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!
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Sign up to join this communityI 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!
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 $Mp_{2n}$.