I'm trying to understand the Weil representation and hope there are some experts around who can set me straight. Let $F$ be a non-Archimedean local field (I don't mind assuming that the characteristic of $F$ is not $2$ if it simplifies things; also the situation over $\mathbb{R}$ is similar but obviously I need a proof which does not rely on any covering space theory.) and fix a nontrivial continuous additive character $\psi : F \to \mathbb{C}^{\times}$. Then by the Stone-von Neumann theorem there is a unique up to isomorphism irreducible smooth representation of the Heisenberg group with central character $\psi$, and one sees the existence of a projective representation of the symplectic group in this space by the uniqueness part of that theorem.


So my question is: why does this not lift to an ordinary representation? All the articles I have read refer this claim to Weil's original paper, which is quite long, and my French is not so good. I think one can see this from the fact that the metaplectic group (I'm talking about the two-sheeted cover) is not a trivial extension of the symplectic group by using the fact that the latter group is perfect. So if one constructs the metaplectic group via Maslov cocycles, one needs to show a certain cocycle is not a coboundary. But how?


Thanks for the help. I hope my question is clear enough.



metaplectic group does not split
I would like the credit to go to Peter Woit for suggesting Section I.6 of Stephen Kudla's "Notes on the Local Theta Correspondence," which contains a nice proof, but he posted this only as a comment and the bounty ends today.


The text of [Woit's comment](https://mathoverflow.net/questions/64326/metaplectic-group-does-not-split#comment261326_64326) is



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> There's a relatively straight-forward argument in Section I.6 of Stephen Kudla's "Notes on the Local Theta Correspondence", available at www.math.toronto.edu/~skudla/castle.pdf This may just be a restatement of the Rao argument. As mentioned elsewhere, it comes down to invoking non-triviality of the Hilbert symbol
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