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A method for studying unitary representations of Lie groups. The theory of unitary representations (cf. [[Unitary representation|Unitary representation]]) of nilpotent Lie groups was developed using the orbit method, and it has been shown that this method can also be used for other groups (see [[#References|[1]]]).
 
A method for studying unitary representations of Lie groups. The theory of unitary representations (cf. [[Unitary representation|Unitary representation]]) of nilpotent Lie groups was developed using the orbit method, and it has been shown that this method can also be used for other groups (see [[#References|[1]]]).
  
The orbit method is based on the following  "experimental"  fact: A close connection exists between unitary irreducible representations of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700201.png" /> and its orbits in the [[Coadjoint representation|coadjoint representation]]. The solution of basic problems in the theory of representations using the orbit method is achieved in the following way (see [[#References|[2]]]).
+
The orbit method is based on the following  "experimental"  fact: A close connection exists between unitary irreducible representations of a Lie group $  G $
 +
and its orbits in the [[Coadjoint representation|coadjoint representation]]. The solution of basic problems in the theory of representations using the orbit method is achieved in the following way (see [[#References|[2]]]).
  
 
==Construction and classification of irreducible unitary representations.==
 
==Construction and classification of irreducible unitary representations.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700202.png" /> be an [[Orbit|orbit]] of a real Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700203.png" /> in the coadjoint representation, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700204.png" /> be a point of this orbit (which is a linear functional on the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700205.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700206.png" />), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700207.png" /> be the [[Stabilizer|stabilizer]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700208.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o0700209.png" /> be the Lie algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002010.png" />. A complex subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002012.png" /> is called a polarization of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002013.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002014.png" /> is the complexification of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002015.png" />, cf. [[Complexification of a Lie algebra|Complexification of a Lie algebra]]) if and only if it possesses the following properties:
+
Let $  \Omega $
 +
be an [[Orbit|orbit]] of a real Lie group $  G $
 +
in the coadjoint representation, let $  F $
 +
be a point of this orbit (which is a linear functional on the Lie algebra $  \mathfrak g $
 +
of $  G $),  
 +
let $  G( F  ) $
 +
be the [[Stabilizer|stabilizer]] of $  F $,  
 +
and let $  \mathfrak g ( F  ) $
 +
be the Lie algebra of the group $  G( F  ) $.  
 +
A complex subalgebra $  \mathfrak h $
 +
in $  \mathfrak g _ {\mathbf C }  $
 +
is called a polarization of the point $  F $(
 +
$  \mathfrak g _ {\mathbf C }  $
 +
is the complexification of the Lie algebra $  \mathfrak g $,  
 +
cf. [[Complexification of a Lie algebra|Complexification of a Lie algebra]]) if and only if it possesses the following properties:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002016.png" />;
+
1) $  \mathop{\rm dim} _ {\mathbf C }  \mathfrak h = \mathop{\rm dim}  \mathfrak g - ( 1/2)  \mathop{\rm dim}  \Omega $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002017.png" /> is contained in the kernel of the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002019.png" />;
+
2) $  [ \mathfrak h, \mathfrak h ] $
 +
is contained in the kernel of the functional $  F $
 +
on $  \mathfrak g $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002020.png" /> is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002021.png" />.
+
3) $  \mathfrak h $
 +
is invariant with respect to $  \mathop{\rm Ad}  G( F  ) $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002023.png" />. The polarization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002024.png" /> is called real if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002025.png" /> and purely complex if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002026.png" />. The functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002027.png" /> defines a character (a one-dimensional unitary representation) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002028.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002029.png" /> according to the formula
+
Let $  H  ^ {0} = \mathop{\rm exp} ( \mathfrak h \cap \mathfrak g) $
 +
and $  H = G( F  ) \cdot H  ^ {0} $.  
 +
The polarization $  \mathfrak h $
 +
is called real if $  \mathfrak h = \overline{ {\mathfrak h }}\; $
 +
and purely complex if $  \mathfrak h \cap \overline{ {\mathfrak h }}\; = \mathfrak g ( F  ) $.  
 +
The functional $  F $
 +
defines a character (a one-dimensional unitary representation) $  \chi _ {F}  ^ {0} $
 +
of the group $  H  ^ {0} $
 +
according to 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/o/o070/o070020/o07002030.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm exp}  X  \rightarrow  \mathop{\rm exp}  2 \pi i \langle  F, X\rangle.
 +
$$
  
