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We consider the commuting variety $\\mathcal C(\\mathfrak p) = \\{(X,Y) \\in \\mathfrak p \\times \\mathfrak p \\mid [X,Y] = 0\\}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $\\mathcal C(\\mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $\\mathfrak p$. 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Goodwin","submitted_at":"2016-06-07T18:53:41Z","abstract_excerpt":"Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\\mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $\\mathcal C(\\mathfrak p) = \\{(X,Y) \\in \\mathfrak p \\times \\mathfrak p \\mid [X,Y] = 0\\}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $\\mathcal C(\\mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $\\mathfrak p$. 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