Abstract
AbstractFor a connected reductive groupGover a finite field, we study automorphic vector bundles on the stack ofG-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack ofG-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.
Highlights
The stack of -zips was introduced by Pink-Wedhorn-Ziegler [PWZ11] and [PWZ15] based on the notion of F-zip defined in the work of Moonen-Wedhorn ([MW04])
The second author and Wedhorn have used the stack -Zip to construct -ordinary Hasse invariants in [KW18], and this result was later generalised to all Ekedahl-Oort strata with Goldring [GK19a]
In [Kos19], the second author studied the space of global sections of the family of vector bundles (V ( )) ∈ ∗ ( )
Summary
The stack of -zips was introduced by Pink-Wedhorn-Ziegler [PWZ11] and [PWZ15] based on the notion of F-zip defined in the work of Moonen-Wedhorn ([MW04]). In [Kos19], the second author studied the space of global sections of the family of vector bundles (V ( )) ∈ ∗ ( ). For any algebraic -representation ( , ) over , there is a naturally attached vector bundle V( ) of rank dim( ) on -Zip modelled on ( , ) (see Subsection 2.4). In the context of Shimura varieties, there are many interesting vector bundles other than the family ( ( )) , which may not always arise from representations in Rep( ). Shimura varieties after choosing a Siegel embedding It extends to a vector bundle on the integral model of Kisin and Vasiu. There is a natural functor Rep( ) → ( ), where ( ) denotes the category of vector bundles on. The results of this article will be used in the follow-up articles [IK21] and [GIK21], where we study partial Hasse invariants for Shimura varieties of Hodge type
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