Following this paper by Bringmann, Kane, Kohnen, the kernel of the Shimura and Shintani lifts is the function $$\Omega(\tau,z) = \frac{1}{\binom{2k-2}{k-1}\pi} \sum_{D=1}^{\infty} \sum_{b^2 - 4ac = D} D^{k-1/2} (a\tau^2 + b\tau + c)^{-k}e^{2\pi i Dz}.$$ The Shimura and Shintani correspondence between cusp forms of weight $k+1/2$ and $2k$ is given essentially by integrating against $\Omega$ and comes down to the facts
(1) For fixed $z$, $\Omega(\tau,z)$ is a modular form of weight $2k$ in the variable $\tau$;
(2) For fixed $\tau$, $\Omega(\tau,z)$ is a modular form of weight $k+1/2$ in the variable $z$.
Showing that $\Omega(\tau,z)$ is modular with respect to $\tau$ for fixed $z$ is straightforward. Fact (2) seems less obvious and the only proof I have seen involves calculating the Fourier coefficients of $\Omega(\tau,z)$ (which turn out to involve Bessel functions and Kloosterman sums), recognizing that these appear in the Fourier expansions of Poincaré series, and concluding that $\Omega(\tau,z)$ is an infinite linear combination of Poincaré series and therefore a modular form in the variable $z$.
This proof is not particularly insightful so I would like to know if any better argument is known for why $\Omega(\tau,z)$ transforms nicely under $z \mapsto -1/z.$
