We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family π:X→Z\documentclass[12pt]{minimal}
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\begin{document}$$\pi :{{\mathcal {X}}}\rightarrow {{\mathcal {Z}}}$$\end{document} of Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, is π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} an S-fibration? The cases of Picard number one were answered by Hwang and Mok. The manifold F(1,Q5)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {F}}}(1, Q^5)$$\end{document} is the unique rational homogeneous space of Picard number one that is not rigid under Fano deformation, and a Fano degeneration of it is constructed by Pasquier and Perrin. For higher Picard number cases, one notices that the Picard number of a rational homogeneous space G/P satisfies ρ(G/P)≤rank(G)\documentclass[12pt]{minimal}
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\begin{document}$$\rho (G/P)\le \mathrm{rank}(G)$$\end{document}. Weber and Wiśniewski proved that the rational homogeneous spaces G/P with ρ(G/P)=rank(G)\documentclass[12pt]{minimal}
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\begin{document}$$\rho (G/P)=\mathrm{rank}(G)$$\end{document} (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper, we show that the rational homogeneous spaces G/P with ρ(G/P)=rank(G)-1\documentclass[12pt]{minimal}
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\begin{document}$$\rho (G/P)=\mathrm{rank}(G)-1$$\end{document} are rigid under Fano deformation, provided that G is a simple algebraic group of type ADE, and G/P is not biholomorphic to F(1,2,P3)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)$$\end{document} or F(1,2,Q6)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {F}}}(1, 2, Q^6)$$\end{document}. We also show that F(1,2,P3)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {F}}}(1, 2, {{\mathbb {P}}}^3)$$\end{document} has a unique Fano degeneration, which is explicitly constructed. Furthermore, the structure of possible Fano degenerations of F(1,2,Q6)\documentclass[12pt]{minimal}
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\begin{document}$${{\mathbb {F}}}(1, 2, Q^6)$$\end{document} is also described explicitly. Our main result is obtained by applying the theory of Cartan connections and symbol algebras.
