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For questions regarding harmonic functions.

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Harmonic functions (eigenfunctions of the Laplace-Beltrami operator) of SO(2n)/U(n)

Have the eigenfunctions of the Laplace-Beltrami operator on $SO(2n)/U(n)$ been worked out explicitly? If not, how does one approach finding them? (I'm thinking of this as in analogy with the spherica …