Equivariant scaling asymptotics for Poisson and Szeg\H{o} kernels on Grauert tube boundaries

TL;DR

This paper investigates the asymptotic behavior of equivariant eigenfunctions and kernels on Grauert tube boundaries of real-analytic manifolds with symmetry, revealing how they concentrate along certain geometric structures.

Contribution

It provides new asymptotic formulas for equivariant Poisson and Szegő kernels on Grauert tube boundaries, extending understanding of symmetry and complexification effects.

Findings

Eigenfunctions concentrate asymptotically along specific geometric structures.

Equivariant kernels split over irreducible representations of the symmetry group.

Results apply to complexified eigenfunctions and Hardy space eigenfunctions with symmetry.

Abstract

Let $(M,\kappa)$ be a closed and connected real-analytic Riemannian manifold, acted upon by a compact Lie group of isometries $G$. We consider the following two kinds of equivariant asymptotics along a fixed Grauer tube boundary $X^\tau$ of $(M,\kappa)$. 1): Given the induced unitary representation of $G$ on the eigenspaces of the Laplacian of $(M,\kappa)$, these split over the irreducible representations of $G$. On the other hand, the eigenfunctions of the Laplacian of $(M,\kappa)$ admit a simultaneous complexification to some Grauert tube. We study the asymptotic concentration along $X^\tau$ of the complexified eigenfunctions pertaining to a fixed isotypical component. 2): There are furthermore an induced action of $G$ as a group of CR and contact automorphisms on $X^\tau$, and a corresponding unitary representation on the Hardy space $H(X^\tau)$. The action of $G$ on $X^\tau$ commutes with the homogeneous \lq geogesic flow\rq\, and the representation on the Hardy space commutes with the elliptic self-adjoint Toeplitz operator induced by the generator of the goedesic flow. Hence each eigenspace of the latter also splits over the irreducible representations of $G$. We study the asymptotic concentration of the eigenfunctions in a given isotypical component. We also give some applications of these asymptotics.