Schwartz $\kappa$-densities for the moduli stack of rank $2$ bundles on a curve over a local field

Let $\mathrm{B}\mathrm{u}\mathrm{n}$ be the moduli stack of rank $2$ bundles with fixed determinant on a smooth proper curve $C$ over a local field $F$. We show how to associate with a Schwartz $\kappa$-density, for $\mathrm{R}\mathrm{e}(\kappa )\ge 1\mathrm{/}2$, a smooth function on the corresponding coarse moduli space of very stable bundles. In the non-archimedean case we also prove that the stack $\mathrm{B}\mathrm{u}\mathrm{n}$ is $\kappa$-bounded in the sense of Definition 2.10 of [arXiv:2112.08139] for any $\kappa \in \mathbb{C}$.