Integrable representation
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) |
Integrable representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrable_representation&oldid=47362