zbMATH — the first resource for mathematics

Homogeneous spaces of compact connected Lie groups which admit nontrivial invariant algebras. (English) Zbl 1023.22013
Let $$M=G/H$$ be a homogeneous space of a compact Lie group $$G$$, $$C(M)$$ be the Banach algebra of all complex valued continuous functions on $$M$$ endowed with the $$\sup$$-norm. An invariant algebra $$A$$ on $$M$$ is a closed $$G$$-invariant subalgebra of $$C(M)$$. If $$A$$ is self-adjoint with respect to complex conjugation then, according to the Stone-Weierstrass theorem, there exists a closed subgroup $$H'\supseteq H$$ such that $$A\cong C(M')$$, where $$M'=G/H'$$. Let us consider $$G$$ as the homogeneous space of the group $$G\times G$$, with $$G$$ acting on itself by left and right translations. In this setting, J. A. Wolf [Pac. J. Math. 15, 1093-1099 (1965; Zbl 0141.31802)] characterized compact groups $$G$$ which admit only self-adjoint invariant algebras. Among the connected Lie groups, this property distinguishes semisimple groups (this was independently proved by R. Gangolli [Bull. Am. Math. Soc. 71, 634-637 (1965; Zbl 0137.31602)]). In this note, under the assumption that $$G$$ and $$H$$ are connected, it is proved that each invariant algebra on $$M$$ is self-adjoint if and only if the decomposition of the isotropy representation of $$H$$ does not have a trivial component. Another equivalent condition is the following: the group $$N/H$$, where $$N$$ is the normalizer of $$H$$, is finite. This generalizes the result of Wolf and Gangolli. Reviewer’s remark. It was noted by M. Raïs that the result remains true if $$H$$ and $$G$$ are not assumed to be connected. He proved this using his earlier results [C. R. Acad. Sci., Paris, Sér. I 305, 713-716 (1987; Zbl 0696.22008)]. Another proof is contained in the joint paper of the author and the reviewer [Transformation Groups 6, 321-331 (2001; Zbl 0999.22020)].
Reviewer: V.Gichev (Omsk)

MSC:
 22F30 Homogeneous spaces 43A85 Harmonic analysis on homogeneous spaces 22E46 Semisimple Lie groups and their representations 46J10 Banach algebras of continuous functions, function algebras
Full Text: