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A mapping of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814201.png" /> into the group of homeomorphisms of a topological space. Most often such a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814202.png" /> is understood to be a [[Linear representation|linear representation]], moreover, a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814203.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814204.png" /> into a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814205.png" /> such that the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814207.png" />, defines for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814208.png" /> a continuous mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r0814209.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142010.png" />. In particular, every [[Continuous representation|continuous representation]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142011.png" /> is a representation of the topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142012.png" />.
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The theory of representations of topological groups is strongly connected with the representation theory of various topological group algebras (cf. [[Group algebra|Group algebra]]). The most important among these is the symmetric Banach measure algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142013.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142014.png" /> (the algebra of all regular Borel measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142015.png" /> with finite total variation, in which multiplication is defined as convolution). Often one also uses the topological algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142016.png" /> of all regular Borel measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142017.png" /> with finite total variation and with compact support. Multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142018.png" /> is defined as convolution, and the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142020.png" />, is defined by
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/r/r081/r081420/r08142021.png" /></td> </tr></table>
+
A mapping of the group  $  G $
 +
into the group of homeomorphisms of a topological space. Most often such a representation of  $  G $
 +
is understood to be a [[Linear representation|linear representation]], moreover, a linear representation  $  \pi $
 +
of  $  G $
 +
into a topological vector space  $  E $
 +
such that the vector function  $  g \rightarrow \pi (g) x $,
 +
$  g \in G $,
 +
defines for any  $  x \in E $
 +
a continuous mapping of  $  G $
 +
into  $  E $.  
 +
In particular, every [[Continuous representation|continuous representation]] of the group  $  G $
 +
is a representation of the topological group  $  G $.
  
The topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142022.png" /> is compatible with the duality between this algebra and the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142023.png" /> (of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142024.png" />), equipped with the compact-open topology. Various subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142026.png" /> also play an important role. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142027.png" /> is a quasi-complete barrelled or complete locally convex space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142028.png" /> is a continuous representation of the topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142029.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142030.png" />, then the formula
+
The theory of representations of topological groups is strongly connected with the representation theory of various topological group algebras (cf. [[Group algebra|Group algebra]]). The most important among these is the symmetric Banach measure algebra $  M (G) $
 +
of the group  $  G $(
 +
the algebra of all regular Borel measures on  $  G $
 +
with finite total variation, in which multiplication is defined as convolution). Often one also uses the topological algebra  $  C  ^  \prime  (G) $
 +
of all regular Borel measures on $  G $
 +
with finite total variation and with compact support. Multiplication in  $  C  ^  \prime  (G) $
 +
is defined as convolution, and the involution  $  \mu \rightarrow \mu  ^ {*} $,
 +
$  \mu \in C  ^  \prime  (G) $,  
 +
is defined by
  
<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/r/r081/r081420/r08142031.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { G }
 +
f (g)  d \mu  ^ {*} (g)  = \
 +
\int\limits _ { G }
 +
\overline{ {f (g  ^ {-1} ) }}\; \
 +
d \mu (g),\ \
 +
f \in C (G).
 +
$$
  
defines a weakly-continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142033.png" />, and the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142034.png" /> is a representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142035.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142036.png" />, uniquely defining the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142037.png" /> of the topological group. Here, a representation of a topological group, a (topologically) [[Irreducible representation|irreducible representation]], an [[Operator-irreducible representation|operator-irreducible representation]], a totally irreducible representation, is equivalent to another representation of the topological group, etc., if and only if the corresponding representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142038.png" /> have the corresponding property.
+
The topology of  $  C  ^  \prime  (G) $
 +
is compatible with the duality between this algebra and the algebra  $  C (G) $(
 +
of all continuous functions on  $  G $),
 +
equipped with the compact-open topology. Various subalgebras of $  M (G) $
 +
and  $  C  ^  \prime  (G) $
 +
also play an important role. In particular, if  $  E $
 +
is a quasi-complete barrelled or complete locally convex space and  $  \pi $
 +
is a continuous representation of the topological group $  G $
 +
into  $  E $,  
 +
then the formula
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142039.png" /> be a representation of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142040.png" /> in a locally convex vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142041.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142042.png" /> be the space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142043.png" />. Functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142044.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142047.png" />, are called matrix elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142048.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142049.png" /> is a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142051.png" />, then functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142053.png" />, are called spherical functions, corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142054.png" />.
+
$$
 +
\pi ( \mu )  = \
 +
\int\limits _ { G }
 +
\pi (g)  d \mu (g),\ \
 +
\mu \in C  ^  \prime  (G),
 +
$$
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142055.png" /> are dual locally convex spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142056.png" /> be a representation of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142057.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142058.png" />. The formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142059.png" /> defines a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142060.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142061.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142062.png" />, called the adjoint, or contragredient, representation to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142063.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142064.png" /> are representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142065.png" /> in locally convex spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142066.png" />, respectively, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142067.png" /> be the direct sum and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142069.png" />, be the continuous linear operator into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142070.png" /> defined by
+
defines a weakly-continuous linear operator  $  \pi ( \mu ) $
 +
on  $  E $,
 +
and the correspondence  $  \mu \rightarrow \pi ( \mu ) $
 +
is a representation of the algebra  $  C  ^  \prime  (G) $
 +
in  $  E $,
 +
uniquely defining the representation  $  \pi $
 +
of the topological group. Here, a representation of a topological group, a (topologically) [[Irreducible representation|irreducible representation]], an [[Operator-irreducible representation|operator-irreducible representation]], a totally irreducible representation, is equivalent to another representation of the topological group, etc., if and only if the corresponding representations of the algebra  $  C  ^  \prime  (G) $
 +
have the corresponding property.
  
