Reductive dual pair

In the mathematical field of representation theory, a reductive dual pair is a pair of subgroups (G, G′) of the isometry group Sp(W) of a symplectic vector space W, such that G is the centralizer of G′ in Sp(W) and vice versa, and these groups act reductively on W. Somewhat more loosely, one speaks of a dual pair whenever two groups are the mutual centralizers in a larger group, which is frequently a general linear group. The concept was introduced by Roger Howe in an influential preprint of the 1970s, which was ultimately published as Howe (1989a).

The notion of a reductive dual pair makes sense over any field F, which we assume to be fixed throughout. Thus W is a symplectic vector space over F.

If W₁ and W₂ are two symplectic vector spaces and (G₁, G′₁), (G₂, G′₂) are two reductive dual pairs in the corresponding symplectic groups, then we may form a new symplectic vector space W = W₁ ⊕ W₂ and a pair of groups G = G₁ × G₂, G′ = G′₁ × G′,₂ acting on W by isometries. It turns out that (G, G′) is a reductive dual pair. A reductive dual pair is called reducible if it can be obtained in this fashion from smaller groups, and irreducible otherwise. A reducible pair can be decomposed into a direct product of irreducible ones, and for many purposes, it is enough to restrict one's attention to the irreducible case.

Several classes of reductive dual pairs had appeared earlier in the work of André Weil. Roger Howe proved a classification theorem, which states that in the irreducible case, those pairs exhaust all possibilities. An irreducible reductive dual pair (G, G′) in Sp(W) is said to be of type II if there is a lagrangian subspace X in W that is invariant under both G and G′, and of type I otherwise.

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