A rigidity result for dimension data

TL;DR

This paper proves a rigidity property for the sequence of dimension data of closed subgroups in compact Lie groups, showing that such sequences have convergent subsequences and the space of all such data is sequentially compact.

Contribution

It establishes a new rigidity theorem for dimension data sequences, ensuring their convergence properties and compactness in the space of subgroups.

Findings

Any sequence of dimension data has a converging subsequence.

The space of dimension data of closed subgroups is sequentially compact.

The limit of a converging sequence corresponds to a related subgroup.

Abstract

The dimension datum of a closed subgroup of a compact Lie group is a sequence by assigning the invariant dimension of each irreducible representation restricting to the subgroup. We prove that any sequence of dimension data contains a converging sequence with limit the dimension datum of a subgroup interrelated to subgroups giving this sequence. This rigidity has an immediate corollary that the space of dimension data of closed subgroups in a given compact Lie group is sequentially compact.