<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/r/r081/r081420/r08142071.png" /></td> </tr></table>
+
Let  $  \pi $
 +
be a representation of a topological group  $  G $
 +
in a locally convex vector space  $  E $
 +
and let  $  E  ^  \prime  $
 +
be the space dual to  $  E $.  
 +
Functions on  $  G $
 +
of the form  $  g \rightarrow \phi ( \pi (g) \xi ) $,
 +
$  \xi \in E $,
 +
$  \phi \in E  ^  \prime  $,
 +
are called matrix elements of  $  \pi $.  
 +
If  $  E $
 +
is a Hilbert space and  $  \xi \in E $,
 +
$  \| \xi \| = 1 $,
 +
then functions of the form  $  g \rightarrow \langle  \pi (g) \xi , \xi \rangle $,
 +
$  g \in G $,
 +
are called spherical functions, corresponding to  $  \pi $.
  
The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142072.png" /> is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142073.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142074.png" />, called the direct sum of the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142076.png" />. In certain situations (in particular for unitary representations) one can define the tensor product of representations of a topological group and the direct sum of an infinite family of such representations. By restricting or extending the field of scalars, one introduces the operations of "realification" or complexification of representations.
+
Suppose that  $  E, E  ^ {*} $
 +
are dual locally convex spaces and let  $  \pi $
 +
be a representation of a topological group  $  G $
 +
in  $  E $.
 +
The formula  $  \pi  ^ {*} (g) = \pi (g  ^ {-1} )  ^ {*} $
 +
defines a representation $  \pi  ^ {*} $
 +
of $  G $
 +
in $  E  ^ {*} $,  
 +
called the adjoint, or contragredient, representation to  $  \pi $.  
 +
Suppose that  $  \pi _ {1} , \pi _ {2} $
 +
are representations of $  G $
 +
in locally convex spaces  $  E _ {1} , E _ {2} $,
 +
respectively, let  $  E = E _ {1} + E _ {2} $
 +
be the direct sum and let  $  \pi (g) $,
 +
$  g \in G $,  
 +
be the continuous linear operator into $ E $
 +
defined by
  
A representation of a topological group is called completely reducible if every closed invariant subspace has a complementary closed invariant subspace. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142077.png" /> of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142078.png" /> in a locally convex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142079.png" /> is called split (decomposable) if there exist closed invariant subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142080.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142081.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142082.png" /> is equivalent to the direct sum of the subrepresentations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142084.png" /> corresponding to the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142085.png" />, respectively; in the opposite case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142086.png" /> is called non-split (indecomposable). A non-split reducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142087.png" /> is not solely determined by its subrepresentation and quotient representation corresponding to a given invariant subspace, but also requires for its characterization certain one-dimensional cohomology classes of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142088.png" /> with coefficients in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142089.png" />-module of bounded linear operators from the space of the quotient representation into the space of the representation.
+
$$
 +
\pi (g) (x _ {1} + x _ {2} )  = \
 +
\pi _ {1} (g) x _ {1} +
 +
\pi _ {2} (g) x _ {2} ,\ \
 +
x _ {1} \in E _ {1} ,\ \
 +
x _ {2} \in E _ {2} .
 +
$$
 +
 
 +
The mapping  $  g \rightarrow \pi (g) $
 +
is a representation of  $  G $
 +
in  $  E $,
 +
called the direct sum of the representations  $  \pi _ {1} $
 +
and  $  \pi _ {2} $.
 +
In certain situations (in particular for unitary representations) one can define the tensor product of representations of a topological group and the direct sum of an infinite family of such representations. By restricting or extending the field of scalars, one introduces the operations of "realification" or complexification of representations.
 +
 
 +
A representation of a topological group is called completely reducible if every closed invariant subspace has a complementary closed invariant subspace. A representation $  \pi $
 +
of a topological group $  G $
 +
in a locally convex space $  E $
 +
is called split (decomposable) if there exist closed invariant subspaces $  E _ {1} , E _ {2} $
 +
in $  E $
 +
such that $  \pi $
 +
is equivalent to the direct sum of the subrepresentations $  \pi _ {1} , \pi _ {2} $
 +
of $  \pi $
 +
corresponding to the subspaces $  E _ {1} , E _ {2} $,  
 +
respectively; in the opposite case $  \pi $
 +
is called non-split (indecomposable). A non-split reducible representation $  \pi $
 +
is not solely determined by its subrepresentation and quotient representation corresponding to a given invariant subspace, but also requires for its characterization certain one-dimensional cohomology classes of the group $  G $
 +
with coefficients in the $  G $-
 +
module of bounded linear operators from the space of the quotient representation into the space of the representation.
  
