Induced unitary representation - Imported Knowlpedia Content

Definition

Let $G$ be a second-countable locally compact Hausdorff group, $H\subseteq G$ a closed subgroup, and $(\sigma,V)$ a strongly continuous unitary representation of $H$ . Form the Hilbert bundle $G\times_HV\to G/H$ , where $(x,v)\sim(xh,\sigma(h)^{-1}v)$ . Given a quasi-invariant measure $\mu$ on $G/H$ , the induced unitary representation $\operatorname{Ind}_H^G\sigma$ acts on the Hilbert space of square-integrable measurable sections. Its action is left translation of sections, multiplied by the square root of the appropriate Radon–Nikodym derivative so that every operator is unitary.

Explicit action and measure independence

Write $r_g=d(g_*\mu)/d\mu$ , with $g_*\mu(E)=\mu(g^{-1}E)$ . If $g\cdot-$ denotes the natural map from the fiber over $g^{-1}xH$ to the fiber over $xH$ , then

\bigl(\operatorname{Ind}_H^G\sigma(g)s\bigr)(xH) =r_g(xH)^{1/2}\,g\cdot s(g^{-1}xH).

The Radon–Nikodym cocycle gives the representation law, and a change to an equivalent quasi-invariant measure is implemented by multiplication by the square root of its density. Hence the unitary-equivalence class is independent of the chosen measure within the canonical measure class Folland, §6.1.

Equivariant-function model

After choosing measurable trivializations, vectors can instead be represented by measurable functions $F:G\to V$ satisfying

F(xh)=\sigma(h)^{-1}F(x).

Their norm is an integral over $G/H$ , with a density or modular correction determined by the chosen quotient-measure convention. This model is useful for calculations, but omitting that correction is legitimate only in invariant measure situations. The bundle model packages the same data without making a choice of coset representatives.

Basic examples and structure

Induction from the trivial representation of $H$ gives the quasi-regular representation on $L^2(G/H)$ . Taking $H=\{e\}$ gives the left regular representation of $G$ . Induction respects Hilbert direct sums and is transitive along chains of closed subgroups. It is also characterized by the system of imprimitivity carried by multiplication operators on the base $G/H$ .

Conventions and scope

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 6, §6.1 on the inducing construction.
George W. Mackey, “Induced Representations of Locally Compact Groups I,” Annals of Mathematics 55 (1952), 101–139. DOI record. Relevant: the Hilbert-space induction construction and its measure-class invariance.