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, i) be the Bott-Samelson-Demazure-Hansen variety corresponding to a reduced expression i of w.In this article, we compute the connected component Aut 0 (Z(w, i)) of the automorphism group of Z(w, i) containing the identity automorphism. We show that Aut 0 (Z(w, i)) contains a closed subgroup isomorphic to B if and only if w −1 (α 0 ) < 0, where α 0 is the highest root. If w 0 denotes the longest element of W , then we prove that Aut 0 (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 Aut 0 (Z(w 0 , i)). If G is simply laced, then we show that for every w ∈ W , and for every reduced expression i of w, Aut 0 (Z(w, i)) is a quotient of the parabolic subgroup Aut 0 (Z(w 0 , j)) of G for a suitable choice of a reduced expression j of w 0 (see Theorem 7.3).

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In this paper we study the geometry of toric degeneration of a Bott-Samelson-Demazure-Hansen (BSDH) variety, which was algebraically constructed in [Pas10] and [PK16]. We give some applications to BSDH varieties. Precisely, we classify Fano, weak Fano and log Fano BSDH varieties and their toric limits in Kac-Moody setting. We prove some vanishing theorems for the cohomology of tangent bundle (and line bundles) on BSDH varieties. We also recover the results in [PK16], by toric methods.