 
The most important general problems in the representation theory of topological groups are the description of all non-split representations of a given topological group and the study of the description (decomposition) of arbitrary representations of a topological group in terms of non-split ones. In both cases the problems are far from being completely solved (1991), but the results obtained still suffice to make the representation theory of topological groups a basis for abstract harmonic analysis (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]), generalizing the theory of Fourier series and integrals, the spectral theory of unitary operators, the theory of Jordan normal forms and systems of ordinary differential equations with constant coefficients, as well as a basis for certain branches of ergodic theory, quantum mechanics, statistical physics, and field theory.
 
The most important general problems in the representation theory of topological groups are the description of all non-split representations of a given topological group and the study of the description (decomposition) of arbitrary representations of a topological group in terms of non-split ones. In both cases the problems are far from being completely solved (1991), but the results obtained still suffice to make the representation theory of topological groups a basis for abstract harmonic analysis (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]), generalizing the theory of Fourier series and integrals, the spectral theory of unitary operators, the theory of Jordan normal forms and systems of ordinary differential equations with constant coefficients, as well as a basis for certain branches of ergodic theory, quantum mechanics, statistical physics, and field theory.
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The most important branch of the representation theory of topological groups is the theory of unitary representations (cf. [[Unitary representation|Unitary representation]]), which has many applications. A number of special properties simplify their study. In particular, the orthogonal complement to an invariant subspace of a unitary representation is invariant, and therefore every unitary representation is completely reducible. For unitary representations, the conditions of total irreducibility, (topological) irreducibility and operator-irreducibility are equivalent (but, in general, are weaker than the condition of algebraic irreducibility).
 
The most important branch of the representation theory of topological groups is the theory of unitary representations (cf. [[Unitary representation|Unitary representation]]), which has many applications. A number of special properties simplify their study. In particular, the orthogonal complement to an invariant subspace of a unitary representation is invariant, and therefore every unitary representation is completely reducible. For unitary representations, the conditions of total irreducibility, (topological) irreducibility and operator-irreducibility are equivalent (but, in general, are weaker than the condition of algebraic irreducibility).
  
Another class of representations of topological groups which has various applications is that of finite-dimensional representations (cf. [[Finite-dimensional representation|Finite-dimensional representation]]). The study of representations of this class is greatly facilitated by the relative simplicity of the functional-analytic problems as compared to the general case; in particular, an irreducible finite-dimensional representation is totally irreducible. However, the theory of finite-dimensional representations of topological groups has been developed satisfactorily (1991) only for certain classes of such groups (in particular, for semi-simple Lie groups and for the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142091.png" />). For many classes of groups, including the class of connected Lie groups, there is a complete description of the irreducible finite-dimensional representations.
+
Another class of representations of topological groups which has various applications is that of finite-dimensional representations (cf. [[Finite-dimensional representation|Finite-dimensional representation]]). The study of representations of this class is greatly facilitated by the relative simplicity of the functional-analytic problems as compared to the general case; in particular, an irreducible finite-dimensional representation is totally irreducible. However, the theory of finite-dimensional representations of topological groups has been developed satisfactorily (1991) only for certain classes of such groups (in particular, for semi-simple Lie groups and for the groups $  \mathbf R $
 +
and $  \mathbf Z $).  
 +
For many classes of groups, including the class of connected Lie groups, there is a complete description of the irreducible finite-dimensional representations.
  
The theory of representations has been mostly developed for locally compact groups. A most important property of the class of locally compact groups is that it coincides with the class of complete topological groups on which there is a non-zero right-invariant regular Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142092.png" /> (cf. [[Haar measure|Haar measure]]). This allows one to add to the useful group algebras of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142093.png" />, the symmetric Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142094.png" /> (under convolution), which plays a decisive role in the theory of bounded representations of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142095.png" /> in Banach spaces (i.e. representations having bounded image). The formula
+
The theory of representations has been mostly developed for locally compact groups. A most important property of the class of locally compact groups is that it coincides with the class of complete topological groups on which there is a non-zero right-invariant regular Borel measure $  m $(
 +
cf. [[Haar measure|Haar measure]]). This allows one to add to the useful group algebras of a locally compact group $  G $,  
 +
the symmetric Banach algebra $  L _ {1} (G) = L _ {1} (G, m) $(
 +
under convolution), which plays a decisive role in the theory of bounded representations of a topological group $  G $
 +
in Banach spaces (i.e. representations having bounded image). The formula
  
<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/r/r081/r081420/r08142096.png" /></td> </tr></table>
+
$$
 +
\pi (f)  = \
 +
\int\limits _ { G }
 +
f (g) \pi (g)  dm (g),\ \
 +
f \in L _ {1} (G),
 +
$$
  
establishes a one-to-one correspondence between the bounded representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142097.png" /> of a locally compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142098.png" /> and the (continuous) representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r08142099.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420100.png" /> with the property that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420101.png" /> is dense in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420102.png" /> of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420103.png" />. Here, unitary representations of the group correspond to symmetric representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420104.png" />. Another property of locally compact groups is that their representations in barrelled locally convex spaces are jointly continuous.
+
establishes a one-to-one correspondence between the bounded representations $  \pi $
 +
of a locally compact group $  G $
 +
and the (continuous) representations $  \widetilde \pi  $
 +
of the algebra $  L _ {1} (G) $
 +
with the property that $  \widetilde \pi  (L _ {1} (G)) H _ {\widetilde \pi  }  $
 +
is dense in the space $  H _ {\widetilde \pi  }  $
 +
of the representation $  \widetilde \pi  $.  
 +
Here, unitary representations of the group correspond to symmetric representations of $  L _ {1} (G) $.  
 +
Another property of locally compact groups is that their representations in barrelled locally convex spaces are jointly continuous.
  
