Remarks on equivariant asymptotics for complexified toric eigenfunction

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 [arXiv:2409.04753]. 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.