What question did this study set out to answer?

The aim is to explore invariant function algebras on homogeneous spaces of compact connected Lie groups and their properties.

March 29, 2026Open Access

Homogeneous Spaces of Compact Connected Lie Groups Which Admit Nontrivial Invariant Algebras

Key Points

The aim is to explore invariant function algebras on homogeneous spaces of compact connected Lie groups and their properties.
Examined properties of compact semisimple and connected Lie groups.
Discussed homogeneous spaces and their stable subgroups.
Analyzed the isotropy representation and fixed vector conditions.
Homogeneous spaces can have only self-conjugated invariant algebras if the isotropy representation lacks nonzero fixed vectors.
This extends previous findings on compact semisimple groups to a broader class of spaces.

Abstract

In 1965 Wolf 10 and Gangolli 1 proved that compact semisimple groups are distinguished in the class of all compact connected Lie groups by the following property: every uniformly closed function algebra which is invariant with respect to left and right translations is also invariant with respect to the complex conjugation.In this article we extend this result to the class of homogeneous spaces of compact connected Lie groups with connected stable subgroups: a homogeneous space admits only self-conjugated invariant function algebras if and only if the isotropy representation has no nonzero fixed vectors.

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