A generalization of cyclic shift classes

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.