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Partial Hasse invariants for Shimura varieties of Hodge-type

Naoki Imai, Jean-Stefan Koskivirta

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.

Partial Hasse invariants for Shimura varieties of Hodge-type

Abstract

For a connected reductive group over a finite field, we define partial Hasse invariants on the stack of -zip flags. We obtain similar sections on the flag space of Shimura varieties of Hodge-type. They are mod 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.

Paper Structure

This paper contains 28 sections, 35 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Definitions
  4. Cocharacter datum
  5. Frames
  6. Representation theory
  7. General theory
  8. Definition
  9. Schubert sections, partial Hasse invariants
  10. Definition
  11. Partial Hasse invariants
  12. Zip partial Hasse invariants
  13. Flag partial Hasse invariants
  14. Verschiebung homomorphism
  15. Consequences
  16. ...and 13 more sections

Key Result

Lemma 2.2.1

Let $\mu \colon \mathbb{G}_{\mathrm{m},k}\to G_k$ be a cocharacter, and let ${\mathcal{Z}}_\mu$ be the attached zip datum. Assume that $(B,T)$ is a Borel pair defined over $\mathbb{F}_q$ such that $B\subset P$. Define the element Then $(B,T,z)$ is a frame for ${\mathcal{Z}}_\mu$.

Theorems & Definitions (73)

  • Lemma 2.2.1: Goldring-Koskivirta-zip-flags
  • Theorem 2.2.2: Pink-Wedhorn-Ziegler-zip-data
  • Lemma 2.3.1
  • proof
  • Theorem 2.3.2
  • proof
  • Lemma 2.5.1: Koskivirta-Wedhorn-Hasse
  • Lemma 2.5.2: Imai-Koskivirta-vector-bundles
  • Proposition 2.5.3
  • Theorem 2.6.1: Imai-Koskivirta-vector-bundles
  • ...and 63 more