Title: Stability of the Poincaré bundle,
Abstract: 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
Poincaré 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.