Partial Hasse invariants for Shimura varieties of Hodge-type

TL;DR

This paper introduces partial Hasse invariants on the stack of G-zip flags and Shimura varieties of Hodge-type, revealing their properties and their relation to automorphic vector bundles and automorphic forms in characteristic p.

Contribution

It defines partial Hasse invariants for G-zip stacks and Shimura varieties of Hodge-type, and explores their factorization through automorphic vector bundles.

Findings

Partial Hasse invariants cut out a codimension one stratum.

These invariants factor through higher rank automorphic vector bundles.

Partial Hasse invariants lie in the socle of automorphic vector bundles.

Abstract

For a connected reductive group $G$ over a finite field, we define partial Hasse invariants on the stack of $G$-zip flags. We obtain similar sections on the flag space of Shimura varieties of Hodge-type. They are mod $p$ automorphic forms which cut out a single codimension one stratum. We study their properties and show that such invariants admit a natural factorization through higher rank automorphic vector bundles. We define the socle of an automorphic vector bundle, and show that partial Hasse invariants lie in this socle.