View source for Infinite-dimensional representation

''of a Lie group''

A representation of a [[Lie group|Lie group]] (cf. [[Representation of a topological group|Representation of a topological group]]) in an infinite-dimensional vector space. The theory of representations of Lie groups is part of the general theory of representations of topological groups. The specific features of Lie groups make it possible to employ analytical tools in this theory (in particular, infinitesimal methods), and also to considerably enlarge the class of "natural" group algebras (function algebras with respect to convolution, cf. [[Group algebra|Group algebra]]), the study of which connects this theory with abstract harmonic analysis, i.e. with part of the general theory of topological algebras (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]; [[Topological algebra|Topological algebra]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508401.png" /> be a Lie group. A representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508402.png" /> in a general sense is any homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508403.png" />, where GL<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508404.png" /> is the group of all invertible linear transformations of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508405.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508406.png" /> is a topological vector space, the homomorphisms which are usually considered are those with values in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508407.png" /> of all continuous linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508408.png" /> or in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i0508409.png" /> of all weakly-continuous transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084010.png" />. The algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084012.png" /> have one of the standard topologies (for example, the weak or the strong). A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084013.png" /> is said to be continuous (separately continuous) if the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084014.png" /> is continuous (separately continuous) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084016.png" /> is a quasi-complete [[Barrelled space|barrelled space]], any separately continuous representation is continuous. A continuous representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084017.png" /> is called differentiable (analytic) if the operator function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084018.png" /> is differentiable (analytic) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084019.png" />. The dimension of a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084020.png" /> is the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084021.png" />. The most important example of a representation of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084022.png" /> is its [[Regular representation|regular representation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084024.png" />, which can be defined on some class of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084027.png" /> is a Lie group, its regular representation is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084028.png" /> and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084029.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084030.png" /> is defined with respect to the [[Haar measure|Haar measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084031.png" />), and is differentiable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084032.png" /> (with respect to the standard topology in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084033.png" />: the topology of compact convergence). Every continuous [[Finite-dimensional representation|finite-dimensional representation]] of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084034.png" /> is analytic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084035.png" /> is a complex Lie group, it is natural to consider its complex-analytic (holomorphic) representations as well. As a rule, only continuous representations are considered in the theory of representations of Lie groups, and the continuity condition is not explicitly stipulated. If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084036.png" /> is compact, all its irreducible (continuous) representations are finite-dimensional. Similarly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084037.png" /> is a semi-simple complex Lie group, all its irreducible holomorphic representations are finite-dimensional.

==Relation to representations of group algebras.==
The most important group algebras for Lie groups are the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084038.png" />; the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084039.png" />, which is the completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084040.png" /> in the smallest regular norm (cf. [[Algebra of functions|Algebra of functions]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084041.png" /> — the algebra of all infinitely-differentiable functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084042.png" /> with compact support; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084043.png" /> — the algebra of all complex Radon measures with compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084044.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084045.png" /> — the algebra of all generalized functions (Schwarz distributions) on G with compact support; and also, for a complex Lie group, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084046.png" /> of all analytic functionals over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084047.png" />. The linear spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084050.png" /> are dual to, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084054.png" /> is the set of all holomorphic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084055.png" /> (with the topology of compact convergence). All these algebras have a natural topology. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084056.png" /> is a [[Banach algebra|Banach algebra]]. The product (convolution) of two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084058.png" /> is one of the group algebras indicated above, is defined by the equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084059.png" /></td> </tr></table>

with respect to a right-invariant measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084060.png" />, with a natural extension of this operation to the class of generalized functions. The integral 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/i/i050/i050840/i05084061.png" /></td> </tr></table>