The theory of unitary representations of locally compact groups is the most fully developed branch of the representation theory of topological groups. Related to the existence of a Haar measure on locally compact groups is the possibility of studying the [[Regular representation|regular representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420105.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420106.png" />, which leads, in particular, to an analogue of the [[Plancherel formula|Plancherel formula]] for such groups, as well as to singling out the basic, complementary and discrete series of unitary representations of groups of the class considered (cf. [[Complementary series (of representations)|Complementary series (of representations)]]; [[Discrete series (of representations)|Discrete series (of representations)]]). Important general problems in the theory of unitary representations are the problems of constructing irreducible representations and quotient representations, of decomposing representations into a direct integral, and of studying dual objects, and the problems of the theory of spherical functions, characters and harmonic analysis related to them, including the study of various group algebras.
+
The theory of unitary representations of locally compact groups is the most fully developed branch of the representation theory of topological groups. Related to the existence of a Haar measure on locally compact groups is the possibility of studying the [[Regular representation|regular representation]] of $  G $
 +
in $  L _ {2} (G) $,  
 +
which leads, in particular, to an analogue of the [[Plancherel formula|Plancherel formula]] for such groups, as well as to singling out the basic, complementary and discrete series of unitary representations of groups of the class considered (cf. [[Complementary series (of representations)|Complementary series (of representations)]]; [[Discrete series (of representations)|Discrete series (of representations)]]). Important general problems in the theory of unitary representations are the problems of constructing irreducible representations and quotient representations, of decomposing representations into a direct integral, and of studying dual objects, and the problems of the theory of spherical functions, characters and harmonic analysis related to them, including the study of various group algebras.
  
 
A subclass of the class of locally compact groups that is exceptionally rich in applications is the class of Lie groups. The theory of infinite-dimensional representations (cf. [[Infinite-dimensional representation|Infinite-dimensional representation]]) of Lie groups, including the representation theory of the classical groups, is one of the most quickly developing branches of the general representation theory of topological groups. A powerful method in the study of representations of Lie groups is the [[Orbit method|orbit method]].
 
A subclass of the class of locally compact groups that is exceptionally rich in applications is the class of Lie groups. The theory of infinite-dimensional representations (cf. [[Infinite-dimensional representation|Infinite-dimensional representation]]) of Lie groups, including the representation theory of the classical groups, is one of the most quickly developing branches of the general representation theory of topological groups. A powerful method in the study of representations of Lie groups is the [[Orbit method|orbit method]].
Line 39: Line 174:
 
Another important subclass of the class of locally compact groups is the class of compact groups. The representation theory of compact groups is one of the most complete branches of the general representation theory of topological groups, and is a tool in the study of representations of topological groups containing compact subgroups. An important branch of the representation theory of compact groups concerns the decomposition of restrictions to subgroups, and the decomposition of tensor products of concrete representations of compact Lie groups. A part of the representation theory of compact groups with many applications in algebra and analysis is the theory of representations of finite groups.
 
Another important subclass of the class of locally compact groups is the class of compact groups. The representation theory of compact groups is one of the most complete branches of the general representation theory of topological groups, and is a tool in the study of representations of topological groups containing compact subgroups. An important branch of the representation theory of compact groups concerns the decomposition of restrictions to subgroups, and the decomposition of tensor products of concrete representations of compact Lie groups. A part of the representation theory of compact groups with many applications in algebra and analysis is the theory of representations of finite groups.
  
Like in the above-mentioned study of non-split representations of topological groups, even the simpler problem of describing the intertwining of totally irreducible representations, related with a corresponding cohomology theory, has only been solved (1991) for certain groups, despite its importance in the harmonic analysis on groups. In fact, in terms of non-split representations (more precisely, in terms of representations participating in the analytic extension of the corresponding basic series) for certain Lie groups (respectively, Chevalley groups) one has obtained analogues of the Paley–Wiener theorem, giving a description of the image of the group algebra of infinitely-differentiable (respectively, locally finite) functions with compact support on the group under Fourier transformation (i.e. under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420108.png" />, assigning to a function on the group an operator-valued function on a set of representatives of the space of equivalence classes of representations of this group). The more special problem of describing all totally-irreducible representations of a given group has been solved (1991) only for locally compact groups whose quotient group by the centre is compact (a totally-irreducible representation of such a group is finite-dimensional and the set of these representations suffices for obtaining an analogue to the Paley–Wiener theorem) and for certain linear Lie groups (including the complex semi-simple ones). As in the theory of unitary representations, in the theory of non-unitary representations one has likewise compiled a vast amount of material relating to concrete representations of certain particular groups and relating to applications to individual problems of harmonic analysis on such groups.
+
Like in the above-mentioned study of non-split representations of topological groups, even the simpler problem of describing the intertwining of totally irreducible representations, related with a corresponding cohomology theory, has only been solved (1991) for certain groups, despite its importance in the harmonic analysis on groups. In fact, in terms of non-split representations (more precisely, in terms of representations participating in the analytic extension of the corresponding basic series) for certain Lie groups (respectively, Chevalley groups) one has obtained analogues of the Paley–Wiener theorem, giving a description of the image of the group algebra of infinitely-differentiable (respectively, locally finite) functions with compact support on the group under Fourier transformation (i.e. under the mapping $  f \rightarrow \int _ {G} f (g) \pi (g)  d \mu (g) $,  
 +
$  f \in K (G) $,  
 +
assigning to a function on the group an operator-valued function on a set of representatives of the space of equivalence classes of representations of this group). The more special problem of describing all totally-irreducible representations of a given group has been solved (1991) only for locally compact groups whose quotient group by the centre is compact (a totally-irreducible representation of such a group is finite-dimensional and the set of these representations suffices for obtaining an analogue to the Paley–Wiener theorem) and for certain linear Lie groups (including the complex semi-simple ones). As in the theory of unitary representations, in the theory of non-unitary representations one has likewise compiled a vast amount of material relating to concrete representations of certain particular groups and relating to applications to individual problems of harmonic analysis on such groups.
  