Extend <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002031.png" /> to a character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002032.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002033.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002034.png" /> is a real polarization, then let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002035.png" /> be the representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002036.png" /> induced by the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002037.png" /> of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002038.png" /> (see [[Induced representation|Induced representation]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002039.png" /> is a purely complex polarization, then let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002040.png" /> be the holomorphically induced representation operating on the space of holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002041.png" />.
+
Extend $  \chi _ {F}  ^ {0} $
 +
to a character $  \chi _ {F} $
 +
of $  H $.  
 +
If $  \mathfrak h $
 +
is a real polarization, then let $  T _ {F, \mathfrak h, \chi _ {F}  } $
 +
be the representation of the group $  G $
 +
induced by the character $  \chi _ {F} $
 +
of the subgroup $  H $(
 +
see [[Induced representation|Induced representation]]). If $  \mathfrak h $
 +
is a purely complex polarization, then let $  T _ {F, \mathfrak h , \chi _ {F}  } $
 +
be the holomorphically induced representation operating on the space of holomorphic functions on $  G/H $.
  
The first basic hypothesis is that the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002042.png" /> is irreducible (cf. [[Irreducible representation|Irreducible representation]]) and its equivalence class depends only on the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002043.png" /> and the choice of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002044.png" /> of the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002045.png" />. This hypothesis is proved for nilpotent groups [[#References|[1]]] and for solvable Lie groups [[#References|[5]]]. For certain orbits of the simple special group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002046.png" /> the hypothesis does not hold [[#References|[7]]]. The possibility of an extension and its degree of ambiguity depend on topological properties of the orbit: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002047.png" />-dimensional cohomology classes act as obstacles to the extension, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002048.png" />-dimensional cohomology classes of the orbit can be used as a parameter for enumerating different extensions. More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002049.png" /> be a canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002050.png" />-form on the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002051.png" />. For an extension to exist, it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002052.png" /> belongs to the integer homology classes (i.e. that its integral along any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002053.png" />-dimensional cycle is an integer); if this condition is fulfilled, then the set of extensions is parametrized by the characters of the fundamental group of the orbit.
+
The first basic hypothesis is that the representation $  T _ {F, \mathfrak h , \chi _ {F}  } $
 +
is irreducible (cf. [[Irreducible representation|Irreducible representation]]) and its equivalence class depends only on the orbit $  \Omega $
 +
and the choice of the extension $  \chi _ {F} $
 +
of the character $  \chi _ {F}  ^ {0} $.  
 +
This hypothesis is proved for nilpotent groups [[#References|[1]]] and for solvable Lie groups [[#References|[5]]]. For certain orbits of the simple special group $  G _ {2} $
 +
the hypothesis does not hold [[#References|[7]]]. The possibility of an extension and its degree of ambiguity depend on topological properties of the orbit: $  2 $-
 +
dimensional cohomology classes act as obstacles to the extension, while $  1 $-
 +
dimensional cohomology classes of the orbit can be used as a parameter for enumerating different extensions. More precisely, let $  B _  \Omega  $
 +
be a canonical $  2 $-
 +
form on the orbit $  \Omega $.  
 +
For an extension to exist, it is necessary and sufficient that $  B _  \Omega  $
 +
belongs to the integer homology classes (i.e. that its integral along any $  2 $-
 +
dimensional cycle is an integer); if this condition is fulfilled, then the set of extensions is parametrized by the characters of the fundamental group of the orbit.
  