establishes a natural connection between the representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084062.png" /> and the representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084063.png" /> (if the integral is correctly defined): If the integral is weakly convergent and defines an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084064.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084065.png" />, then the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084066.png" /> is a homomorphism. One then says that the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084067.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084068.png" /> is extended to the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084069.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084070.png" />, or that it is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084072.png" />-representation. Conversely, all weakly-continuous non-degenerate representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084073.png" /> are determined, in accordance with the formula above, by some representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084074.png" /> (weakly continuous for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084075.png" />, weakly differentiable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084076.png" />, weakly analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084077.png" />). This correspondence preserves all natural relations between the representations, such as topological irreducibility or equivalence. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084078.png" /> is a unimodular group, its unitary representations (in Hilbert spaces, cf. [[Unitary representation|Unitary representation]]) correspond to symmetric representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084079.png" /> with respect to the involution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084080.png" /> (cf. [[Group algebra|Group algebra]]; [[Involution representation|Involution representation]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084081.png" /> is a sequentially complete, locally convex Hausdorff space, any continuous representation of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084082.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084083.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084084.png" />-representation. If, moreover, the representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084085.png" /> is differentiable, it is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084086.png" />-representation. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084087.png" /> is a reflexive or a quasi-complete barrelled space, any separately-continuous representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084088.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084089.png" />-representation, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084090.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084091.png" />.

==The infinitesimal method.==
If a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084092.png" /> is differentiable, it is infinitely often differentiable, and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084093.png" /> has the structure of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084094.png" />-module, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084095.png" /> is the [[Lie algebra|Lie algebra]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084096.png" />, by considering the Lie infinitesimal operators:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084097.png" /></td> </tr></table>

The operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084098.png" /> form a representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i05084099.png" />, called the differential representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840100.png" />. A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840101.png" /> is said to be differentiable (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840102.png" />) if the vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840103.png" /> is differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840104.png" />. A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840105.png" /> is said to be analytic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840106.png" /> is an analytic function in a neighbourhood of the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840107.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840108.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840109.png" />-representation, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840110.png" /> of all infinitely-differentiable vectors is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840111.png" />. In particular, this is true for all continuous representations in a Banach space; moreover, in this case [[#References|[4]]] the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840112.png" /> of analytic vectors is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840113.png" />. The differential representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840114.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840115.png" /> may be reducible, even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840116.png" /> is topologically irreducible in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840117.png" />. To two equivalent representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840118.png" /> correspond equivalent differential representations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840119.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840120.png" />); the converse is, generally speaking, not true. For unitary representations in Hilbert spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840122.png" /> it follows from the equivalence of differential representations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840124.png" /> that the representations are equivalent [[#References|[7]]]. In the finite-dimensional case a representation of a connected Lie group can be uniquely reproduced from its differential representation. A representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840125.png" /> is said to be integrable (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840127.png" />-integrable) if it coincides with a differential representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840128.png" /> in a subspace which is everywhere-dense in the representation space. Integrability criteria are now (1988) known only in isolated cases [[#References|[4]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840129.png" /> is simply connected, all finite-dimensional representations of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840130.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840131.png" />-integrable.

