  We construct and study in detail various dual pairs acting on some Fock
representations between a finite dimensional Lie group and a completed infinite
rank affine algebra associated to an infinite affine Cartan matrix. We give
explicit decompositions of a Fock representation into a direct sum of
irreducible isotypic subspaces with respect to the action of a dual pair,
present explicit formulas for the common highest weight vectors and calculate
the corresponding highest weights. We further outline applications of these
dual pairs to the study of tensor products of modules of such an infinite
dimensional Lie algebra.