 
A number of problems of the representation theory of topological groups is related to representations in spaces with an indefinite metric (cf. [[Space with an indefinite metric|Space with an indefinite metric]]). A complete description of the irreducible representations in such spaces has been obtained for certain semi-simple Lie groups (this includes, in particular, their irreducible finite-dimensional representations). For these groups one has also found a decomposition of tensor products of certain irreducible representations of this type into irreducible unitary representations. The theory of operator-irreducible representations of semi-simple Lie groups in such spaces and the determination of the structures of their invariant subspaces is closely related with the analytic extension of the basic series of representations of these groups.
 
A number of problems of the representation theory of topological groups is related to representations in spaces with an indefinite metric (cf. [[Space with an indefinite metric|Space with an indefinite metric]]). A complete description of the irreducible representations in such spaces has been obtained for certain semi-simple Lie groups (this includes, in particular, their irreducible finite-dimensional representations). For these groups one has also found a decomposition of tensor products of certain irreducible representations of this type into irreducible unitary representations. The theory of operator-irreducible representations of semi-simple Lie groups in such spaces and the determination of the structures of their invariant subspaces is closely related with the analytic extension of the basic series of representations of these groups.
Line 46: Line 183:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.O. Barut,   R. Raçzka,   "Theory of group representations and applications" , '''1–2''' , PWN (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.Ya. Vilenkin,   "Special functions and the theory of group representations" , Amer. Math. Soc. (1968) (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top"> E. Jaquet,   R. Langlands,   "Automorphic forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420109.png" />" , '''1–2''' , Springer (1970–1972)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D.P. Zhelobenko,   "Compact Lie groups and their representations" , Amer. Math. Soc. (1973) (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> D.P. Zhelobenko,   "Harmonic analysis of functions on semi-simple complex Lie groups" , Moscow (1974) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D.P. Zhelobenko,   A.I. Shtern,   "Representations of Lie groups" , Moscow (1983) (In Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.A. Kirillov,   "Elements of the theory of representations" , Springer (1976) (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.U. Klimyk,   "Matrix elements and Clebsch–Gordan coefficients of group representations" , Kiev (1979) (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> S. Lang,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420110.png" />" , Addison-Wesley (1975)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> M.A. Naimark,   "Normed rings" , Reidel (1984) (Translated from Russian)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> M.A. Naimark,   "Theory of group representations" , Springer (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> S.A. Gaal,   "Linear analysis and representation theory" , Springer (1973)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> I.M. Gel'fand (ed.) , ''Lie groups and their representations'' , A. Hilger (1975)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> G.W. Mackey,   "Unitary group representations in physics, probability and number theory" , Benjamin/Cummings (1978)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> G. Carmona (ed.) M. Vergne (ed.) , ''Non-commutative harmonic analysis (Marseille, 1978)'' , ''Lect. notes in math.'' , '''728''' , Springer (1979)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.O. Barut, R. Raçzka, "Theory of group representations and applications" , '''1–2''' , PWN (1977) {{MR|0495836}} {{ZBL|0644.22011}} {{ZBL|0546.22015}} {{ZBL|0471.22021}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N.Ya. Vilenkin, "Special functions and the theory of group representations" , Amer. Math. Soc. (1968) (Translated from Russian) {{MR|0229863}} {{ZBL|0172.18404}} </TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top"> E. Jaquet, R. Langlands, "Automorphic forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420109.png" />" , '''1–2''' , Springer (1970–1972)</TD></TR><TR><TD valign="top">[5]</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">[6]</TD> <TD valign="top"> D.P. Zhelobenko, "Harmonic analysis of functions on semi-simple complex Lie groups" , Moscow (1974) (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> D.P. Zhelobenko, A.I. Shtern, "Representations of Lie groups" , Moscow (1983) (In Russian) {{MR|0709598}} {{ZBL|0521.22006}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) {{MR|0412321}} {{ZBL|0342.22001}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A.U. Klimyk, "Matrix elements and Clebsch–Gordan coefficients of group representations" , Kiev (1979) (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> S. Lang, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420110.png" />" , Addison-Wesley (1975)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian) {{MR|1292007}} {{MR|0355601}} {{MR|0355602}} {{MR|0205093}} {{MR|0110956}} {{MR|0090786}} {{MR|0026763}} {{ZBL|0218.46042}} {{ZBL|0137.31703}} {{ZBL|0089.10102}} {{ZBL|0073.08902}} </TD></TR><TR><TD valign="top">[12]</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">[13]</TD> <TD valign="top"> S.A. Gaal, "Linear analysis and representation theory" , Springer (1973) {{MR|0447465}} {{ZBL|0275.43008}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> I.M. Gel'fand (ed.) , ''Lie groups and their representations'' , A. Hilger (1975)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> G.W. Mackey, "Unitary group representations in physics, probability and number theory" , Benjamin/Cummings (1978) {{MR|0515581}} {{ZBL|0401.22001}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> G. Carmona (ed.) M. Vergne (ed.) , ''Non-commutative harmonic analysis (Marseille, 1978)'' , ''Lect. notes in math.'' , '''728''' , Springer (1979)</TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
See also the references to [[Representation of a group|Representation of a group]]; [[Representation of a compact group|Representation of a compact group]].
 
