Topological groups, Lie groups Definition

Induced representation of a locally compact group

Builds on Locally compact group Haar measure Unitary representation Hilbert space Left coset

Definition

Let $G$ be a second countable locally compact group, and let $H$ be a closed subgroup of $G$ . Assume that the coset space $G/H$ carries a nonzero $G$ -invariant Radon measure $\nu$ . Let $\sigma$ be a unitary representation of $H$ on a Hilbert space $K$ . Let $\mathcal{H}$ be the space of measurable functions $f \colon G \to K$ such that:

$f(gh) = \sigma(h)^{-1} f(g)$ for all $g \in G$ and $h \in H$ ;
$\int_{G/H} \norm{f(g)}^2 \, d\nu(gH) < \infty$ .

Functions that agree almost everywhere are identified. The norm $\norm{f(g)}$ depends only on the coset $gH$ , because $\sigma$ is unitary, so clause 2 makes sense. The induced representation $\operatorname{Ind}_H^G \sigma$ is the representation of $G$ on $\mathcal{H}$ given by $(\lambda(x) f)(g) = f(x^{-1} g)$ .

Remarks

Example: take $H = \set{e}$ and $\sigma$ trivial on $K = \C$ . Clause 1 is empty, $G/H = G$ , and $\nu$ is a left Haar measure. So $\operatorname{Ind}_{\set{e}}^G 1$ is the left regular representation of $G$ on $L^2(G)$ .

Example: take $G = \R$ , $H = \Z$ , and $\sigma(n) = e^{2 \pi i \theta n}$ for a fixed $\theta \in \R$ . Clause 1 asks that $f(x + n) = e^{-2 \pi i \theta n} f(x)$ , and clause 2 asks that $\int_0^1 \abs{f(x)}^2 \, dx < \infty$ .

Non-example: in the second example, the constant function $f = 1$ fails clause 1 when $\theta \notin \Z$ .

When $G/H$ has no invariant measure, one uses a quasi-invariant measure instead. The action then includes the square root of a Radon–Nikodym factor, so that $\lambda(x)$ stays unitary. For finite index and finite groups, the result agrees with the induced representation of finite group theory.

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Haar measure concept
Hilbert space concept
Left coset concept
Locally compact group concept
Unitary representation concept
Induced representation of a locally compact group definition

Tag	`5J603A6DFH`
URI	`m2://5J603A6DFH`
Kind	definition
Writing	W0 stub
MSC	22D30, 22D10
Defines	Induced representation of a locally compact group
Uses	Locally compact group, Haar measure, Unitary representation, Hilbert space, Left coset
Changed	2 November 2026

Induced representation of a locally compact group

Definition

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