A generalization of cyclic shift classes

TL;DR

This paper generalizes cyclic shift classes to combinatorial pieces involving Weyl group elements and subsets of simple reflections, with applications to symmetry and induction functors in parabolic character sheaves.

Contribution

It introduces a new generalization of cyclic shift classes for combinatorial pieces related to Weyl groups, extending previous notions.

Findings

Partial cyclic shift classes have well-behaved representatives.

Proves left-right symmetry in the context of combinatorial pieces.

Establishes compatibility of induction functors for parabolic character sheaves.

Abstract

Motivated by Lusztig's $G$-stable pieces, we consider the combinatorial pieces: the pairs $(w, K)$ for elements $w$ in the Weyl group and subsets $K$ of simple reflections that are normalized by $w$. We generalize the notion of cyclic shift classes on the Weyl groups to the set of combinatorial pieces. We show that the partial cyclic shift classes of combinatorial pieces associated with minimal-length elements have nice representatives. As applications, we prove the left-right symmetry and the compatibility of the induction functors of the parabolic character sheaves.