Dual reductive pair

introduciton

In the mid-1970s, Howe [8] 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 $W = U \times V$ 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 $\Omega$.