Quasi-regular representation - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact Hausdorff group, let $H$ be a closed subgroup, and let $\mu$ be a quasi-invariant regular measure on the locally compact homogeneous space $G/H$ . Write $g_*\mu(E)=\mu(g^{-1}E)$ and $r_g=d(g_*\mu)/d\mu$ . The quasi-regular representation of $G$ on $L^2(G/H,\mu)$ is

(\lambda_{G/H}(g)\xi)(xH) =r_g(xH)^{1/2}\xi(g^{-1}xH).

Quasi-invariance makes the derivative exist, and the Radon–Nikodym factor makes every $\lambda_{G/H}(g)$ unitary. The cocycle identity for $r_g$ gives the representation law.

Invariant-measure case

If $G/H$ has a $G$ -invariant measure, then $r_g=1$ almost everywhere and the formula reduces to translation:

(\lambda_{G/H}(g)\xi)(xH)=\xi(g^{-1}xH).

By the Weil integration formula, such an invariant measure exists exactly when $\Delta_G|_H=\Delta_H$ . In particular, it exists when $H$ is compact.

Relationship to induction

The quasi-regular representation is the unitary representation induced from the trivial one-dimensional representation of $H$ . This interpretation explains why it is central to harmonic analysis on quotients: its matrix coefficients and invariant vectors record how $G$ acts on $G/H$ , while induction connects those data to representations of the subgroup Folland, Chapter 6.

Examples and measure-class independence

For $H=\{e\}$ , the construction is the left regular representation on $L^2(G)$ . For $H=G$ , it is the trivial representation on a one-dimensional Hilbert space. Replacing $\mu$ by an equivalent quasi-invariant measure changes the displayed realization but yields a unitarily equivalent representation through multiplication by the square root of the density.

References

G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: homogeneous spaces, quasi-invariant measures, and induced representations.
B. Bekka, P. de la Harpe, and A. Valette, Kazhdan's Property (T), Cambridge University Press, 2008. DOI record. Relevant: quasi-regular representations and invariant vectors.