A generalized PGL(2) Petersson/Bruggeman-Kuznetsov formula for analytic applications

We develop generalized Petersson/Bruggeman-Kuznetsov (PBK) formulas for specified local components at non-archimedean places. In fact, we introduce two hypotheses on non-archimedean test function pairs $f\leftrightarrow \pi (f)$, called geometric and spectral hypotheses, under which one obtains `nice’ PBK formulas by the adelic relative trace function approach. Then, given a supercuspidal representation $\sigma$ of ${\mathrm{P}\mathrm{G}\mathrm{L}}_2({\mathbb{Q}}_p)$, we study extensively the case that $\pi (f)$ is a projection onto the line of the newform if $\pi$ is isomorphc to $\sigma$ or its unramified quadratic twist, and $\pi (f)=0$ otherwise. As a first application, we prove an optimal large sieve inequality for families of automorphic representations that arise in our framework.