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Abstract
The eigenfunctions of the Laplace-Beltrami operator on a real-analytic Riemannian manifold admit a symultaneous holomorphic extension to a sufficiently small Grauert tube. If the manifold is endowed with an isometric action of a compact Lie group, one can decompose each eigenspace into isotypical components associated to the irreducible representations of the group. The local asymptotics of the complexified eigenfunctions in a fixed isotypical component and (heuristically speaking) belonging to a spectral band drifting to infinity have been studied recently in the work "Equivariant scaling asymptotics for Poisson and Szegő kernels on Grauert tube boundaries" by Gallivanone and Paoletti (2024) . In this note, we illustrate these results in the special case where the base manifold is a d-dimensional torus with the standard metric, acted upon by a proper subtorus.
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