On the dimension datum problem and the linear dependence problem

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.