2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/119118We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's $G$-stable pieces and the generalization of $G$-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in \cite{GKP}.35 pagesRepresentation Theory20F55, 20G99text