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Character sheaves for classical symmetric pairs

Kari Vilonen, Ting Xue, with an appendix by Dennis Stanton

TL;DR

This work develops a Springer-theoretic framework for classical symmetric pairs, providing a complete description of the character sheaves ${\operatorname{Char}}_K({\mathfrak g}_1)$, including both full-support and nilpotent-support cases, and a classification of cuspidal objects. The authors integrate dual strata, nearby cycle methods, and parabolic induction, with Hecke-algebra representations and endoscopy providing explicit labeling and construction of the full family. They give explicit parametrizations via signed Young diagrams and partitions, and establish precise correspondences between geometric objects and representation-theoretic data in types A, C, and D (BD). The results unify geometric representation theory of symmetric spaces with dual-group and endoscopic perspectives, yielding concrete counting formulas and explicit CS constructions with potential applications to endoscopy and related areas.

Abstract

We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.

Character sheaves for classical symmetric pairs

TL;DR

This work develops a Springer-theoretic framework for classical symmetric pairs, providing a complete description of the character sheaves , including both full-support and nilpotent-support cases, and a classification of cuspidal objects. The authors integrate dual strata, nearby cycle methods, and parabolic induction, with Hecke-algebra representations and endoscopy providing explicit labeling and construction of the full family. They give explicit parametrizations via signed Young diagrams and partitions, and establish precise correspondences between geometric objects and representation-theoretic data in types A, C, and D (BD). The results unify geometric representation theory of symmetric spaces with dual-group and endoscopic perspectives, yielding concrete counting formulas and explicit CS constructions with potential applications to endoscopy and related areas.

Abstract

We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.

Paper Structure

This paper contains 39 sections, 24 theorems, 250 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Character sheaves
  4. Restricted roots and the little Weyl group
  5. The equivariant fundamental group
  6. The classical symmetric pairs
  7. Nilpotent orbits and component groups of the centralizers
  8. Weyl groups and Hecke algebras
  9. General strategy
  10. Dual strata for symmetric spaces and supports of character sheaves
  11. Full support character sheaves
  12. The nearby cycle construction in the split case and endoscopy
  13. The groups $W_{{\mathfrak a},\chi}$ and $W_{{\mathfrak a},\chi}^0$ for a general symmetric pair
  14. Nilpotent support character sheaves
  15. Induced character sheaves
  16. ...and 24 more sections

Key Result

Theorem 1.1

(Corollary cuspidal corollary) The cuspidal character sheaves consist of

Theorems & Definitions (43)

  • Theorem 1.1
  • Remark
  • Lemma 3.1
  • Remark 3.2
  • Proposition 4.1: T
  • Theorem 4.2
  • Remark 4.3
  • Theorem 4.4
  • Proposition 4.5: BZ, T2
  • Proposition 4.6
  • ...and 33 more