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A continuous irreducible [[Unitary representation|unitary representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513201.png" /> of a locally compact [[Unimodular group|unimodular group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513202.png" /> in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513203.png" /> such that for some non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513204.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513206.png" />, is integrable with respect to the [[Haar measure|Haar measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513207.png" />. In this case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513208.png" /> is a square-integrable representation and there exists a dense vector subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i0513209.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132011.png" />, is an integrable function with respect to the Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132013.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132014.png" />, the unitary equivalence class of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132015.png" />, denotes the corresponding element of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132016.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132017.png" />, then the singleton set containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132018.png" /> is both open and closed in the support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132019.png" /> of the [[Regular representation|regular representation]].
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A continuous irreducible [[Unitary representation|unitary representation]]  $  \pi $
 +
of a locally compact [[Unimodular group|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|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|regular representation]].
  
 
====Comments====
 
====Comments====
Instead of integrable representation one usually finds square-integrable representation in the literature. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132021.png" /> be two square-integrable representations; then the following orthogonality relations hold:
+
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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132022.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { G } ( \pi ( g) \xi , \eta )
 +
\overline{ {( \pi  ^  \prime  ( g) \xi  ^  \prime  , \eta  ^  \prime  ) }}\; \
 +
d g =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132023.png" /></td> </tr></table>
+
$$
 +
= \
 +
\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}
  
where the integral is with respect to Haar measure. The scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132024.png" /> is called the formal degree or formal dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132025.png" />. It depends on the normalization of the Haar measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132027.png" /> is compact, then every irreducible unitary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132028.png" /> is square integrable and finite dimensional, and if Haar measure is normalized so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132029.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132030.png" /> is its dimension.
+
\right .$$
  
The square-integrable representations are precisely the irreducible subrepresentations of the left (or right) regular representation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051320/i05132031.png" /> and occur as discrete direct summands.
+
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====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Wanner,  "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)  pp. Sect. 4.5.9</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.A. Gaal,  "Linear analysis and representation theory" , Springer  (1973)  pp. Chapt. VII</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  pp. 138  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Wanner,  "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer  (1972)  pp. Sect. 4.5.9</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.A. Gaal,  "Linear analysis and representation theory" , Springer  (1973)  pp. Chapt. VII</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  pp. 138  (Translated from Russian)</TD></TR></table>

Revision as of 22:12, 5 June 2020


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)
How to Cite This Entry:
Integrable representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integrable_representation&oldid=47362
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article