==Irreducible representations.==
One of the main tasks of the theory of representations is the classification of all irreducible representations (cf. [[Irreducible representation|Irreducible representation]]) of a given group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840132.png" />, defined up to an equivalence, using a suitable definition of the concepts of irreducibility and equivalence. Thus, the following two problems are of interest: 1) the description of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840133.png" /> of all unitary equivalence classes of irreducible unitary representations of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840134.png" />; and 2) the description of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840135.png" /> of all Fell equivalence classes [[#References|[7]]] of totally-irreducible representations (also called completely-irreducible representations) of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840136.png" />. For semi-simple Lie groups with a finite centre, Fell equivalence is equivalent to Naimark equivalence [[#References|[7]]], and the natural imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840137.png" /> holds. The sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840138.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840139.png" /> have a natural topology, and their topologies are not necessarily Hausdorff [[#References|[5]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840140.png" /> is a compact Lie group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840141.png" /> is a discrete space. The description of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840142.png" /> in such a case is due to E. Cartan and H. Weyl. The linear envelope <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840143.png" /> of matrix entries of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840144.png" /> (i.e. of matrix entries of the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840145.png" />) here forms a subalgebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840146.png" /> (the algebra of spherical functions) which is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840147.png" /> and in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840148.png" />. The matrix entries form a basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840149.png" />. If the matrices of all representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840150.png" /> are defined in a basis with respect to which they are unitary, the corresponding matrix entries form an orthogonal basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840151.png" /> ( the Peter–Weyl theorem). If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840152.png" /> is not compact, its irreducible representations are usually infinite-dimensional. A method for constructing such representations analogous to the classical matrix groups was proposed by I.M. Gel'fand and M.A. Naimark [[#References|[1]]], and became the starting point of an intensive development of the theory of unitary infinite-dimensional representations. G.W. Mackey's [[#References|[5]]] theory of induced representations is a generalization of this method to arbitrary Lie groups. The general theory of non-unitary representations in locally convex vector spaces, which began to develop in the 1950's, is based to a great extent on the theory of topological vector spaces and on the theory of generalized functions. A detailed description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840153.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840154.png" />) is known (1988) for isolated classes of Lie groups (semi-simple complex, nilpotent and certain solvable Lie groups, as well as for their semi-direct products).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840155.png" /> be a semi-simple Lie group with a finite centre, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840156.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840157.png" />-representation in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840158.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840159.png" /> be a compact subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840160.png" />. A vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840161.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840163.png" />-finite if its cyclic envelope is finite-dimensional with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840164.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840165.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840166.png" />-finite vectors is everywhere-dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840167.png" /> and is the direct (algebraic) sum of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840168.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840169.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840170.png" /> is the maximal subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840171.png" /> in which the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840172.png" /> is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840173.png" />. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840174.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840176.png" />-finite if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840177.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840178.png" />. A subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840179.png" /> is said to be massive (large or rich) if every totally-irreducible representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840180.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840181.png" />-finite. The following fact is of paramount importance in the theory of representations: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840182.png" /> is a maximal compact subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840183.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840184.png" /> is massive. If the vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840185.png" /> are differentiable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840186.png" /> is invariant with respect to the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840187.png" /> of the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840188.png" />. The representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840189.png" /> is said to be normal if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840190.png" />-finite and if the vectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840191.png" /> are weakly analytic. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840192.png" /> is normal, there exists a one-to-one mapping (defined by restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840193.png" />) between closed submodules of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840194.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840195.png" /> and submodules of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840196.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840197.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840198.png" /> is the Lie algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840199.png" /> [[#References|[7]]]. Thus, the study of normal representations can be algebraized by the infinitesimal method. An example of a normal representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840200.png" /> is its principal series representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840201.png" />. This representation is totally irreducible for points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840202.png" /> in general position. In the general case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840203.png" /> can be decomposed into a finite composition series the factors of which are totally irreducible. Any quasi-simple irreducible representation of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840204.png" /> in a Banach space is infinitesimally equivalent to one of the factors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840205.png" /> for a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840206.png" />. This is also true for totally-irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840207.png" /> in quasi-complete locally convex spaces. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840208.png" /> is real or complex, it is sufficient to consider subrepresentations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840209.png" /> instead of its factors [[#References|[7]]]. In the simplest case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840210.png" />, the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840211.png" /> is defined by a pair of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840212.png" /> with integral difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840213.png" />, and operates in accordance with the right-shift formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840214.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840215.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840216.png" />, on the space of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840217.png" /> which satisfy the homogeneity condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840218.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840219.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840220.png" /> are positive integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840221.png" /> contains the irreducible finite-dimensional subrepresentation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840222.png" /> (in the class of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840223.png" />), the factors of which are totally irreducible. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840224.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840225.png" /> are negative integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840226.png" /> has a dual structure. In all other cases the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840227.png" /> is totally irreducible. In such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840228.png" /> is in one-to-one correspondence with the set of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840229.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840230.png" /> is an integer, factorized with respect to the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840231.png" />. The subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840232.png" /> consists of the representations of the basis series (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840233.png" /> is purely imaginary) (cf. [[Series of representations|Series of representations]]), the complementary series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840234.png" /> and the trivial (unique) representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840235.png" />, which results if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840236.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840237.png" /> be a semi-simple connected complex Lie group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840238.png" /> be its maximal solvable (Borel) subgroup, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840239.png" /> be a maximal torus, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840240.png" /> be a Cartan subgroup, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840241.png" /> be a character of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840242.png" /> (extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840243.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840244.png" /> is in one-to-one correspondence with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840245.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840246.png" /> is the set of all characters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840247.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840248.png" /> is the [[Weyl group|Weyl group]] of the complex algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840249.png" /> [[#References|[7]]]. For characters in "general position" the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840250.png" /> is totally irreducible. The description of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840251.png" /> is reduced to the study of the positive definiteness of certain bilinear forms, but the ultimate description is as yet (1988) unknown. Of special interest to real groups are the so-called [[Discrete series (of representations)|discrete series (of representations)]] (direct sums in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840252.png" />). All irreducible representations of the discrete series are classified [[#References|[3]]] by describing the characters of these representations.

