metaplectic group does not split 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.
 A: Take a look at Proposition 5.8 in Rao, On some explicit formulas in the theory of Weil representation, Pacific Journal of Mathematics 157 (1993), 335-371. This is actually a paper that dates back to 1978. Basically, it comes down to the Hilbert symbol being non-trivial in the p-adic case. 
A: Have a look at
Lion & Vergne, The Weil representation, Maslov index, and Theta series.
I haven't got it with me but I think I remember that it contains what you want.
Also there is
Teruji Thomas, The trace of the Weil representation (on arXiv).
This paper is quite explicit but not "as explicit" as Rao (I know what you mean). It's got constructions of the metaplectic groups, and again I think I remember that what you're asking is an easy consequence of these.
Some random comments now. The Maslov index gives you an extension of $Sp_{2n}(k)$ by the Witt group $W(k)$. For $k=\mathbb{R}$ the connected component of 1 in this extension is the simply-connected extension of $Sp_{2n}(\mathbb{R})$ (this fact is in Lion-Vergne), so it's certainly non-trivial. The metaplectic group is a quotient of this. Note that for $n=1$ you get an extension of $SL_2(\mathbb{R})$, and the inverse image of $SL_2(\mathbb{Z})$ is the braid group $B_3$, proving again that the extension is non-trivial (see Kassel & Turaev, Braid groups, appendix, and Milnor's book on algebraic K-theory).
A: Hi All,
The topic is old but in case some of you may be interested in the 
following additional comments, I may recommend to read 
Hans Reiter: Lecture Notes in Mathematics 1382
"Metaplectic Groups and Segal Algebras".
Basically, it is a translation in english of Weil seminal paper (Acta Math 111),
but placed in a slightly generalized context. So it may help if you want 
to understand the way Weil introduced the metaplectic group.
To answer the original post. The projective representation lifts to an ordinary 
representation of the metaplectic group "by construction", because the 
metaplectic group is actually a central extension of the symplectic group 
and the cocycle used in this central extensionm is actually the 2-cocycle of 
the projective representation. But of course is you define the metaplectic group 
through assuming the existence of a 2-fold cover this may not be direct, 
but it is not the way followed by Weil in his paper.
ps: I also recommend Ranga Rao's paper which is still a good reference on this subject...
A: 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 is

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

