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Bounds of restriction of characters to submanifolds

Yunfeng Zhang

TL;DR

The paper develops sharp $L^p$ restriction bounds for Laplace--Beltrami eigenfunctions on compact Lie groups, focusing on two geometrically meaningful submanifold classes: maximal flats and torus-generated conjugation-invariant submanifolds. It introduces a barycentric-semiclassical subdivision of the Weyl alcove, a root-system peeling mechanism, and a detailed character-theoretic analysis to obtain precise exponents and uniformity results, including for characters and general sums of matrix coefficients. The results improve upon general restriction bounds by providing explicit power savings in high-rank cases, with sharpness demonstrated on carefully chosen alcove facets and submanifolds. The work also extends to torus-generated conjugation-invariant submanifolds and yields a coherent framework for singular submanifolds, culminating in corollaries for restriction to regular points. Overall, the methods fuse representation theory, semiclassical analysis, and Weyl geometry to achieve sharp, uniform, and saturable restriction estimates with broad implications for eigenfunction concentration on Lie groups.

Abstract

A fruitful approach to studying the concentration of Laplace--Beltrami eigenfunctions on a compact manifold, as the eigenvalue tends to infinity, is to bound their restriction to submanifolds. In this paper, we adopt this approach in the setting of compact Lie groups and provide sharp restriction bounds for general Laplace--Beltrami eigenfunctions, as well as for important special cases such as sums of matrix coefficients and, in particular, characters of irreducible representations. We prove sharp asymptotic $L^p$ bounds for the restriction of general Laplace--Beltrami eigenfunctions to maximal flats and all of their submanifolds, for all $p \geq 2$. Furthermore, we establish sharp asymptotic $L^p$ bounds for the restriction of characters to maximal tori and all of their submanifolds for all $p>0$, and to torus-generated conjugation-invariant submanifolds for all $p \geq 2$. We also obtain sharp $L^p$ bounds for the restriction of general sums of matrix coefficients to maximal flats and all of their submanifolds, for all $p \geq 2$.

Bounds of restriction of characters to submanifolds

TL;DR

The paper develops sharp restriction bounds for Laplace--Beltrami eigenfunctions on compact Lie groups, focusing on two geometrically meaningful submanifold classes: maximal flats and torus-generated conjugation-invariant submanifolds. It introduces a barycentric-semiclassical subdivision of the Weyl alcove, a root-system peeling mechanism, and a detailed character-theoretic analysis to obtain precise exponents and uniformity results, including for characters and general sums of matrix coefficients. The results improve upon general restriction bounds by providing explicit power savings in high-rank cases, with sharpness demonstrated on carefully chosen alcove facets and submanifolds. The work also extends to torus-generated conjugation-invariant submanifolds and yields a coherent framework for singular submanifolds, culminating in corollaries for restriction to regular points. Overall, the methods fuse representation theory, semiclassical analysis, and Weyl geometry to achieve sharp, uniform, and saturable restriction estimates with broad implications for eigenfunction concentration on Lie groups.

Abstract

A fruitful approach to studying the concentration of Laplace--Beltrami eigenfunctions on a compact manifold, as the eigenvalue tends to infinity, is to bound their restriction to submanifolds. In this paper, we adopt this approach in the setting of compact Lie groups and provide sharp restriction bounds for general Laplace--Beltrami eigenfunctions, as well as for important special cases such as sums of matrix coefficients and, in particular, characters of irreducible representations. We prove sharp asymptotic bounds for the restriction of general Laplace--Beltrami eigenfunctions to maximal flats and all of their submanifolds, for all . Furthermore, we establish sharp asymptotic bounds for the restriction of characters to maximal tori and all of their submanifolds for all , and to torus-generated conjugation-invariant submanifolds for all . We also obtain sharp bounds for the restriction of general sums of matrix coefficients to maximal flats and all of their submanifolds, for all .
Paper Structure (12 sections, 22 theorems, 163 equations, 5 figures, 4 tables)

This paper contains 12 sections, 22 theorems, 163 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The Weyl alcove and the characters
  3. Structure of compact Lie groups and their alcoves
  4. Barycentric-semiclassical subdivision
  5. Behavior of characters across the alcove
  6. Peeling the root system
  7. Proof of Theorem \ref{['characterrestriction']}
  8. Sharpness of Theorem \ref{['characterrestriction']}
  9. Proof of Theorem \ref{['jointrestriction']}
  10. Proof of Theorem \ref{['gef']}
  11. Torus-generated conjugation-invariant submanifolds
  12. Proof of Theorem \ref{['singular']} and \ref{['singular2']}

Key Result

Theorem 1.1

Let $\chi$ be the character of an irreducible representation of $U$ such that $\Delta \chi=-N^2\chi$, $N>1$. Let $S$ be a compact smooth $k$-dimensional submanifold of a maximal torus $T$ of $U$, $k=0,1,2,\ldots,r$. Let $p_0=0$ and $p_r={2r}/{(d-r)}$. For $k=1,2,\ldots,r-1$, let $p_k$ be as given in Moreover, the above bound is sharp, in the sense that for any $k=0,1,2,\ldots,r$, there exists a co

Figures (5)

  • Figure 1: Extended Dynkin diagrams
  • Figure 2: (a) Barycentric subdivision (b) Semiclassical subdivision (c) Barycentric-semiclassical subdivision $A=\bigsqcup P_{K,J}$
  • Figure 3: A submanifold $S=\bigsqcup_{1\leq i\leq 7} S_i$ of the alcove
  • Figure 4: $P_{K,J}\subset P_J^K\times\mathcal{N}^{K\perp}$
  • Figure 5: The nodal set of $\delta_J(N\cdot)$ on $A_J$

Theorems & Definitions (38)

  • Theorem 1.1
  • Remark 1
  • Theorem 1.2
  • Theorem 1.3
  • Definition
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 2: Limitation of our methods and future directions
  • Remark 3
  • ...and 28 more