  We study the question whether rational homogeneous spaces are rigid under
Fano deformation. In other words, given any smooth connected family f:X -> Zof
Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space
S, whether is f an S-fibration? The cases of Picard number one were studied in
a series of papers by J.-M. Hwang and N. Mok. For higher Picard number cases,
we notice that the Picard number of a rational homogeneous space G/P is less or
equal to the rank of G. Recently A. Weber and J. A. Wisniewski proved that
rational homogeneous spaces G/P with Picard numbers equal to the rank of G
(i.e. complete flag manifolds) are rigid under Fano deformation. In this paper
we show that the rational homogeneous space G/P is rigid under Fano
deformation, providing that G is a simple algebraic group of type ADE, the
Picard number equal to rank(G)-1 and G/P is not biholomorphic to F(1, 2, P^3)
or F(1, 2, Q^6). The variety F(1, 2, P^3) is the set of flags of projective
lines and planes in P^3, and F(1, 2, Q^6) is the set of flags of projective
lines and planes in 6-dimensional smooth quadric hypersurface. We show that
F(1, 2, P^3) have a unique Fano degeneration, which is explicitly constructed.
The structure of possible Fano degeneration of F(1, 2, Q^6) is also described
explicitly. To prove our rigidity result, we firstly show that the Fano
deformation rigidity of a homogeneous space of type ADE can be implied by that
property of suitable homogeneous submanifolds. Then we complete the proof via
the study of Fano deformation rigidity of rational homogeneous spaces of small
Picard numbers. As a byproduct, we also show the Fano deformation rigidity of
other manifolds such as F(0, 1, ..., k_1, k_2, k_2+1, ..., n-1, P^n) and F(0,
1, ..., k_1, k_2, k_2+1, ..., n, Q^{2n+2}) with 0 <= k_1 < k_2 <= n-1.
