# Integrable representation

A continuous irreducible unitary representation $\pi$ of a locally compact unimodular group $G$ in a Hilbert space $H$ such that for some non-zero vector $\xi \in H$ the function $g \mapsto ( \pi ( g) \xi , \xi )$, $g \in G$, is integrable with respect to the Haar measure on $G$. In this case, $\pi$ is a square-integrable representation and there exists a dense vector subspace $H ^ \prime \subset H$ such that $g \mapsto ( \pi ( g) \xi , \eta )$, $g \in G$, is an integrable function with respect to the Haar measure on $G$ for all $\xi , \eta \in H ^ \prime$. If $\{ \pi \}$, the unitary equivalence class of the representation $\pi$, denotes the corresponding element of the dual space $\widehat{G}$ of $G$, then the singleton set containing $\{ \pi \}$ is both open and closed in the support $\widehat{G} _ {r}$ of the regular representation.

Instead of integrable representation one usually finds square-integrable representation in the literature. Let $\pi$ and $\pi ^ \prime$ be two square-integrable representations; then the following orthogonality relations hold:

$$\int\limits _ { G } ( \pi ( g) \xi , \eta ) \overline{ {( \pi ^ \prime ( g) \xi ^ \prime , \eta ^ \prime ) }}\; \ d g =$$

$$= \ \left \{ \begin{array}{ll} 0 &\textrm{ if } \pi \textrm{ and } \pi ^ \prime \textrm{ are not equivalent , } \\ d _ \pi ^ {-1} ( \xi , \xi ^ \prime ) ( \eta , \eta ^ \prime ) &\textrm{ if } \pi = \pi ^ \prime , \\ \end{array} \right .$$

where the integral is with respect to Haar measure. The scalar $d _ \pi$ is called the formal degree or formal dimension of $\pi$. It depends on the normalization of the Haar measure $d g$. If $G$ is compact, then every irreducible unitary representation $\pi$ is square integrable and finite dimensional, and if Haar measure is normalized so that $\int _ {G} dg = 1$, then $d _ \pi$ is its dimension.

The square-integrable representations are precisely the irreducible subrepresentations of the left (or right) regular representation on $L _ {2} ( G)$ and occur as discrete direct summands.

#### References

 [a1] G. Warner, "Harmonic analysis on semi-simple Lie groups" , 1 , Springer (1972) pp. Sect. 4.5.9 Zbl 0265.22020 [a2] S.A. Gaal, "Linear analysis and representation theory" , Springer (1973) pp. Chapt. VII [a3] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) pp. 138 (Translated from Russian)
How to Cite This Entry:
Integrable representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrable_representation&oldid=52797
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article