The second basic hypothesis is that all unitary irreducible representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002054.png" /> in question are obtained in the way shown. Up to 1983, the only examples which contradicted this hypothesis were the so-called complementary series of representations of semi-simple Lie groups.
+
The second basic hypothesis is that all unitary irreducible representations of the group $  G $
 +
in question are obtained in the way shown. Up to 1983, the only examples which contradicted this hypothesis were the so-called complementary series of representations of semi-simple Lie groups.
  
 
==Functional properties of the relation between orbits and representations.==
 
==Functional properties of the relation between orbits and representations.==
In the theory of representations great significance is attached to questions of decomposition into irreducible components of a representation: Given a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002055.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002056.png" />, how are such decompositions obtained by restricting an irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002057.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002058.png" /> and by inducing an irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002059.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002060.png" />? The orbit method gives answers to these questions in terms of a natural projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002061.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002062.png" /> signifies a transfer to the adjoint space; the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002063.png" /> consists of restriction of a functional from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002064.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002065.png" />). Indeed, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002066.png" /> be an exponential Lie group (for such groups the relation between orbits and representations is a one-to-one relation, cf. [[Lie group, exponential|Lie group, exponential]]). The irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002067.png" /> corresponding to the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002068.png" />, when restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002069.png" />, decomposes into irreducible components corresponding to those orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002070.png" /> which ly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002071.png" />, while a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002072.png" /> induced by an irreducible representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002073.png" />, corresponding to the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002074.png" />, decomposes into irreducible components corresponding to the orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002075.png" /> which have a non-empty intersection with the pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002076.png" />. These results have two important consequences: If the irreducible representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002077.png" /> correspond to the orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002079.png" />, then the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002080.png" /> decomposes into irreducible components corresponding to those orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002081.png" /> which ly in the arithmetic sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002082.png" />; a quasi-regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002083.png" /> in a space of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002084.png" /> decomposes into irreducible components corresponding to those orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002085.png" /> for which the image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002086.png" /> contains zero.
+
In the theory of representations great significance is attached to questions of decomposition into irreducible components of a representation: Given a subgroup $  H $
 +
of a group $  G $,  
 +
how are such decompositions obtained by restricting an irreducible representation of $  G $
 +
to $  H $
 +
and by inducing an irreducible representation of $  H $
 +
to $  G $?  
 +
The orbit method gives answers to these questions in terms of a natural projection $  p: \mathfrak g  ^  \star  \rightarrow \mathfrak h  ^  \star  $(
 +
where $  {}  ^  \star  $
 +
signifies a transfer to the adjoint space; the projection $  p $
 +
consists of restriction of a functional from $  \mathfrak g $
 +
onto $  \mathfrak h $).  
 +
Indeed, let $  G $
 +
be an exponential Lie group (for such groups the relation between orbits and representations is a one-to-one relation, cf. [[Lie group, exponential|Lie group, exponential]]). The irreducible representation of $  G $
 +
corresponding to the orbit $  \Omega \subset  \mathfrak g  ^  \star  $,  
 +
when restricted to $  H $,  
 +
decomposes into irreducible components corresponding to those orbits $  \omega \in \mathfrak h  ^  \star  $
 +
which ly in $  p( \Omega ) $,  
 +
while a representation of $  G $
 +
induced by an irreducible representation of the group $  H $,  
 +
corresponding to the orbit $  \omega \subset  \mathfrak h  ^  \star  $,  
 +
decomposes into irreducible components corresponding to the orbits $  \Omega \subset  \mathfrak g  ^  \star  $
 +
which have a non-empty intersection with the pre-image $  p  ^ {-} 1 ( \omega ) $.  
 +
These results have two important consequences: If the irreducible representations $  T _ {i} $
 +
correspond to the orbits $  \Omega _ {i} $,
 +
$  i = 1, 2 $,  
 +
then the tensor product $  T _ {1} \otimes T _ {2} $
 +
decomposes into irreducible components corresponding to those orbits $  \Omega $
 +
which ly in the arithmetic sum $  \Omega _ {1} + \Omega _ {2} $;  
 +
a quasi-regular representation of $  G $
 +
in a space of functions on $  G/H $
 +
decomposes into irreducible components corresponding to those orbits $  \Omega \subset  \mathfrak g  ^  \star  $
 +
for which the image $  p( \Omega ) \subset  \mathfrak h  ^  \star  $
 +
contains zero.
  
