Dual pairs and infinite dimensional Lie algebras

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.