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A [[Homomorphism|homomorphism]] of a [[Compact group|compact group]] into the group of continuous linear automorphisms of a (complex) Banach space that is continuous with respect to the strong operator topology.
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A [[homomorphism]] of a [[compact group]] into the group of continuous linear automorphisms of a (complex) Banach space that is continuous with respect to the strong operator topology.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813701.png" /> be a compact group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813702.png" /> be a Banach space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813703.png" /> be a representation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813704.png" /> is a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813705.png" /> is a unitary operator for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813706.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813707.png" /> is called a [[Unitary representation|unitary representation]]. There always is an equivalent norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813708.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r0813709.png" /> is unitary.
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Let $  G $
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be a compact group, let $  V $
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be a Banach space and let $  \phi : \  G \rightarrow  \mathop{\rm GL}\nolimits (V) $
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be a representation. If $  V = H $
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is a Hilbert space and $  \phi (g) $
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is a unitary operator for every $  g \in G $ ,  
 +
then $  \phi $
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is called a [[Unitary representation|unitary representation]]. There always is an equivalent norm in $  H $
 +
for which $  \phi $
 +
is unitary.
  
Every irreducible unitary representation (cf. [[Irreducible representation|Irreducible representation]]) of a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137010.png" /> is finite-dimensional. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137011.png" /> be the family of all possible pairwise inequivalent irreducible unitary representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137012.png" />. Every unitary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137013.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137014.png" /> is an orthogonal sum of unique representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137016.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137017.png" /> is an orthogonal sum, possibly zero, of a set of representations equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137018.png" />.
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Every irreducible unitary representation (cf. [[Irreducible representation|Irreducible representation]]) of a compact group $  G $
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is finite-dimensional. Let $  \{ {\rho ^ \alpha } : {\alpha \in I} \} $
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be the family of all possible pairwise inequivalent irreducible unitary representations of the group $  G $ .  
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Every unitary representation $  \phi $
 +
of $  G $
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is an orthogonal sum of unique representations $  \phi ^ \alpha  $ ,  
 +
$  \alpha \in I $ ,  
 +
such that $  \phi ^ \alpha  $
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is an orthogonal sum, possibly zero, of a set of representations equivalent to $  \rho ^ \alpha  $ .
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137019.png" /> is finite, then the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137020.png" /> is also finite and contains as many elements as there are distinct conjugacy classes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137021.png" /> (moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137022.png" />). The problem of studying these representations (computing their characters, finding explicit realizations, etc.) is the subject of an extensive theory (cf. [[Finite group, representation of a|Finite group, representation of a]]).
 
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137023.png" /> is a connected, simply-connected, compact Lie group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137024.png" /> is its complexification (cf. [[Complexification of a Lie group|Complexification of a Lie group]]), then the description of the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137025.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137026.png" /> amounts (by restricting the representations to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137027.png" />) to the description of the family of all irreducible pairwise inequivalent finite-dimensional rational representations of the reductive algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137028.png" />. The latter family, in turn, allows of a complete description by considering highest weights (cf. [[Representation with a highest weight vector|Representation with a highest weight vector]]).
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If $  G $
 +
is finite, then the family $  \{ \rho ^ \alpha  \} $
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is also finite and contains as many elements as there are distinct conjugacy classes in $  G $ (
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moreover, $  \sum _ {\alpha \in I} ( \mathop{\rm dim}\nolimits \  \rho ^ \alpha  ) ^{2} = | G | $ ).  
 +
The problem of studying these representations (computing their characters, finding explicit realizations, etc.) is the subject of an extensive theory (cf. [[Finite group, representation of a|Finite group, representation of a]]).
  
In modern number theory and algebraic geometry one considers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137029.png" />-adic representations of compact totally-disconnected groups (cf. [[#References|[5]]], [[#References|[6]]]).
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If $  G $
 
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is a connected, simply-connected, compact Lie group and $  G _ {\mathbf C} $
====References====
+
is its complexification (cf. [[Complexification of a Lie group|Complexification of a Lie group]]), then the description of the family $  \{ {\rho ^ \alpha } : {\alpha \in I} \} $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Lang, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081370/r08137030.png" />" , Addison-Wesley (1975)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.M. Gel'fand, M.I. Graev, I.I. Pyatetskii-Shapiro, "Generalized functions" , '''6. Representation theory and automorphic functions''' , Saunders (1969) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0181643}} {{ZBL|0143.05901}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946) {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </TD></TR></table>
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for $  G $
 
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amounts (by restricting the representations to $  G $ )
 
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to the description of the family of all irreducible pairwise inequivalent finite-dimensional rational representations of the reductive algebraic group $  G _ {\mathbf C} $ .  
 