 
==Character theory.==
 
==Character theory.==
 
For characters of irreducible representations (as generalized functions on a group) the following universal formula has been proposed (see [[#References|[2]]]):
 
For characters of irreducible representations (as generalized functions on a group) the following universal formula has been proposed (see [[#References|[2]]]):
  
<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/o/o070/o070020/o07002087.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
\chi (  \mathop{\rm exp}  X)  =
 +
\frac{1}{p( X) }
 +
\int\limits _  \Omega  e ^ {2 \pi i\langle  F, X\rangle } \beta ( ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002088.png" /> is the exponential mapping of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002089.png" /> into the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002090.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002091.png" /> is the square root of the density of the invariant Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002092.png" /> in canonical coordinates and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002093.png" /> is the volume form on the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002094.png" /> connected to the canonical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002095.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002096.png" /> by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002098.png" />. This formula is correct for nilpotent groups, solvable groups of type 1, compact groups, discrete series of representations of semi-simple real groups, and principal series of representations of complex semi-simple groups. For certain degenerate series of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o07002099.png" /> the formula does not hold. Formula (*) provides a simple formula for the calculation of the infinitesimal character of the irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020100.png" /> corresponding to the orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020101.png" />; moreover, to each Laplace operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020102.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020103.png" /> an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020104.png" />-invariant polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020105.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020106.png" /> is related, such that the value of the infinitesimal character of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020107.png" /> at the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020108.png" /> is equal to the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020109.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020110.png" />.
+
where $  \mathop{\rm exp} : \mathfrak g \rightarrow G $
 +
is the exponential mapping of the Lie algebra $  \mathfrak g $
 +
into the group $  G $,  
 +
where $  p( X) $
 +
is the square root of the density of the invariant Haar measure on $  G $
 +
in canonical coordinates and where $  \beta $
 +
is the volume form on the orbit $  \Omega $
 +
connected to the canonical $  2 $-
 +
form $  B _  \Omega  $
 +
by the relation $  \beta = B _  \Omega  ^ {k} /k! $,  
 +
$  k = (  \mathop{\rm dim}  \Omega ) / 2 $.  
 +
This formula is correct for nilpotent groups, solvable groups of type 1, compact groups, discrete series of representations of semi-simple real groups, and principal series of representations of complex semi-simple groups. For certain degenerate series of representations of $  \mathop{\rm SL} ( 3, \mathbf R ) $
 +
the formula does not hold. Formula (*) provides a simple formula for the calculation of the infinitesimal character of the irreducible representation $  T _  \Omega  $
 +
corresponding to the orbit $  \Omega $;  
 +
moreover, to each Laplace operator $  \Delta $
 +
on $  G $
 +
an $  \mathop{\rm Ad}  ^  \star  G $-
 +
invariant polynomial $  P _  \Delta  $
 +
on $  \mathfrak g  ^  \star  $
 +
is related, such that the value of the infinitesimal character of the representation $  T _  \Omega  $
 +
at the element $  \Delta $
 +
is equal to the value of $  P _  \Delta  $
 +
at $  \Delta $.
  
