Moduli of $G$-bundles on rigid gerbes over affine curves

We geometrize the basic cohomology set $H^1({\text{Kal}}_F,G{)}_{\text{basic}}$ for a global function field $F$. We do this by constructing a v-stack ${\text{Bun}}_{G,F}^e$ which has localization maps to Fargues’ analogous stack ${\text{Bun}}_{G,F_v}^e$ for all places $v$ of $F$ and whose semistable locus is the disjoint union of ${\text{Bun}}_{G_b,F}$ for all $b\in H^1({\text{Kott}}_F{\times }_F{\text{Kal}}_F,G{)}_{\text{basic}}$. We also prove a version of Tate-Nakayama duality for $H^1({\text{Kott}}_F{\times }_F{\text{Kal}}_F,G{)}_{\text{basic}}$, which lets us state a conjectural multiplicity formula for discrete automorphic representations of $G({\mathbb{A}}_F)$ adapted to this new cohomology set.