Torus orbit closures in flag varieties and retractions on Weyl groups

Abstract

A finite Coxeter group W has a natural metric d and if [Formula: see text] is a subset of W, then for each [Formula: see text], there is [Formula: see text] such that [Formula: see text]. Such q is not unique in general but if [Formula: see text] is a Coxeter matroid, then it is unique, and we define a retraction [Formula: see text] so that [Formula: see text]. The T-fixed point set [Formula: see text] of a T-orbit closure Y in a flag variety G/B is a Coxeter matroid, where G is a semi-simple algebraic group, B is a Borel subgroup, and T is a maximal torus of G contained in B. We define a retraction [Formula: see text] geometrically, where W is the Weyl group of [Formula: see text], and show that [Formula: see text]. We introduce another retraction [Formula: see text] algebraically for an arbitrary subset [Formula: see text] of W when W is a Weyl group of classical Lie type, and show that [Formula: see text] when [Formula: see text] is a Coxeter matroid.