See also the references to [[Representation of a group|Representation of a group]]; [[Representation of a compact group|Representation of a compact group]].
  
A linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420111.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420112.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420113.png" />, the continuous linear operators on a topological vector space, is called algebraically irreducible if there are no non-trivial invariant subspaces; it is called irreducible or, to stress the topological context, topologically irreducible, if there are no non-trivial closed invariant subspaces; it is called totally irreducible, also called completely irreducible, if every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420114.png" /> is the weak limit of a net consisting of linear combinations of operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081420/r081420116.png" />; cf. [[#References|[8]]], §7, [[#References|[1]]], Chapt. V, §3; [[#References|[13]]], Chapt. IV, §2.
+
A linear representation $  \pi $
 +
of $  G $
 +
into $  {\mathcal L} (H) $,  
 +
the continuous linear operators on a topological vector space, is called algebraically irreducible if there are no non-trivial invariant subspaces; it is called irreducible or, to stress the topological context, topologically irreducible, if there are no non-trivial closed invariant subspaces; it is called totally irreducible, also called completely irreducible, if every element of $  {\mathcal L} (H) $
 +
is the weak limit of a net consisting of linear combinations of operators $  \pi (g) $,  
 +
$  g \in G $;  
 +
cf. [[#References|[8]]], §7, [[#References|[1]]], Chapt. V, §3; [[#References|[13]]], Chapt. IV, §2.

Latest revision as of 16:40, 31 March 2020


A mapping of the group $ G $ into the group of homeomorphisms of a topological space. Most often such a representation of $ G $ is understood to be a linear representation, moreover, a linear representation $ \pi $ of $ G $ into a topological vector space $ E $ such that the vector function $ g \rightarrow \pi (g) x $, $ g \in G $, defines for any $ x \in E $ a continuous mapping of $ G $ into $ E $. In particular, every continuous representation of the group $ G $ is a representation of the topological group $ G $.

The theory of representations of topological groups is strongly connected with the representation theory of various topological group algebras (cf. Group algebra). The most important among these is the symmetric Banach measure algebra $ M (G) $ of the group $ G $( the algebra of all regular Borel measures on $ G $ with finite total variation, in which multiplication is defined as convolution). Often one also uses the topological algebra $ C ^ \prime (G) $ of all regular Borel measures on $ G $ with finite total variation and with compact support. Multiplication in $ C ^ \prime (G) $ is defined as convolution, and the involution $ \mu \rightarrow \mu ^ {*} $, $ \mu \in C ^ \prime (G) $, is defined by

$$ \int\limits _ { G } f (g) d \mu ^ {*} (g) = \ \int\limits _ { G } \overline{ {f (g ^ {-1} ) }}\; \ d \mu (g),\ \ f \in C (G). $$

The topology of $ C ^ \prime (G) $ is compatible with the duality between this algebra and the algebra $ C (G) $( of all continuous functions on $ G $), equipped with the compact-open topology. Various subalgebras of $ M (G) $ and $ C ^ \prime (G) $ also play an important role. In particular, if $ E $ is a quasi-complete barrelled or complete locally convex space and $ \pi $ is a continuous representation of the topological group $ G $ into $ E $, then the formula

$$ \pi ( \mu ) = \ \int\limits _ { G } \pi (g) d \mu (g),\ \ \mu \in C ^ \prime (G), $$

defines a weakly-continuous linear operator $ \pi ( \mu ) $ on $ E $, and the correspondence $ \mu \rightarrow \pi ( \mu ) $ is a representation of the algebra $ C ^ \prime (G) $ in $ E $, uniquely defining the representation $ \pi $ of the topological group. Here, a representation of a topological group, a (topologically) irreducible representation, an operator-irreducible representation, a totally irreducible representation, is equivalent to another representation of the topological group, etc., if and only if the corresponding representations of the algebra $ C ^ \prime (G) $ have the corresponding property.

