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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 spherical harmonics $Y(\theta,\phi)$ being the eigenfunctions of the Laplace-Beltrami operator on $SU(2)/U(1) = \mathbb S^2$.)

Context: Brif and Mann (1998) gave a nearly explicit construction for phase space representations of quantum systems with Lie algebraic dynamical symmetries. However, their construction is not all that useful in practice because it lacks some ingredients from representation theory.

This is evident in their examples which start with the statements "Group elements can be parameterized in the following way..." and "The harmonic functions on [some coset space] are..." What follows the ellipses are well-known facts and the connection to the general theory is lost on me.

I would love to follow their construction for the algebra I'm interested in: $\mathfrak{so}(2n)$. Following their construction leads to the coset space $SO(2n)/U(n)$. Now this is where I am stuck since in the examples they give the parameterization of the group elements and harmonic functions come from elsewhere.

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The pair $(SO(2n), U(n))$ is a symmetric pair. You can have a look at Kraemer's paper (Comp. Math. 1979), where in the table the author lists explicitly the complex representations of $SO(2n)$ on which the fixed point set of $U(n)$ is nontrivial. There you'll find (combinations of) fundamental weights of $so(2n)$.

These are the representations that appear in the Peter-Weyl theorem about $L^2(SO(2n)/U(n))$.

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