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

Let Bun 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 κ-density, for Re(κ)≥ 1/2, a smooth function on the corresponding coarse moduli space of very stable bundles. In the non-archimedean case we also prove that the stack Bun is κ-bounded in the sense of Definition 2.10 of [arXiv:2112.08139] for any κ∈ℂ.