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 I think the situation over real numbers is similar enough for the purposes of this question.) 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.

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.