INVARIANT-THEORY OF THE DUAL PAIRS (SO-ASTERISK(2N), SP(2K,C)) AND (SP(2N,R), O(N))

Let G = Sp(2k, C) or O(N) and G' = SO(2n) or Sp(2n, R). The adjoint r epresentation of G' on its Lie algebra G' gives rise to the coadjoint representation of G' on the symmetric algebra of ah polynomial functio ns on G'. The polynomials that are fixed by the restriction of the coa djoint representation to a block diagonal subgroup K' of G' form a sub algebra called the algebra of K'-invariants. Using the theory of invar iants of Procesi for the ''dual pair'' (G', G), a finite set of genera tors of this algebra is explicitly determined.