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

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

设$\rm{Bun}$为在局部域$F$上的光滑恰当曲线$C$上具有固定行列式的秩为$2$的丛的模空间层。我们展示了对于$\rm{Re}(\kappa)\geq1/2$，如何将一个施瓦茨$\kappa$-密度与非常稳定丛的相应粗模空间上的一个光滑函数相关联。在非阿基米德情形下，我们还证明了对于任意$\kappa\in\mathbb{C}$，层$\rm{Bun}$在[arXiv:2112.08139]的定义2.10的意义下是$\kappa$-有界的。