Equivariant asymptotics on Grauert tubes

We report on recent work on the scaling asymptotics of the equivariant components of Poisson and Szegö kernels on the Grauert tube boundaries associated to a real-analytic Riemannian manifold acted upon by a compact Lie group. Building largely on techniques of Zelditch and Chang and Rabinowitz, we describe the asymptotic concentration along the zero locus of the moment map of the equivariant eigenfunctions of a Toeplitz operator associated to the homogeneous geodesic flow and of the complexified equivariant eigenfunctions of the Laplacian. We also digress on some applications.