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The basic locus of unitary splitting Rapoport-Zink spaces with vertex stabilizer level

Ioannis Zachos, Zhihao Zhao

TL;DR

The paper analyzes the reduced basic locus of ramified unitary Rapoport–Zink spaces at a maximal vertex level and constructs a Bruhat–Tits stratification whose strata are shown to correspond to (modified) Deligne–Lusztig varieties. To study local properties, the authors introduce strata splitting models—linear-algebraic analogues of local models—proving they are normal, Cohen–Macaulay, and that BT strata are etale locally isomorphic to these models. Globally, they establish explicit isomorphisms between BT strata and modified DL varieties in both the symplectic and orthogonal settings, and describe the intersections in terms of DL-type data. The results unify and extend known descriptions of basic loci, provide tools for higher-systems of vertex levels, and offer a framework (via strata splitting models) for analyzing BT strata at deeper level structures.

Abstract

We construct the Bruhat-Tits stratification of the ramified unitary splitting Rapoport-Zink space, with the level being the stabilizer of a vertex lattice. To determine certain local properties of the Bruhat-Tits strata, we develop a theory of the strata splitting models. To study their global structure, we establish an explicit isomorphism between the Bruhat-Tits strata and certain (modified) Deligne-Lusztig varieties.

The basic locus of unitary splitting Rapoport-Zink spaces with vertex stabilizer level

TL;DR

The paper analyzes the reduced basic locus of ramified unitary Rapoport–Zink spaces at a maximal vertex level and constructs a Bruhat–Tits stratification whose strata are shown to correspond to (modified) Deligne–Lusztig varieties. To study local properties, the authors introduce strata splitting models—linear-algebraic analogues of local models—proving they are normal, Cohen–Macaulay, and that BT strata are etale locally isomorphic to these models. Globally, they establish explicit isomorphisms between BT strata and modified DL varieties in both the symplectic and orthogonal settings, and describe the intersections in terms of DL-type data. The results unify and extend known descriptions of basic loci, provide tools for higher-systems of vertex levels, and offer a framework (via strata splitting models) for analyzing BT strata at deeper level structures.

Abstract

We construct the Bruhat-Tits stratification of the ramified unitary splitting Rapoport-Zink space, with the level being the stabilizer of a vertex lattice. To determine certain local properties of the Bruhat-Tits strata, we develop a theory of the strata splitting models. To study their global structure, we establish an explicit isomorphism between the Bruhat-Tits strata and certain (modified) Deligne-Lusztig varieties.
Paper Structure (22 sections, 29 theorems, 104 equations)

This paper contains 22 sections, 29 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Rapoport-Zink spaces
  3. Preliminaries
  4. Unitary RZ spaces
  5. Bruhat-Tits strata
  6. Strata splitting models
  7. Review of strata local models
  8. Strata splitting models
  9. Affine charts
  10. $\mathcal{Z}^{\rm spl}$-strata splitting models
  11. $\mathcal{Y}^{\rm spl}$-strata splitting models
  12. Intersection of $\mathcal{Z}^{\rm spl}$-strata and $\mathcal{Y}^{\rm spl}$-strata splitting models
  13. Local properties of Bruhat-Tits strata
  14. Bruhat-Tits stratification
  15. Global properties of Bruhat-Tits strata
  16. ...and 7 more sections

Key Result

Theorem 1.1

The Bruhat-Tits stratification of the reduced basic locus of the splitting RZ space $\mathcal{N}_{n, {\rm red}}^{\rm spl}$ is

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.5
  • Proposition 2.6
  • proof
  • ...and 53 more