==Construction of an irreducible unitary representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020111.png" /> along its orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020112.png" /> in the coadjoint representation.==
+
==Construction of an irreducible unitary representation of the group $  G $along its orbit $  \Omega $in the coadjoint representation.==
This construction can be considered as a quantization operation of a Hamiltonian system for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020113.png" /> plays the role of phase space, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020114.png" /> plays the role of a multi-dimensional non-commutative time (or group of symmetries). Under these conditions, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020115.png" />-orbits in the coadjoint representation are all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020116.png" />-homogeneous symplectic manifolds which admit quantization. The second basic hypothesis can therefore be reformulated thus: Every elementary quantum system with time (or group of symmetries) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070020/o070020117.png" /> is obtained by quantization from the corresponding classical system (see [[#References|[2]]]).
+
This construction can be considered as a quantization operation of a Hamiltonian system for which $  \Omega $
 +
plays the role of phase space, while $  G $
 +
plays the role of a multi-dimensional non-commutative time (or group of symmetries). Under these conditions, the $  G $-
 +
orbits in the coadjoint representation are all $  G $-
 +
homogeneous symplectic manifolds which admit quantization. The second basic hypothesis can therefore be reformulated thus: Every elementary quantum system with time (or group of symmetries) $  G $
 +
is obtained by quantization from the corresponding classical system (see [[#References|[2]]]).
  
 
A connection has also been discovered with the theory of completely-integrable Hamiltonian systems (see [[#References|[11]]]).
 
A connection has also been discovered with the theory of completely-integrable Hamiltonian systems (see [[#References|[11]]]).

Latest revision as of 08:04, 6 June 2020


A method for studying unitary representations of Lie groups. The theory of unitary representations (cf. Unitary representation) of nilpotent Lie groups was developed using the orbit method, and it has been shown that this method can also be used for other groups (see [1]).

The orbit method is based on the following "experimental" fact: A close connection exists between unitary irreducible representations of a Lie group $ G $ and its orbits in the coadjoint representation. The solution of basic problems in the theory of representations using the orbit method is achieved in the following way (see [2]).

Construction and classification of irreducible unitary representations.

Let $ \Omega $ be an orbit of a real Lie group $ G $ in the coadjoint representation, let $ F $ be a point of this orbit (which is a linear functional on the Lie algebra $ \mathfrak g $ of $ G $), let $ G( F ) $ be the stabilizer of $ F $, and let $ \mathfrak g ( F ) $ be the Lie algebra of the group $ G( F ) $. A complex subalgebra $ \mathfrak h $ in $ \mathfrak g _ {\mathbf C } $ is called a polarization of the point $ F $( $ \mathfrak g _ {\mathbf C } $ is the complexification of the Lie algebra $ \mathfrak g $, cf. Complexification of a Lie algebra) if and only if it possesses the following properties:

1) $ \mathop{\rm dim} _ {\mathbf C } \mathfrak h = \mathop{\rm dim} \mathfrak g - ( 1/2) \mathop{\rm dim} \Omega $;

2) $ [ \mathfrak h, \mathfrak h ] $ is contained in the kernel of the functional $ F $ on $ \mathfrak g $;

3) $ \mathfrak h $ is invariant with respect to $ \mathop{\rm Ad} G( F ) $.

Let $ H ^ {0} = \mathop{\rm exp} ( \mathfrak h \cap \mathfrak g) $ and $ H = G( F ) \cdot H ^ {0} $. The polarization $ \mathfrak h $ is called real if $ \mathfrak h = \overline{ {\mathfrak h }}\; $ and purely complex if $ \mathfrak h \cap \overline{ {\mathfrak h }}\; = \mathfrak g ( F ) $. The functional $ F $ defines a character (a one-dimensional unitary representation) $ \chi _ {F} ^ {0} $ of the group $ H ^ {0} $ according to the formula

$$ \mathop{\rm exp} X \rightarrow \mathop{\rm exp} 2 \pi i \langle F, X\rangle. $$

Extend $ \chi _ {F} ^ {0} $ to a character $ \chi _ {F} $ of $ H $. If $ \mathfrak h $ is a real polarization, then let $ T _ {F, \mathfrak h, \chi _ {F} } $ be the representation of the group $ G $ induced by the character $ \chi _ {F} $ of the subgroup $ H $( see Induced representation). If $ \mathfrak h $ is a purely complex polarization, then let $ T _ {F, \mathfrak h , \chi _ {F} } $ be the holomorphically induced representation operating on the space of holomorphic functions on $ G/H $.

