On Poincaré series of half-integral weight

Abstract

We use Poincaré 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 $m\in \frac{5}{2}+{\mathbb{Z}}_{\ge 0}$. 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 Poincaré 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 $k\in {\mathbb{Z}}_{\ge 0}$. 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 Poincaré series.

21 pages