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Witt affine Springer theory

Noam Nissan, Yakov Varshavsky

TL;DR

This work extends the affine Springer correspondence to mixed characteristic by constructing Witt-vector arc spaces and perfect $\infty$-stacks, developing a dimension theory and GKM-type stratification in the Witt setting. It proves a Witt-analogue of the Chevalley morphism being flat on arc spaces and builds the Witt-affine Grothendieck–Springer fibration, along with perverse $t$-structures, on infinite-dimensional perfect stacks. The main construction yields a perverse Witt Springer sheaf with End$\,(\mathcal{S})$ isomorphic to $\overline{\mathbb{Q}_\ell}[\widetilde{W}]$, mirroring the equal-characteristic theory in a mixed-characteristic framework. These results generalize the Bouthier–Kazhdan–Vieh affine Springer theory to Witt-and-mixed characteristic, enabling new tools for geometric representation theory in this setting.

Abstract

This paper extends the affine Springer theory developed by Bouthier, Kazhdan, and the second author (see [BKV]) to the mixed characteristic case. In particular, we introduce a theory of perfectly placid perfect infinity stacks and establish their dimension theory. Furthermore, we prove that, in the Witt vector setting, the Chevalley morphism between arc spaces is flat.

Witt affine Springer theory

TL;DR

This work extends the affine Springer correspondence to mixed characteristic by constructing Witt-vector arc spaces and perfect -stacks, developing a dimension theory and GKM-type stratification in the Witt setting. It proves a Witt-analogue of the Chevalley morphism being flat on arc spaces and builds the Witt-affine Grothendieck–Springer fibration, along with perverse -structures, on infinite-dimensional perfect stacks. The main construction yields a perverse Witt Springer sheaf with End isomorphic to , mirroring the equal-characteristic theory in a mixed-characteristic framework. These results generalize the Bouthier–Kazhdan–Vieh affine Springer theory to Witt-and-mixed characteristic, enabling new tools for geometric representation theory in this setting.

Abstract

This paper extends the affine Springer theory developed by Bouthier, Kazhdan, and the second author (see [BKV]) to the mixed characteristic case. In particular, we introduce a theory of perfectly placid perfect infinity stacks and establish their dimension theory. Furthermore, we prove that, in the Witt vector setting, the Chevalley morphism between arc spaces is flat.
Paper Structure (13 sections, 63 theorems, 72 equations)

This paper contains 13 sections, 63 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Perfect schemes.
  3. Perfect infinity-stacks
  4. Perfectly placid infinity-stacks and perfectly smooth morphisms between them.
  5. Dimension theory.
  6. Perfectly placid stratifications and semi-small morphisms.
  7. The GKM stratification.
  8. Proof of Proposition 7.12
  9. The Witt-affine Grothendieck–Springer fibration.
  10. Perverse t-structures on infinite-dimensional perfect infinity-stacks.
  11. The main theorem.
  12. Flatness of the Chevalley morphism
  13. Codimension of locally closed perfect subschemes of lattices and valuations of differentials

Key Result

Theorem 1.3

[codimofstrata:prop:finitechevalleyflat and codimofstrata:prop:chevalleyflat] Suppose $\operatorname{char}\kappa>2h$ where $h$ is the Coxeter number of $\mathfrak{g}$. Then the Chevalley morphisms between the arc spaces (see wittloopandarc:def:arcofnu for definitions) and their truncated versions: are flat.

Theorems & Definitions (157)

  • Theorem 1.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Proposition 2.11
  • proof
  • Proposition 2.12
  • ...and 147 more