Dual reductive pair

introduciton

• In the mid-1970s, Howe introduced the notion of dual pairs in \(Mp(W)\): these are subgroups of \(Mp(W)\) of the form \(G \times H\) where \(G\) and \(H\) are mutual centralisers of each other.
• He gave a classification and construction of all such possible dual pairs. They basically take the following form:
• (i) if \(U\) is a quadratic space with corresponding orthogonal group \(O(U)\) and \(V\) a symplectic space with corresponding metaplectic group \(Mp(V)\), then \(W = U \otimes V\) is naturally a symplectic space, and \(O(U)\times Mp(V)\) is a dual pair in \(Mp(W) = Mp(U \otimes V)\).
• (ii) \(U(V)\times U(V')\), where \(V\) and \(V'\) are Hermitian and skew-Hermitian spaces respectively for a quadratic extension \(E/F\).
• (iii) \(GL(U) \times GL(V)\), where \(U\) and \(V\) are vector spaces over \(F\).
• The dual pairs in (i) and (ii) are called Type I dual pairs, while those in (iii) are called Type II.

Type II dual pairs

• It is particularly easy to describe the Weil representation \(\Omega\) for Type II dual pairs.
• The group \(GL(U) \times GL(V)\) acts naturally on \(U \otimes V\) and hence on the space \(S(U \otimes V)\) of Schwarz functions: this is the Weil representation \(\Omega\).

related items

• Howe duality
• Schur-Weyl duality for general linear groups

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• ID : Q7306388