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.

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 \(\Omega\).