Locally compact homogeneous space - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact Hausdorff group and $H\leq G$ a closed subgroup. The locally compact homogeneous space $G/H$ is the set of left cosets $gH$ , equipped with the quotient topology and the continuous action

x\cdot(gH)=(xg)H.

It is a locally compact Hausdorff space, and the action is transitive. More generally, a locally compact homogeneous $G$ -space means a $G$ -space equivariantly homeomorphic to some $G/H$ with $H$ closed. This topological notion does not require $G$ to be a Lie group or $G/H$ to be a manifold.

Quotient topology

The canonical projection $q:G\to G/H$ is continuous, surjective, and open: for every open $U\subseteq G$ ,

q^{-1}(q(U))=UH=\bigcup_{h\in H}Uh

is open. Closedness of $H$ is exactly what makes the quotient Hausdorff. Local compactness descends through the open quotient map, so compact neighborhoods in $G$ yield relatively compact neighborhoods in $G/H$ .

Stabilizers and recognition

If $G$ acts continuously and transitively on a Hausdorff space $X$ , the stabilizer $G_x$ is closed and the orbit map induces a continuous $G$ -equivariant bijection

G/G_x\longrightarrow X,\qquad gG_x\longmapsto g\cdot x.

It is a homeomorphism when the orbit map $G\to X$ is a quotient map—for example, when it is open. Transitivity alone should not silently be used to claim this topological conclusion without an applicable quotient-map hypothesis.

Measures on the quotient

A quotient $G/H$ carries a natural quasi-invariant measure class under standard locally compact hypotheses. It carries a nonzero invariant Radon measure precisely when the modular functions satisfy

\Delta_G|_H=\Delta_H.

The Weil integration formula describes integration on $G$ in terms of integration along $H$ and over $G/H$ , including the correction needed when an invariant quotient measure does not exist.

Lie-group specialization

When $G$ is a Lie group, every closed subgroup $H$ is a Lie subgroup and $G/H$ has a canonical smooth structure. It is then the smooth homogeneous space associated with the transitive Lie-group action. The locally compact definition is broader: it also includes quotients of totally disconnected, profinite, discrete, and $p$ -adic groups.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §2.6 on homogeneous spaces, rho-functions, and quasi-invariant measures.
Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Volume I, 2nd ed., Springer, 1979. DOI record. Relevant: quotient spaces of locally compact groups and invariant integration.