View source for Representation of a compact group(2)

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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.

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|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 $  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|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|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|Representation with a highest weight vector]]).

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

====References====
<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></table>



====Comments====


====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>