For nilpotent connected Lie groups [[#References|[8]]] the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840253.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840254.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840255.png" /> is the linear space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840256.png" />, and the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840257.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840258.png" /> is conjugate with the adjoint representation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840259.png" /> [[#References|[9]]]. The correspondence is established by the [[Orbit method|orbit method]] [[#References|[8]]]. A subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840260.png" /> is called the polarization of an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840261.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840262.png" /> annihilates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840263.png" /> and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840264.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840265.png" /> is the orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840266.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840267.png" /> (all orbits are even-dimensional). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840268.png" /> is the corresponding analytic subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840269.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840270.png" /> is a character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840271.png" />, the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840272.png" /> corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840273.png" /> is induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840274.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840275.png" /> is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840276.png" /> if and only if the corresponding functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840277.png" /> lie on the same orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840278.png" />. In the simple case of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840279.png" /> of all unipotent matrices with respect to a fixed basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840280.png" />, the orbits of general position in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840281.png" /> are the two-dimensional planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840282.png" /> and the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840283.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840284.png" />. To each orbit in general position corresponds an irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840285.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840286.png" />, determined by 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/i/i050/i050840/i050840287.png" /></td> </tr></table>

in the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840288.png" />. The infinitesimal operators of this representation coincide with the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840289.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840290.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840291.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840292.png" /> is the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840293.png" />. This result is equivalent to the Stone–von Neumann theorem on self-adjoint operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840294.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840295.png" /> with the commutator relationship <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840296.png" />. To each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840297.png" /> corresponds a one-dimensional representation (a character) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840298.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840299.png" /> is then described in an analogous way, with values of the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840300.png" /> in the complex domain. This method of orbits can be naturally generalized to solvable connected Lie groups and even to arbitrary Lie groups; in the general case the orbits to be considered are orbits in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840301.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840302.png" /> is the complexification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840303.png" />), which satisfy certain integer conditions [[#References|[8]]].

The study of the general case is reduced, to a certain extent, to the two cases considered above by means of the theory of induced representations [[#References|[5]]], which permits one to describe the irreducible unitary representations of a semi-direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840304.png" /> with normal subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840305.png" /> in terms of irreducible representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840306.png" /> and of certain subgroups of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840307.png" /> (in view of the Levi–Mal'tsev theorem, cf. [[Levi–Mal'tsev decomposition|Levi–Mal'tsev decomposition]]). In practice, this method is only effective if the radical is commutative. Another method for studying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840308.png" /> (and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840309.png" />) is the description of the characters of the irreducible unitary representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840310.png" />; the set of such characters is in one-to-one correspondence with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840311.png" />. The validity of the general formula for characters, proposed by A.A. Kirillov [[#References|[8]]], has been verified (1988) only for a few special classes of Lie groups.

