Construction of Poincar\'e-type series by generating kernels

Let $\Gamma\subset \textrm{PSL}_2({\mathbb R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\widetilde\Gamma$ be its cover in $\textrm{SL}_2({\mathbb R})$. Consider the space of twice continuously differentiable, square-integrable functions on the hyperbolic upper half-plane, which transform in a suitable way with respect to a multiplier system of weight $k\in{\mathbb R}$ under the action of $\widetilde\Gamma$. The space of such functions admits the action of the hyperbolic Laplacian $\Delta_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlovi\'c (where $k=0$), we use the spectral expansion associated to $\Delta_k$ to construct a wave distribution and then identify the conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincar\'e-type series. An advantage of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of $\Delta_k$.