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.

`$\left( \begin{smallmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \right)$`

in $SL_2(\mathbb{R})$. If you look at the action of the corresponding Lie algebra on the metaplectic rep, the weights are in $\mathbb{Z}+1/2$. (I'm using the presentation in Section 2 of Woit's notes math.columbia.edu/~woit/notes21.pdf ). So the preimage in the metaplectic group is the nontrivial double cover of this one and, in particular, the preimages of $\theta =0$ and $\pi$ form a cyclic group of order $4$. $\endgroup$3more comments