Torus orbit closures in flag varieties and retractions on Weyl groups

Abstract

A finite Coxeter group W has a natural metric d and if M is a subset of W, then for each u ∈ W, there is q ∈ M such that d(u,q) = d(u,M). Such q is not unique in general but if M is a Coxeter matroid, then it is unique, and we define a retraction Rm: W → M ⊂ W so that Rm(u) = q. The T-fixed point set YT 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 RYg: W → YT W geometrically, where W is the Weyl group of G, and show that RYg = R YTm. We introduce another retraction Ra: W → W algebraically for an arbitrary subset M of W when W is a Weyl group of classical Lie type, and show that Ra = Rm when M is a Coxeter matroid.