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The basic locus of ramified unitary Rapoport-Zink space at maximal vertex level

Qiao He, Yu Luo, Yousheng Shi

Abstract

We construct the Bruhat-Tits stratification of the ramified unitary Rapoport-Zink space, with the level being the stabilizer of a vertex lattice. We develop the local model theory for Bruhat-Tits strata, proving their normality and Cohen-Macaulayness, and provide precise dimension formulas. Additionally, we establish an explicit isomorphism between Bruhat-Tits strata and Deligne-Lusztig varieties, revealing new phenomena beyond the previously studied Coxeter-type cases.

The basic locus of ramified unitary Rapoport-Zink space at maximal vertex level

Abstract

We construct the Bruhat-Tits stratification of the ramified unitary Rapoport-Zink space, with the level being the stabilizer of a vertex lattice. We develop the local model theory for Bruhat-Tits strata, proving their normality and Cohen-Macaulayness, and provide precise dimension formulas. Additionally, we establish an explicit isomorphism between Bruhat-Tits strata and Deligne-Lusztig varieties, revealing new phenomena beyond the previously studied Coxeter-type cases.

Paper Structure

This paper contains 39 sections, 40 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Notation
  4. Statements of the main results
  5. Bruhat-Tits strata
  6. Bruhat-Tits stratification
  7. Relation to Deligne-Lusztig variety
  8. Kottwitz-Rapoport stratification
  9. Rapoport-Zink spaces
  10. Review of strict $O_{F_0}$-modules
  11. Framing object
  12. Two Rapoport-Zink spaces
  13. Local models of basic locus strata
  14. Standard polarized lattice chain
  15. Local models
  16. ...and 24 more sections

Key Result

Lemma 2.1

Let $\kappa$ be any perfect field over $\mathbb{F}$ and let $M\subset\Lambda\otimes W_O(\kappa)$ be a $O_{F}\otimes W_O(\kappa)$-lattice such that $M\subseteq M^\sharp$. Then $M$ and $M^\sharp$ are stable under ${\mathrm{F}},{\mathrm{V}}$ and $\Pi$.∎

Theorems & Definitions (89)

  • Lemma 2.1: RTW
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Example 2.6
  • Example 2.7
  • Theorem 2.8
  • Corollary 2.9
  • Remark 2.10
  • ...and 79 more