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Let G be a simple linear algebraic group defined over an algebraically closed field k of characteristic p ≥ 0, and let P be a maximal proper parabolic subgroup of G. If p > 0, then we will assume that dimG/P ≤ p. Let ι : H ↪ G/P be a reduced smooth hypersurface in G/P of degree d. We will assume that the pullback homomorphism 
is an isomorphism (this assumption is automatically satisfied when dimH ≥ 3). We prove that the tangent bundle of H is stable if the two conditions τ(G/P) ≠ d and 
hold; here n = dimH, and τ(G/P) ∈ 
is the index of G/P which is defined by the identity 
= 
where L is the ample generator of Pic(G/P) and is the anti–canonical line bundle of G/P. If d = τ(G/P), then the tangent bundle TH is proved to be semistable. If p > 0, and 
then TH is strongly stable. If p > 0, and d = τ(G/P), then TH is strongly semistable.
