Eigenfunction asymptotics in the complex domain for a compact Lie group

TL;DR

This paper investigates the asymptotic behavior of eigenfunctions on compact Lie groups in the complex domain, using Grauert tube techniques to analyze operator kernels related to representation theory.

Contribution

It introduces a novel application of Grauert tube methods to study eigenfunction asymptotics in the complexified setting of compact Lie groups.

Findings

Derived near-diagonal scaling asymptotics for operator kernels

Connected kernels to Poisson and Szegő kernels on complexified tangent bundles

Provided new insights into eigenfunction behavior in the complex domain

Abstract

Let $(G,\kappa)$ be a compact connected Lie group endowed with a biinvariant Riemannian metric, and let $\tilde{G}$ be the complexification of $G$. We apply Grauert tube techniques to the near-diagonal scaling asymptotics of certain operator kernels, which are defined in terms of the matrix elements of an irreducuble representation drifting to infinity along a ray in weight space. These kernels are the equivariant components of Poisson and Szeg\H{o} kernels on a fixed sphere bundle in $\tilde{G}$, when the latter is identified with the tangent bundle of $G$ in an appropriate way.