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Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d in pi_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X times mathcal{M}^d_G$ is stable with respect to any polarization on $X times mathcal{M}^d_G$. A similar result is proved for the Poincare adjoint bundle over $X times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.
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A theorem of the first author states that the cotangent bundle of the type $A$ Grassmannian variety can be embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety. We extend this result to cotangent bundles of cominuscule generalized Grassmannians of arbitrary Lie type.
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