Let $G$ be a connected semisimple Lie group with finite center, and let $\pi$ be an integrable discrete series representation of $G$. Miličić showed that the Poincaré series of $K$-finite matrix coefficients of $\pi$, constructed with respect to a discrete subgroup $\Gamma$ of $G$, span a dense subspace of the $\pi$-isotypic component of $L^2(\Gamma\backslash G)$. Extending the analytical framework introduced by Muić, we study the inner product and non-vanishing properties of these Poincaré series, along with applications to various families of modular and automorphic forms.
This work was supported by the Croatian Science Foundation under the project number HRZZ-IP-2022-10-4615.