+
The latter family, in turn, allows of a complete description by considering highest weights (cf. [[Representation with a highest weight vector|Representation with a highest weight vector]]).
====Comments====
 
  
 +
In modern number theory and algebraic geometry one considers $  \ell $-adic representations of compact totally-disconnected groups (cf. [[#References|[5]]], [[#References|[6]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , ''Eléments de mathématiques'' , Masson (1982) pp. Chapt. 9. Groupes de Lie réels compacts {{MR|0682756}} {{ZBL|0505.22006}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) {{MR|0781344}} {{ZBL|0581.22009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , '''II''' , Springer (1970) {{MR|0262773}} {{ZBL|0213.40103}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Wawrzyńczyk, "Group representations and special functions" , Reidel &amp; PWN (1984) {{MR|0750113}} {{ZBL|0545.43001}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) {{MR|0793377}} {{ZBL|0484.22018}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) {{MR|0473097}} {{MR|0473098}} {{ZBL|0228.22013}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Lang, "${\rm SL}_2({\bf R})$", Addison-Wesley (1975)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> I.M. Gel'fand, M.I. Graev, I.I. Pyatetskii-Shapiro, "Generalized functions" , '''6. Representation theory and automorphic functions''' , Saunders (1969) (Translated from Russian) {{MR|}} {{ZBL|0801.33020}} {{ZBL|0699.33012}} {{ZBL|0159.18301}} {{ZBL|0355.46017}} {{ZBL|0144.17202}} {{ZBL|0115.33101}} {{ZBL|0108.29601}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0181643}} {{ZBL|0143.05901}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> C. Chevalley, "Theory of Lie groups" , '''1''' , Princeton Univ. Press (1946) {{MR|0082628}} {{MR|0015396}} {{ZBL|0063.00842}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Groupes et algèbres de Lie" , ''Eléments de mathématiques'' , Masson (1982) pp. Chapt. 9. Groupes de Lie réels compacts {{MR|0682756}} {{ZBL|0505.22006}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) {{MR|0781344}} {{ZBL|0581.22009}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , '''II''' , Springer (1970) {{MR|0262773}} {{ZBL|0213.40103}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Wawrzyńczyk, "Group representations and special functions" , Reidel &amp; PWN (1984) {{MR|0750113}} {{ZBL|0545.43001}} </TD></TR>
 +
</table>

Latest revision as of 06:35, 19 May 2024

A homomorphism of a compact group into the group of continuous linear automorphisms of a (complex) Banach space that is continuous with respect to the strong operator topology.

Let $ G $ be a compact group, let $ V $ be a Banach space and let $ \phi : \ G \rightarrow \mathop{\rm GL}\nolimits (V) $ be a representation. If $ V = H $ is a Hilbert space and $ \phi (g) $ is a unitary operator for every $ g \in G $ , then $ \phi $ is called a unitary representation. There always is an equivalent norm in $ H $ for which $ \phi $ is unitary.

Every irreducible unitary representation (cf. Irreducible representation) of a compact group $ G $ is finite-dimensional. Let $ \{ {\rho ^ \alpha } : {\alpha \in I} \} $ be the family of all possible pairwise inequivalent irreducible unitary representations of the group $ G $ . Every unitary representation $ \phi $ of $ G $ is an orthogonal sum of unique representations $ \phi ^ \alpha $ , $ \alpha \in I $ , such that $ \phi ^ \alpha $ is an orthogonal sum, possibly zero, of a set of representations equivalent to $ \rho ^ \alpha $ .


If $ G $ is finite, then the family $ \{ \rho ^ \alpha \} $ is also finite and contains as many elements as there are distinct conjugacy classes in $ G $ ( moreover, $ \sum _ {\alpha \in I} ( \mathop{\rm dim}\nolimits \ \rho ^ \alpha ) ^{2} = | G | $ ). The problem of studying these representations (computing their characters, finding explicit realizations, etc.) is the subject of an extensive theory (cf. Finite group, representation of a).

If $ G $ is a connected, simply-connected, compact Lie group and $ G _ {\mathbf C} $ is its complexification (cf. Complexification of a Lie group), then the description of the family $ \{ {\rho ^ \alpha } : {\alpha \in I} \} $ for $ G $ amounts (by restricting the representations to $ G $ ) to the description of the family of all irreducible pairwise inequivalent finite-dimensional rational representations of the reductive algebraic group $ G _ {\mathbf C} $ . The latter family, in turn, allows of a complete description by considering highest weights (cf. Representation with a highest weight vector).

In modern number theory and algebraic geometry one considers $ \ell $-adic representations of compact totally-disconnected groups (cf. [5], [6]).

References

[1] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) MR0201557 Zbl 0022.17104
[2] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[3] D.P. Zhelobenko, "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian) MR0473097 MR0473098 Zbl 0228.22013
[4] S. Lang, "${\rm SL}_2({\bf R})$", Addison-Wesley (1975)
[5] I.M. Gel'fand, M.I. Graev, I.I. Pyatetskii-Shapiro, "Generalized functions" , 6. Representation theory and automorphic functions , Saunders (1969) (Translated from Russian) Zbl 0801.33020 Zbl 0699.33012 Zbl 0159.18301 Zbl 0355.46017 Zbl 0144.17202 Zbl 0115.33101 Zbl 0108.29601
[6] J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0181643 Zbl 0143.05901 Zbl 0128.26303
[7] C. Chevalley, "Theory of Lie groups" , 1 , Princeton Univ. Press (1946) MR0082628 MR0015396 Zbl 0063.00842
[a1] N. Bourbaki, "Groupes et algèbres de Lie" , Eléments de mathématiques , Masson (1982) pp. Chapt. 9. Groupes de Lie réels compacts MR0682756 Zbl 0505.22006
[a2] Th. Bröcker, T. Tom Dieck, "Representations of compact Lie groups" , Springer (1985) MR0781344 Zbl 0581.22009
[a3] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , II , Springer (1970) MR0262773 Zbl 0213.40103
[a4] A. Wawrzyńczyk, "Group representations and special functions" , Reidel & PWN (1984) MR0750113 Zbl 0545.43001
How to Cite This Entry:
Representation of a compact group(2). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_a_compact_group(2)&oldid=21924
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article