I am aware that the correct way to look at modular forms of half-integral weight is on the metaplectic cover of $\mathrm{SL}_2(\mathbb R)$. Assume however that we insist on considering them as functions from the upper half-plane to $\mathbb C$. I have two questions concerning the slash operator. Let $k$ be a half-integer and $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ a matrix in $\mathrm{SL}_2(\mathbb Z)$.
1) Assume that we naively define $$f|_k\gamma(\tau)=(c\tau+d)^{-k}f(\gamma(\tau))\;,$$ where as usual $(c\tau+d)^{1/2}$ is the principal determination (argument in $]-\pi/2,\pi/2]$), and $(c\tau+d)^{-k}=((c\tau+d)^{1/2})^{-2k}$. Then $f|_k\gamma|_k\gamma'=\epsilon f|_k\gamma\gamma'$ for a suitable sign $\varepsilon$. I managed to compute the cocycle $\varepsilon$ explicitly, but is this done somewhere in the literature ?
2) Is it possible (still not using the metaplectic cover) to modify the definition of the slash operator so that we always have $\varepsilon=1$, for instance perhaps by setting $$f|_k\gamma(\tau)=v(\gamma)(c\tau+d)^{-k}f(\gamma(\tau))$$ for a suitable $v(\gamma)$? (In other words, is $\varepsilon$ a coboundary ?)
Note that I require a complete action by $\mathrm{SL}_2(\mathbb Z)$ (or even $\mathrm{GL}_2^+(\mathbb Q)$) and not only by $\Gamma_0(4)$, in which case the answer to this question is trivially yes.
