metaplectic group does not split

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.

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).

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...