The first basic hypothesis is that the representation $ T _ {F, \mathfrak h , \chi _ {F} } $ is irreducible (cf. Irreducible representation) and its equivalence class depends only on the orbit $ \Omega $ and the choice of the extension $ \chi _ {F} $ of the character $ \chi _ {F} ^ {0} $. This hypothesis is proved for nilpotent groups [1] and for solvable Lie groups [5]. For certain orbits of the simple special group $ G _ {2} $ the hypothesis does not hold [7]. The possibility of an extension and its degree of ambiguity depend on topological properties of the orbit: $ 2 $- dimensional cohomology classes act as obstacles to the extension, while $ 1 $- dimensional cohomology classes of the orbit can be used as a parameter for enumerating different extensions. More precisely, let $ B _ \Omega $ be a canonical $ 2 $- form on the orbit $ \Omega $. For an extension to exist, it is necessary and sufficient that $ B _ \Omega $ belongs to the integer homology classes (i.e. that its integral along any $ 2 $- dimensional cycle is an integer); if this condition is fulfilled, then the set of extensions is parametrized by the characters of the fundamental group of the orbit.

The second basic hypothesis is that all unitary irreducible representations of the group $ G $ in question are obtained in the way shown. Up to 1983, the only examples which contradicted this hypothesis were the so-called complementary series of representations of semi-simple Lie groups.

Functional properties of the relation between orbits and representations.

In the theory of representations great significance is attached to questions of decomposition into irreducible components of a representation: Given a subgroup $ H $ of a group $ G $, how are such decompositions obtained by restricting an irreducible representation of $ G $ to $ H $ and by inducing an irreducible representation of $ H $ to $ G $? The orbit method gives answers to these questions in terms of a natural projection $ p: \mathfrak g ^ \star \rightarrow \mathfrak h ^ \star $( where $ {} ^ \star $ signifies a transfer to the adjoint space; the projection $ p $ consists of restriction of a functional from $ \mathfrak g $ onto $ \mathfrak h $). Indeed, let $ G $ be an exponential Lie group (for such groups the relation between orbits and representations is a one-to-one relation, cf. Lie group, exponential). The irreducible representation of $ G $ corresponding to the orbit $ \Omega \subset \mathfrak g ^ \star $, when restricted to $ H $, decomposes into irreducible components corresponding to those orbits $ \omega \in \mathfrak h ^ \star $ which ly in $ p( \Omega ) $, while a representation of $ G $ induced by an irreducible representation of the group $ H $, corresponding to the orbit $ \omega \subset \mathfrak h ^ \star $, decomposes into irreducible components corresponding to the orbits $ \Omega \subset \mathfrak g ^ \star $ which have a non-empty intersection with the pre-image $ p ^ {-} 1 ( \omega ) $. These results have two important consequences: If the irreducible representations $ T _ {i} $ correspond to the orbits $ \Omega _ {i} $, $ i = 1, 2 $, then the tensor product $ T _ {1} \otimes T _ {2} $ decomposes into irreducible components corresponding to those orbits $ \Omega $ which ly in the arithmetic sum $ \Omega _ {1} + \Omega _ {2} $; a quasi-regular representation of $ G $ in a space of functions on $ G/H $ decomposes into irreducible components corresponding to those orbits $ \Omega \subset \mathfrak g ^ \star $ for which the image $ p( \Omega ) \subset \mathfrak h ^ \star $ contains zero.

Character theory.

