On Poincar\'e series of half-integral weight

We use Poincar\'e series of $K$-finite matrix coefficients of genuine integrable representations of the metaplectic cover of $\mathrm{SL}_2(\mathbb R)$ to construct a spanning set for the space of cusp forms $S_m(\Gamma,\chi)$, where $\Gamma$ is a discrete subgroup of finite covolume in the metaplectic cover of $\mathrm{SL}_2(\mathbb R)$, $\chi$ is a character of $\Gamma$ of finite order, and $m\in\frac52+\mathbb Z_{\geq0}$. We give a result on the non-vanishing of the constructed cusp forms and compute their Petersson inner product with any $f\in S_m(\Gamma,\chi)$. Using this last result, we construct a Poincar\'e series $\Delta_{\Gamma,k,m,\xi,\chi}\in S_m(\Gamma,\chi)$ that corresponds, in the sense of the Riesz representation theorem, to the linear functional $f\mapsto f^{(k)}(\xi)$ on $S_m(\Gamma,\chi)$, where $\xi\in\mathbb C_{\Im(z)>0}$ and $k\in\mathbb Z_{\geq0}$. Under some additional conditions on $\Gamma$ and $\chi$, we provide the Fourier expansion of cusp forms $\Delta_{\Gamma,k,m,\xi,\chi}$ and their expansion in a series of classical Poincar\'e series.