Equivariant scaling asymptotics for Poisson and Szegő 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.

We have added detail on a symbolic computation going back to S. Zelditch and intervening in the asymptotic expansion of the complexified eigenfunctions (Section 5), and corrected a power of $\pi$ in one of the statements (Theorem 29); some typos corrected