$L^2$-property for algebraic stacks over local non-archimedean fields

We introduce an $L^2$-norm on the space of Schwartz half-densities over algebraic stacks over local non-archimedean fields. We show that these $L^2$-norms are finite for the stacks of $PGL_2$-bundles on $\mathbb{P}^1$ with parabolic structures at $\ge 3$ points. The latter property was conjectured in the context of the analytic Langlands correspondence of arXiv:2103.01509.