Moduli of bundles and semiorthogonal decomposition

In this paper we construct semiorthogonal decompositions of moduli of principal bundles on a curve into its symmetric powers, for both the moduli stack of all $G$-bundles and the coarse moduli space of semistable $G$-bundles. The essential ingredients in the proof include Borel-Weil-Bott theory for loop groups, highest weight structure of current group representation, derived $\Theta$-stratification and local-global compatibility of Kac-Moody localization.