Reductive Dual Pair - Examples

Examples

• The full symplectic group G = Sp(W) and the two-element group G′, the center of Sp(W), form a reductive dual pair. The double centralizer property is clear from the way these groups were defined: the centralizer of the group G in G is its center, and the centralizer of the center of any group is the group itself. The group G′, consists of the identity transformation and its negative, and can be interpreted as the orthogonal group of a one-dimensional vector space. It emerges from the subsequent development of the theory that this pair is a first instance of a general family of dual pairs consisting of a symplectic group and an orthogonal group, which are known as type I irreducible reductive dual pairs.

• Let X be an n-dimensional vector space, Y be its dual, and W be the direct sum of X and Y. Then W can be made into a symplectic vector space in a natural way, so that (X, Y) is its lagrangian polarization. The group G is the general linear group GL(X), which acts tautologically on X and contragrediently on Y. The centralizer of G in the symplectic group is the group G′, consisting of linear operators on W that act on X by multiplication by a non-zero scalar λ and on Y by scalar multiplication by its inverse λ−1. Then the centralizer of G′, is G, these two groups act completely reducibly on W, and hence form a reductive dual pair. The group G′, can be interpreted as the general linear group of a one-dimensional vector space. This pair is a member of a family of dual pairs consisting of general linear groups known as type II irreducible reductive dual pairs.