For characters of irreducible representations (as generalized functions on a group) the following universal formula has been proposed (see [2]):

$$ \tag{* } \chi ( \mathop{\rm exp} X) = \frac{1}{p( X) } \int\limits _ \Omega e ^ {2 \pi i\langle F, X\rangle } \beta ( F ), $$

where $ \mathop{\rm exp} : \mathfrak g \rightarrow G $ is the exponential mapping of the Lie algebra $ \mathfrak g $ into the group $ G $, where $ p( X) $ is the square root of the density of the invariant Haar measure on $ G $ in canonical coordinates and where $ \beta $ is the volume form on the orbit $ \Omega $ connected to the canonical $ 2 $- form $ B _ \Omega $ by the relation $ \beta = B _ \Omega ^ {k} /k! $, $ k = ( \mathop{\rm dim} \Omega ) / 2 $. This formula is correct for nilpotent groups, solvable groups of type 1, compact groups, discrete series of representations of semi-simple real groups, and principal series of representations of complex semi-simple groups. For certain degenerate series of representations of $ \mathop{\rm SL} ( 3, \mathbf R ) $ the formula does not hold. Formula (*) provides a simple formula for the calculation of the infinitesimal character of the irreducible representation $ T _ \Omega $ corresponding to the orbit $ \Omega $; moreover, to each Laplace operator $ \Delta $ on $ G $ an $ \mathop{\rm Ad} ^ \star G $- invariant polynomial $ P _ \Delta $ on $ \mathfrak g ^ \star $ is related, such that the value of the infinitesimal character of the representation $ T _ \Omega $ at the element $ \Delta $ is equal to the value of $ P _ \Delta $ at $ \Delta $.

Construction of an irreducible unitary representation of the group $ G $along its orbit $ \Omega $in the coadjoint representation.

This construction can be considered as a quantization operation of a Hamiltonian system for which $ \Omega $ plays the role of phase space, while $ G $ plays the role of a multi-dimensional non-commutative time (or group of symmetries). Under these conditions, the $ G $- orbits in the coadjoint representation are all $ G $- homogeneous symplectic manifolds which admit quantization. The second basic hypothesis can therefore be reformulated thus: Every elementary quantum system with time (or group of symmetries) $ G $ is obtained by quantization from the corresponding classical system (see [2]).

A connection has also been discovered with the theory of completely-integrable Hamiltonian systems (see [11]).

References

[1] A.A. Kirillov, "Unitary representations of nilpotent Lie groups" Russian Math. Surveys , 17 : 4 (1962) pp. 53–104 Uspekhi Mat. Nauk , 17 : 4 (1962) pp. 57–110
[2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[3] J. Dixmier, "Enveloping algebras" , North-Holland (1974) (Translated from French)
[4] D.J. Simms, N.M.J. Woodhouse, "Lectures on geometric quantization" , Springer (1976)
[5] L. Auslander, B. Kostant, "Polarization and unitary representations of solvable Lie groups" Invent. Math. , 14 (1971) pp. 255–354
[6] C.C. Moore, "Decomposition of unitary representations defined by discrete subgroups of nilpotent groups" Ann. of Math. , 82 : 1 (1965) pp. 146–182
[7] L.P. Rothschild, J.A. Wolf, "Representations of semisimple groups associated to nilpotent orbits" Ann. Sci. Ecole Norm. Sup. Ser. 4 , 7 (1974) pp. 155–173
[8] P. Bernal, et al., "Représentations des groupes de Lie résolubles" , Dunod (1972)
[9] V.A. Ginzburg, "The method of orbits and perturbation theory" Soviet Math. Dokl. , 20 : 6 (1979) pp. 1287–1291 Dokl. Akad. Nauk SSSR , 249 : 3 (1979) pp. 525–528
[10] A.A. Kirillov, "Infinite dimensional groups, their representations, orbits, invariants" , Proc. Internat. Congress Mathematicians (Helsinki, 1978) , 2 , Acad. Sci. Fennicae (1980) pp. 705–708
[11] A.G. Reyman, M.A. Semenov-Tian-Shansky, "Reduction of Hamiltonian systems, affine Lie algebras and Lax equations" Invent. Math. , 54 : 1 (1979) pp. 81–100
[12] A.A. Kirillov, "Introduction to representation theory and noncommutative analysis" , Springer (to appear) (Translated from Russian)
How to Cite This Entry:
Orbit method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orbit_method&oldid=18205
This article was adapted from an original article by A.A. Kirillov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article