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Close fields, affine Springer fibers and fundamental lemmas

Sebastian Bartling, Kazuhiro Ito

Abstract

We prove a geometric local constancy theorem for affine Springer fibers in families of close local fields. Consequently, stable orbital integrals are locally constant in these families, and both the base change fundamental lemma and the standard endoscopic fundamental lemma transfer from characteristic zero to arbitrary positive characteristic.

Close fields, affine Springer fibers and fundamental lemmas

Abstract

We prove a geometric local constancy theorem for affine Springer fibers in families of close local fields. Consequently, stable orbital integrals are locally constant in these families, and both the base change fundamental lemma and the standard endoscopic fundamental lemma transfer from characteristic zero to arbitrary positive characteristic.
Paper Structure (26 sections, 42 theorems, 81 equations)

This paper contains 26 sections, 42 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Main results and explanation of the method
  3. Applications
  4. Relation to previous work and possible further applications
  5. Structure of the article
  6. Reductive groups in families of close fields
  7. Families of close fields
  8. Tori in families of close fields
  9. Unramified reductive groups and parahoric subgroups in families of close fields
  10. Parahoric Hecke algebras in families of close local fields
  11. Affine Grassmannian in families of close fields
  12. $\mathcal{O}$-Witt vector affine Grassmannian
  13. Schubert varieties
  14. Affine Springer fibers in families of close fields
  15. Perfectly finitely presented algebraic spaces
  16. ...and 11 more sections

Key Result

Theorem 1.1

The sheaf $\mathop{\mathrm{Gr}}\nolimits_{\mathcal{P}}$ is representable by an increasing union of perfections of finitely presented projective schemes over $\mathop{\mathrm{Spec}}\nolimits \mathcal{O}/\pi.$

Theorems & Definitions (100)

  • Theorem 1.1: Corollary \ref{['Corollary:affine Grassmannian of parahoric is representable']}
  • Theorem 1.2: Theorem \ref{['Theorem: ASF is quasi-compact separated pfp algebraic space']}
  • Theorem 1.3: Base change fundamental lemma, Theorem \ref{['Thm: base change fundamental lemma']}
  • Theorem 1.4: Standard endoscopic fundamental lemma, Theorem \ref{['Theorem: Standard Endo FL']}
  • Definition 1: Li-Huerta
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Corollary 1
  • ...and 90 more