Let $E/F$ be a quadratic extension of number field and let $V$ be a Hermitian space over $E$. Then we have Weil representations for the dual pair $U(n,n)\times U(V)$, and we can consider the theta lift from a cuspidal automorphic representation on $U(V)$ to $U(n,n)$.
Locally, we have similar construction. At a split place $v$ of $F$, $E_v$ is just $F_v\oplus F_v$, the corresponding local dual pairs are just $GL(2n)\times GL(m)$.
In Kudla-Sweet's paper, "degenerate principal series on $U(n,n)$", they considered the local dual pair $U(n,n)\times U(V)$, and also $GL(2n)\times GL(m)$.
My question is: do we have a global theory of Weil representations, theta series for the dual pair $GL(2n)\times GL(m)$ so that its local counterpart is the same as the local counterpart of $U(n,n)\times U(V)$ at split places? If it exists, where can I find this? If not, why?
