On the dimension datum problem and the linear dependence problem

TL;DR

This paper classifies closed connected subgroups of compact Lie groups based on their dimension data, exploring their linear dependence and implications for isospectral geometry and automorphic forms.

Contribution

It provides a classification of subgroups with equal or linearly dependent dimension data, extending to non-connected subgroups in the unitary group.

Findings

Classification of subgroups with equal dimension data

Criteria for linear dependence of dimension data

Applications to isospectral geometry and automorphic forms

Abstract

The dimension datum of a closed subgroup of a compact Lie group is the sequence of invariant dimensions of irreducible representations by restriction. In this article we classify closed connected subgroups with equal dimension data or linearly dependent dimension data. This classification should have applications to the isospectral geometry and automorphic form theory. We also study the equality/linear dependence of not necessarily connected subgroups of unitary group acting irreducibly on the natural representation.