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
Nearby in the stack
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 (the desingularization of the Schubert variety
X(w)
) corresponding to a reduced expression
i
of
w
. In this article, we compute 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
w0
denotes the longest element of
W
, then we prove that
Aut0(Z(w0,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
w0
and we describe all parabolic subgroups of
G
that occur as
Aut0(Z(w0,i))
. If
G
is simply laced, then we show that for every
w∈W
and for every reduced expression
i
of
w
,
Aut0(Z(w,i))
is a quotient of the parabolic subgroup
Aut0(Z(w0,j))
of
G
for a suitable choice of a reduced expression
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.