Let $ \pi $ be a representation of a topological group $ G $ in a locally convex vector space $ E $ and let $ E ^ \prime $ be the space dual to $ E $. Functions on $ G $ of the form $ g \rightarrow \phi ( \pi (g) \xi ) $, $ \xi \in E $, $ \phi \in E ^ \prime $, are called matrix elements of $ \pi $. If $ E $ is a Hilbert space and $ \xi \in E $, $ \| \xi \| = 1 $, then functions of the form $ g \rightarrow \langle \pi (g) \xi , \xi \rangle $, $ g \in G $, are called spherical functions, corresponding to $ \pi $.

Suppose that $ E, E ^ {*} $ are dual locally convex spaces and let $ \pi $ be a representation of a topological group $ G $ in $ E $. The formula $ \pi ^ {*} (g) = \pi (g ^ {-1} ) ^ {*} $ defines a representation $ \pi ^ {*} $ of $ G $ in $ E ^ {*} $, called the adjoint, or contragredient, representation to $ \pi $. Suppose that $ \pi _ {1} , \pi _ {2} $ are representations of $ G $ in locally convex spaces $ E _ {1} , E _ {2} $, respectively, let $ E = E _ {1} + E _ {2} $ be the direct sum and let $ \pi (g) $, $ g \in G $, be the continuous linear operator into $ E $ defined by

$$ \pi (g) (x _ {1} + x _ {2} ) = \ \pi _ {1} (g) x _ {1} + \pi _ {2} (g) x _ {2} ,\ \ x _ {1} \in E _ {1} ,\ \ x _ {2} \in E _ {2} . $$

The mapping $ g \rightarrow \pi (g) $ is a representation of $ G $ in $ E $, called the direct sum of the representations $ \pi _ {1} $ and $ \pi _ {2} $. In certain situations (in particular for unitary representations) one can define the tensor product of representations of a topological group and the direct sum of an infinite family of such representations. By restricting or extending the field of scalars, one introduces the operations of "realification" or complexification of representations.

A representation of a topological group is called completely reducible if every closed invariant subspace has a complementary closed invariant subspace. A representation $ \pi $ of a topological group $ G $ in a locally convex space $ E $ is called split (decomposable) if there exist closed invariant subspaces $ E _ {1} , E _ {2} $ in $ E $ such that $ \pi $ is equivalent to the direct sum of the subrepresentations $ \pi _ {1} , \pi _ {2} $ of $ \pi $ corresponding to the subspaces $ E _ {1} , E _ {2} $, respectively; in the opposite case $ \pi $ is called non-split (indecomposable). A non-split reducible representation $ \pi $ is not solely determined by its subrepresentation and quotient representation corresponding to a given invariant subspace, but also requires for its characterization certain one-dimensional cohomology classes of the group $ G $ with coefficients in the $ G $- module of bounded linear operators from the space of the quotient representation into the space of the representation.

The most important general problems in the representation theory of topological groups are the description of all non-split representations of a given topological group and the study of the description (decomposition) of arbitrary representations of a topological group in terms of non-split ones. In both cases the problems are far from being completely solved (1991), but the results obtained still suffice to make the representation theory of topological groups a basis for abstract harmonic analysis (cf. Harmonic analysis, abstract), generalizing the theory of Fourier series and integrals, the spectral theory of unitary operators, the theory of Jordan normal forms and systems of ordinary differential equations with constant coefficients, as well as a basis for certain branches of ergodic theory, quantum mechanics, statistical physics, and field theory.

The most important branch of the representation theory of topological groups is the theory of unitary representations (cf. Unitary representation), which has many applications. A number of special properties simplify their study. In particular, the orthogonal complement to an invariant subspace of a unitary representation is invariant, and therefore every unitary representation is completely reducible. For unitary representations, the conditions of total irreducibility, (topological) irreducibility and operator-irreducibility are equivalent (but, in general, are weaker than the condition of algebraic irreducibility).

Another class of representations of topological groups which has various applications is that of finite-dimensional representations (cf. Finite-dimensional representation). The study of representations of this class is greatly facilitated by the relative simplicity of the functional-analytic problems as compared to the general case; in particular, an irreducible finite-dimensional representation is totally irreducible. However, the theory of finite-dimensional representations of topological groups has been developed satisfactorily (1991) only for certain classes of such groups (in particular, for semi-simple Lie groups and for the groups $ \mathbf R $ and $ \mathbf Z $). For many classes of groups, including the class of connected Lie groups, there is a complete description of the irreducible finite-dimensional representations.

The theory of representations has been mostly developed for locally compact groups. A most important property of the class of locally compact groups is that it coincides with the class of complete topological groups on which there is a non-zero right-invariant regular Borel measure $ m $( cf. Haar measure). This allows one to add to the useful group algebras of a locally compact group $ G $, the symmetric Banach algebra $ L _ {1} (G) = L _ {1} (G, m) $( under convolution), which plays a decisive role in the theory of bounded representations of a topological group $ G $ in Banach spaces (i.e. representations having bounded image). The formula

$$ \pi (f) = \ \int\limits _ { G } f (g) \pi (g) dm (g),\ \ f \in L _ {1} (G), $$

establishes a one-to-one correspondence between the bounded representations $ \pi $ of a locally compact group $ G $ and the (continuous) representations $ \widetilde \pi $ of the algebra $ L _ {1} (G) $ with the property that $ \widetilde \pi (L _ {1} (G)) H _ {\widetilde \pi } $ is dense in the space $ H _ {\widetilde \pi } $ of the representation $ \widetilde \pi $. Here, unitary representations of the group correspond to symmetric representations of $ L _ {1} (G) $. Another property of locally compact groups is that their representations in barrelled locally convex spaces are jointly continuous.

