \mathcal {G}’ . The polynomials that are fixed by the restriction of the coadjoint representation to a block diagonal subgroup K ′ K’ of G ′ G’ form a subalgebra called the algebra of K ′ K’ -invariants. Using the theory of invariants of Procesi for the "dual pair" ( G ′ , G ) (G’,G) , a finite set of generators of this algebra is explicitly determined.