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

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.