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Integrals over the circular ensembles relating to classical domains

Using the theory of classical domains and Jack polynomials, we generalize several integrals over the unitary group, due to Fyodorov and Khoruzhenko, to integrals over the circular orthogonal and circular symplectic ensembles. While in the unitary case a fundamental role is played by complex matrices, it is anti-symmetric and symmetric matrices which play the same role in the latter two cases. In our workings, a generalization of the evaluation of the integral of a Jack polynomial times the Selberg weight sought by Fyodorov and Khoruzhenko is identified.