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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1$\begingroup$ Well, trivially, by projecting to the symplectic group …? Do you have some other action in mind? If so, why? Do you know that it exists, or just hope? Also, over what field? $\endgroup$– LSpiceMar 23, 2020 at 19:23
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$\begingroup$ @LSpice, Oh, the action uses the projection map from the metaplectic group to symplectic group. Since many people use the semi-direct product of metaplectic group and Heisenberg group, I just wondered people’s convention. Thank you! $\endgroup$– MontyMar 23, 2020 at 20:10
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$\begingroup$ Why close this beautiful question with even more beautiful answer? $\endgroup$– Bugs BunnyMar 24, 2020 at 6:06
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$\begingroup$ @BugsBunny Because there was no effort from OP (even after LSpice's comment) to be more precise, e.g., symplectic group over which field, in what dimension, Heisenberg group in which sense? $\endgroup$– YCorMar 24, 2020 at 11:50
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1 Answer
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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 $Mp_{2n}$.
answered Mar 24, 2020 at 6:05
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$\begingroup$ Thank you very much! I learned much from your comments! $\endgroup$– MontyMar 24, 2020 at 9:12