Some new results on dimension datum

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 $({\tau }_1,{\tau }_2)\in \widehat{H_1}\times \widehat{H_2}$ such that ${\mathcal{D}}_{H_1,{\tau }_1}={\mathcal{D}}_{H_2,{\tau }_2}$. 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.

arXiv admin note: substantial text overlap with arXiv:1803.06210