We construct and study various dual pairs between finite dimensional classical Lie groups and infinite dimensional Lie algebras in some Fock representations. The infinite dimensional Lie algebras here can be either a completed infinite rank affine Lie algebra, the $\winf$ algebra or its certain Lie subalgebras. We give a formulation in the framework of vertex algebras. We also formulate several conjectures and open problems.
Categories: math.QA hep-th math.RT