Some new results on dimension datum

TL;DR

This paper presents new results on dimension datum, including examples of isospectral vector bundles, the determination of homomorphism images by dimension data, and an improved compactness theorem for isospectral spaces.

Contribution

It introduces novel examples of isospectral bundles, shows how dimension data determine homomorphism images, and enhances compactness results for isospectral spaces with variable metrics.

Findings

Constructed pairs of irreducible representations with equal dimension data.

Dimension data of 1D representations determine homomorphism images.

Extended compactness results to variable metrics with semisimple groups.

Abstract

In this paper we show three new results concerning dimension datum. Firstly, for two subgroups $H_{1}$($\cong U(2n+1)$) and $H_{2}$($\cong Sp(n)\times SO(2n+2)$) of $SU(4n+2)$, we find a family of pairs of irreducible representations $(\tau_1,\tau_2)\in\hat{H_{1}}\times\hat{H_{2}}$ such that $\mathscr{D}_{H_1,\tau_1}=\mathscr{D}_{H_2,\tau_2}$. With this we construct examples of isospectral hermitian vector bundles. Secondly, we show that: $\tau$-dimension data of one-dimensional representations of a connected compact Lie group $H$ determine the image of homomorphism from $H$ to a given compact Lie group $G$. Lastly, we improve a compactness result for an isospectral set of normal homogeneous spaces $(G/H,m)$ by allowing the Riemannian metric $m$ vary, but posing a constraint that $G$ is semisimple.