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In other words, how to compute $H_{*}(U(n)/O(n))$ as well as $H_{*}(Sp(n)/U(n))$ ?

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Perhaps the Eilenberg-Moore spectral sequence could be of use... realize the Lagrangian Grassmanian as the fiber of a certain map of classifying spaces that you're intimately familiar with. – Dylan Wilson Oct 31 '12 at 3:57

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