AUTOMORPHISM GROUP OF A BOTT–SAMELSON–DEMAZURE–HANSEN VARIETY

Abstract

Let G be a simple algebraic group of adjoint type over the field of complex numbers, B be a Borel subgroup of G containing a maximal torus T of G, w be an element of the Weyl group W and X(w) be the Schubert variety in G/B corresponding to w. Let Z(w; i) be the Bott-Samelson-Demazure-Hansen variety corresponding to a reduced expression i of w. In this article, we compute inductively, the connected component Aut⁰(Z(w; i)) of the automorphism group of Z(w; i) containing the identity automorphism. We show that Aut⁰(Z(w; i)) contains a closed subgroup isomorphic to B if and only if w ^-1(α₀) < 0, where α₀ is the highest root. If w ₀ denotes the longest element of W, then we prove that Aut⁰(Z(w ₀; i)) is a parabolic subgroup of G. It is also shown that this parabolic subgroup depends very much on the chosen reduced expression i of w ₀ and we describe all parabolic subgroups of G that occur as Aut⁰(Z(w ₀; i)). If G is simply laced, then we show that for every w ϵ W, and for every reduced expression i of w, Aut⁰(Z(w; i)) is a quotient of the parabolic subgroup Aut⁰(Z(w ₀; j)) of G for a suitable choice of a reduced expression j of w ₀ (see Theorem 7.3).