Bounds of restriction of characters to submanifolds

Yunfeng Zhang

Published 2024-02-05, updated 2024-03-07Version 2

Matrix coefficients and in particular characters of irreducible representations of a compact Lie group are important special Laplacian eigenfunctions thereon. To investigate concentration of these eigenfunctions as the eigenvalue grows, an important route is to bound their restriction to submanifolds. We focus on two classes of submanifolds, namely, submanifolds of maximal flats and the conjugation-invariant submanifolds. We prove sharp asymptotic $L^p$ bounds of restriction of characters to submanifolds of any maximal torus for all $p>0$, of general matrix coefficients to submanifolds of any maximal flat for all $p\geq 2$, and of characters to the conjugation-invariant submanifolds for all $p\geq 2$.