2015/08/05 by B. Narasimha Chary, S. Senthamarai Kannan, A. J. Parameswaran
Mathematics · #math.AG
published as Transformation Groups 20 (2015), no.5, 665-698 · 34 pages, to appear in Transformation Groups
arxiv created 2015/08/05 · arxiv updated 2018/08/06
Let G be a simple, adjoint, algebraic group 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,\underline i) be the Bott-Samelson-Demazure-Hansen variety (the desingularization of the Schubert variety X(w)) corresponding to a reduced expression \underline i of w. In this article, we compute the connected component Aut0(Z(w, \underline i)) of the automorphism group of Z(w,\underline i) containing the identity automorphism. We show that Aut0(Z(w, \underline i)) contains a closed subgroup isomorphic to B if and only if w-1(α0)<0, where α0 is the highest root. If w0 denotes the longest element of W, then we prove that Aut0(Z(w0, \underline i)) is a parabolic subgroup of G. It is also shown that this parabolic subgroup depends very much on the chosen reduced expression \underline i of w0 and we describe all parabolic subgroups of G that occur as Aut0(Z(w0, \underline i)). If G is simply laced, then we show that for every w∈ W and for every reduced expression \underline i of w, Aut0(Z(w, \underline i)) is a quotient of the parabolic subgroup Aut0(Z(w0, \underline j)) of G for a suitable choice of a reduced expression \underline j of w0. We also prove that the Bott-Samelson-Demazure-Hansen varieties are rigid for simply laced groups and their deformations are unobstructed in general.