I know that Symplectic group has an action on Heisenberg group.


I am wondering how to extend this to non-trivial two fold metaplectic covering?


Thanks in advance!



Semidirect product of metaplectic group and Heisenberg group
I presume you are looking for a faithful action of $Mp\_{2n}$ on something related to the Heisenberg group $H\_{2n+1}$. This is well-known as Weil Representation. 


In the modern language, consider $Mp\_{2n}$ acting on $H\_{2n+1}$ by automorphisms. This action has a kernel. Now consider the action on the category of unitary representations of $H\_{2n+1}$ by twisting representations by automorphisms. This categorical representation still has the same kernel. Finally, choose a skeleton of the category of unitary representations of $H\_{2n+1}$. The modern interpretation of all this Weil and Stone–von-Neumann business is that $Mp\_{2n}$ acts on the skeleton and this yields a faithful categorical representation of $Mp\_{2n}$.


