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Integrable representation

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A continuous irreducible unitary representation of a locally compact unimodular group in a Hilbert space such that for some non-zero vector the function , , is integrable with respect to the Haar measure on . In this case, is a square-integrable representation and there exists a dense vector subspace such that , , is an integrable function with respect to the Haar measure on for all . If , the unitary equivalence class of the representation , denotes the corresponding element of the dual space of , then the singleton set containing is both open and closed in the support of the regular representation.


Comments

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

where the integral is with respect to Haar measure. The scalar is called the formal degree or formal dimension of . It depends on the normalization of the Haar measure . If is compact, then every irreducible unitary representation is square integrable and finite dimensional, and if Haar measure is normalized so that , then is its dimension.

The square-integrable representations are precisely the irreducible subrepresentations of the left (or right) regular representation on 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)
How to Cite This Entry:
Integrable representation. A.I. Shtern (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Integrable_representation&oldid=11835
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098