Character sheaves for symmetric pairs: special linear groups

TL;DR

This work completes the classification of character sheaves for symmetric pairs attached to inner involutions of $SL_n$, by developing an enhanced nearby-cycle framework that accommodates central-character obstructions. The authors systematically organize character sheaves by central character, construct nilpotent-support sheaves, and describe their Fourier transforms via dual strata, microlocal analysis, and parabolic induction from suitable Levi subgroups. They obtain an explicit correspondence between nilpotent-orbit data and character-sheaf data, and they determine exactly when cuspidal character sheaves exist, with precise descriptions in split and quasi-split cases. The results extend Lusztig’s generalized Springer framework to inner-involution symmetric pairs and yield a complete, case-heavy classification that matches prior partial results and clarifies the role of central characters in the geometry of character sheaves.

Abstract

We give an explicit description of character sheaves for the symmetric pairs associated to inner involutions of the special linear groups. We make use of the general strategy given in [VX1] and central character consideration. We also determine the cuspidal character sheaves.

Theorems & Definitions (16)

• Theorem 1.1
• Theorem 3.2
• Lemma 3.3
• proof
• Theorem 4.1
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Corollary 4.4