GIT quotient of a Bott–Samelson–Demazure–Hansen variety by a maximal torus

Abstract

Let G be an almost simple, simply connected algebraic group over the field $$\mathbb {C}$$ of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G, and let W be the Weyl group defined by T. The Borel group B determines a subset of simple reflections in W. For w in W, we let $$Z(w,\underline{i})$$ be the Bott–Samelson–Demazure–Hansen variety corresponding to a reduced expression $$\underline{i}$$ of w as a product of these simple reflections. In this article, we study the geometric invariant theoretic quotient of $$Z(w,\underline{i})$$ for the T-linearized ample line bundles.