Extended Weil representations: the real field case

Let F be the usual real field. Let W be a symplectic vector space over F. It is known that there are two different Weil representations of a Meteplectic covering group $\widetilde{Sp}(W)$. By some twisted actions, we reorganize them into a representation of $\widetilde{Sp}^{\pm}(W)$, a covering group over a subgroup $Sp^{\pm}(W)$ of $GSp(W)$. Based on the works of MVW, Kudla, and Howe on reductive dual pairs in $Sp(W)$, we explore the analogous dual pairs in $Sp^{\pm}(W)$ . Finally, following Lion-Vergne's classical book on Weil representations and theta series, we investigate some simple theta series in $Sp^{\pm}(W)$ where $W$ has dimension two.