The stack of <i>G</i>-zips is a Mori dream space

Abstract We first extend previous results of Koskivirta with Wedhorn and Goldring regarding the existence of $\mu $ -ordinary Hasse invariants for Hodge-type Shimura varieties to other automorphic line bundles. We also determine exactly which line bundles admit nonzero sections on the stack of G -zips of Pink–Wedhorn–Ziegler. Then, we define and study the Cox ring of the stack of G -zips and show that it is always finitely generated. Finally, beyond the case of line bundles, we define a ring of vector-valued automorphic forms on the stack of G -zips and study its properties. We prove that it is finitely generated in certain cases.