{"title":"Equivariant scaling asymptotics for Poisson and Szegő kernels on Grauert tube boundaries","authors":"Simone Gallivanone, Roberto Paoletti","doi":"arxiv-2409.04753","DOIUrl":null,"url":null,"abstract":"Let $(M,\\kappa)$ be a closed and connected real-analytic Riemannian manifold,\nacted upon by a compact Lie group of isometries $G$. We consider the following\ntwo kinds of equivariant asymptotics along a fixed Grauer tube boundary\n$X^\\tau$ of $(M,\\kappa)$. 1): Given the induced unitary representation of $G$ on the eigenspaces of the\nLaplacian of $(M,\\kappa)$, these split over the irreducible representations of\n$G$. On the other hand, the eigenfunctions of the Laplacian of $(M,\\kappa)$\nadmit a simultaneous complexification to some Grauert tube. We study the\nasymptotic concentration along $X^\\tau$ of the complexified eigenfunctions\npertaining to a fixed isotypical component. 2): There are furthermore an induced action of $G$ as a group of CR and\ncontact automorphisms on $X^\\tau$, and a corresponding unitary representation\non the Hardy space $H(X^\\tau)$. The action of $G$ on $X^\\tau$ commutes with the\nhomogeneous \\lq geogesic flow\\rq\\, and the representation on the Hardy space\ncommutes with the elliptic self-adjoint Toeplitz operator induced by the\ngenerator of the goedesic flow. Hence each eigenspace of the latter also splits\nover the irreducible representations of $G$. We study the asymptotic\nconcentration of the eigenfunctions in a given isotypical component. We also give some applications of these asymptotics.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"2 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.04753","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.