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On Moy-Prasad quotients over Laurent series fields

David Yang

Abstract

Let $k$ be an algebraically closed field and $G$ a connected reductive group over $k((t))$ satisfying some conditions. We define a stratification by conjugacy classes of twisted Levi subgroups of $G$ on each Moy-Prasad quotient $\mathfrak{k}_{x,r}/\mathfrak{k}_{x,r+}$ of $G$. We then calculate the strata in terms of the associated twisted Levi subgroups. This calculation is necessary for several followup papers on the local geometric Langlands program.

On Moy-Prasad quotients over Laurent series fields

Abstract

Let be an algebraically closed field and a connected reductive group over satisfying some conditions. We define a stratification by conjugacy classes of twisted Levi subgroups of on each Moy-Prasad quotient of . We then calculate the strata in terms of the associated twisted Levi subgroups. This calculation is necessary for several followup papers on the local geometric Langlands program.
Paper Structure (17 sections, 28 theorems, 76 equations)

This paper contains 17 sections, 28 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Acknowledgements
  4. Groups over $k((t))$
  5. Quasi-split groups and twisted loop algebras
  6. The Bruhat-Tits building
  7. Moy-Prasad subgroups of quasi-split groups
  8. Twisted Levi subgroups
  9. The vector space $\mathfrak{k}_{x,r}/\mathfrak{k}_{x,r+}$
  10. The semistable locus
  11. The map $q_{x,r}$
  12. GIT of $\mathfrak{k}_{x,r}/\mathfrak{k}_{x,r+}$
  13. Stratification by twisted Levi subgroups
  14. Proof of Theorem \ref{['basecase']}
  15. Bijectivity on semisimple elements
  16. ...and 2 more sections

Key Result

Lemma 2.1

If $H$ is the split form over $k$ of $G$, there exists a positive integer $n$ not divisible by the characteristic of $k$, a generator $a:t^{\frac{1}{n}}\mapsto\zeta t^{\frac{1}{n}}$ of $\operatorname{Gal}(k((t^{\frac{1}{n}}))/k((t)))$, an endomorphism $\sigma$ of $H$ preserving $T_H$ and $B_H$, and which intertwines the action of $\operatorname{id}\times a$ on the left and $\sigma\times a$ on the

Theorems & Definitions (59)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 3.1
  • Lemma 3.2
  • ...and 49 more