---
tag: 5J603A6DFH
kind: definition
name: Induced representation of a locally compact group
msc: [22D30, 22D10]
defines: 6Q7AHAZ6Q4
uses: [AHV05ZBCYW, VY0HXD3GC4, 74BPYJNZTM, HM2HHE3J51, 03DEH43F4H]
sources:
  - url: https://en.wikipedia.org/wiki/Induced_representation
---

## Definition

Let $G$ be a second countable [[locally compact group|AHV05ZBCYW]], 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|74BPYJNZTM]] of $H$ on a [[Hilbert space|HM2HHE3J51]] $K$.
Let $\mathcal{H}$ be the space of measurable functions $f \colon G \to K$ such that:

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

Functions that agree almost everywhere are identified.
The norm $\norm{f(g)}$ depends only on the [[coset|03DEH43F4H]] $gH$, 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|VY0HXD3GC4]].
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|JJP4JFH3C6]] of finite group theory.