The theory of unitary representations of locally compact groups is the most fully developed branch of the representation theory of topological groups. Related to the existence of a Haar measure on locally compact groups is the possibility of studying the regular representation of $ G $ in $ L _ {2} (G) $, which leads, in particular, to an analogue of the Plancherel formula for such groups, as well as to singling out the basic, complementary and discrete series of unitary representations of groups of the class considered (cf. Complementary series (of representations); Discrete series (of representations)). Important general problems in the theory of unitary representations are the problems of constructing irreducible representations and quotient representations, of decomposing representations into a direct integral, and of studying dual objects, and the problems of the theory of spherical functions, characters and harmonic analysis related to them, including the study of various group algebras.

A subclass of the class of locally compact groups that is exceptionally rich in applications is the class of Lie groups. The theory of infinite-dimensional representations (cf. Infinite-dimensional representation) of Lie groups, including the representation theory of the classical groups, is one of the most quickly developing branches of the general representation theory of topological groups. A powerful method in the study of representations of Lie groups is the orbit method.

Another important subclass of the class of locally compact groups is the class of compact groups. The representation theory of compact groups is one of the most complete branches of the general representation theory of topological groups, and is a tool in the study of representations of topological groups containing compact subgroups. An important branch of the representation theory of compact groups concerns the decomposition of restrictions to subgroups, and the decomposition of tensor products of concrete representations of compact Lie groups. A part of the representation theory of compact groups with many applications in algebra and analysis is the theory of representations of finite groups.

Like in the above-mentioned study of non-split representations of topological groups, even the simpler problem of describing the intertwining of totally irreducible representations, related with a corresponding cohomology theory, has only been solved (1991) for certain groups, despite its importance in the harmonic analysis on groups. In fact, in terms of non-split representations (more precisely, in terms of representations participating in the analytic extension of the corresponding basic series) for certain Lie groups (respectively, Chevalley groups) one has obtained analogues of the Paley–Wiener theorem, giving a description of the image of the group algebra of infinitely-differentiable (respectively, locally finite) functions with compact support on the group under Fourier transformation (i.e. under the mapping $ f \rightarrow \int _ {G} f (g) \pi (g) d \mu (g) $, $ f \in K (G) $, assigning to a function on the group an operator-valued function on a set of representatives of the space of equivalence classes of representations of this group). The more special problem of describing all totally-irreducible representations of a given group has been solved (1991) only for locally compact groups whose quotient group by the centre is compact (a totally-irreducible representation of such a group is finite-dimensional and the set of these representations suffices for obtaining an analogue to the Paley–Wiener theorem) and for certain linear Lie groups (including the complex semi-simple ones). As in the theory of unitary representations, in the theory of non-unitary representations one has likewise compiled a vast amount of material relating to concrete representations of certain particular groups and relating to applications to individual problems of harmonic analysis on such groups.

A number of problems of the representation theory of topological groups is related to representations in spaces with an indefinite metric (cf. Space with an indefinite metric). A complete description of the irreducible representations in such spaces has been obtained for certain semi-simple Lie groups (this includes, in particular, their irreducible finite-dimensional representations). For these groups one has also found a decomposition of tensor products of certain irreducible representations of this type into irreducible unitary representations. The theory of operator-irreducible representations of semi-simple Lie groups in such spaces and the determination of the structures of their invariant subspaces is closely related with the analytic extension of the basic series of representations of these groups.

The representation theory of topological groups comprises the development of the theory of projective representations (cf. Projective representation), the generalization of the theory of representations of Lie groups (in particular, the orbit method) to locally compact groups of general type, and the theory of representations of topological groups that are not locally compact (the group of smooth functions on a manifold with values in a Lie group, the group of diffeomorphisms of a smooth manifold, infinite-dimensional analogues of the classical and other groups). The study of representations of such groups turned out to be related with probability theory (in particular, with the theory of Markov processes) and with problems in statistical physics. On the other hand, deep connections between the theory of representations of second-order matrix groups over locally compact fields and problems in number theory have been established.

References

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Comments

See also the references to Representation of a group; Representation of a compact group.

A linear representation $ \pi $ of $ G $ into $ {\mathcal L} (H) $, the continuous linear operators on a topological vector space, is called algebraically irreducible if there are no non-trivial invariant subspaces; it is called irreducible or, to stress the topological context, topologically irreducible, if there are no non-trivial closed invariant subspaces; it is called totally irreducible, also called completely irreducible, if every element of $ {\mathcal L} (H) $ is the weak limit of a net consisting of linear combinations of operators $ \pi (g) $, $ g \in G $; cf. [8], §7, [1], Chapt. V, §3; [13], Chapt. IV, §2.

How to Cite This Entry:
Representation of a topological group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_a_topological_group&oldid=18500
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article