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.

Comments

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. Wanner, "Harmonic analysis on semi-simple Lie groups" , 1 , Springer (1972) pp. Sect. 4.5.9
[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)