==Harmonic analysis of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840312.png" />.==
For a compact Lie group, the harmonic analysis is reduced to the expansion of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840313.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840314.png" />, into generalized Fourier series by the matrix entries of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840315.png" /> (the Peter–Weyl theorem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840316.png" /> and its analogues for other function classes). For non-compact Lie groups the foundations of harmonic analysis were laid in [[#References|[1]]] by the introduction of the generalized Fourier transform

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840317.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840318.png" /> is the operator of the elementary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840319.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840320.png" /> is the Haar measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840321.png" />, and by the introduction of the inversion formula (in analogy to the [[Plancherel formula|Plancherel formula]]) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840322.png" /> for the case of classical matrix groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840323.png" />. This result was generalized to locally compact unimodular groups (the abstract [[Plancherel theorem|Plancherel theorem]]). The Fourier transform converts convolution of functions on the group to multiplication of their (operator) Fourier images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840324.png" /> and is accordingly a very important tool in the study of group algebras. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840325.png" /> is a semi-simple Lie group, the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840326.png" /> satisfy structure relations of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840327.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840328.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840329.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840330.png" /> are intertwining operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840331.png" /> is the Weyl group of the symmetric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840332.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840333.png" /> is a maximal compact subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840334.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840335.png" /> is the Weyl group of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840336.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840337.png" /> is the complexification of the Lie algebra of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840338.png" />. If the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840339.png" /> have compact support, the operator functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840340.png" /> are entire functions of the complex parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840341.png" />. For the group algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840342.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840343.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840344.png" /> is a semi-simple connected complex Lie group, analogues of the classical [[Paley–Wiener theorem|Paley–Wiener theorem]] [[#References|[7]]] are known; these are descriptions of the images of these algebras under Fourier transformation. These results permit one to study the structure of a group algebra, its ideals and representations; in particular, they are used in the classification of irreducible representations of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840345.png" />. Analogues of the Paley–Wiener theorem are also known for certain nilpotent (metabelian) Lie groups and for groups of motions of a Euclidean space.

==Problems of spectral analysis.==
For unitary representations of Lie groups a general procedure is known for the decomposition of the representation into a direct integral of irreducible representations [[#References|[5]]]. The problem consists of finding analytical methods which would realize this decomposition for specific classes of groups and their representations, and in the establishment of uniqueness criteria of such a decomposition. For nilpotent Lie groups a method is known for the restriction of an irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840346.png" /> of a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840347.png" /> to a subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840348.png" /> (cf. [[Orbit method|Orbit method]]). For non-unitary representations, the task itself must be formulated more precisely, since the property of total reducibility lacks in the class of such representations. In several cases, not the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840349.png" /> itself is considered, but rather one of its group algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840350.png" />, and the problem of [[Spectral analysis|spectral analysis]] is treated as the study of two-sided ideals of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840351.png" />. The problem of spectral analysis (and [[Spectral synthesis|spectral synthesis]]) is also closely connected with the problem of approximation of functions on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840352.png" /> or on the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840353.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840354.png" /> is a subgroup, by linear combinations of matrix entries of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840355.png" />.

==Applications to mathematical physics.==
Cartan was the first to note the connection between the theory of representations of Lie groups and the special functions of mathematical physics. It was subsequently established that the principal classes of functions are closely connected with the representations of classical matrix groups [[#References|[10]]]. In fact, the existence of this connection throws light on fundamental problems in the theory of special functions: the properties of completeness and orthogonality, differential and recurrence relations, addition theorems, etc., and also makes it possible to detect new relationships and classes of functions. All these functions are matrix entries of classical groups or their modifications (characters, spherical functions). The theory of expansion with respect to these functions forms part of the general harmonic analysis on a homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840356.png" />. The fundamental role played by the theory of Lie groups in mathematical physics, particularly in quantum mechanics and quantum field theory, is due to the presence of a group of symmetries (at least approximately) in the fundamental equations of this theory. Classical examples of such symmetries include Einstein's relativity principle (with respect to the Lorentz group), the connection between Mendeleev's table and the representations of the rotation group, the theory of isotopic spin, unitary symmetry of elementary particles, etc. The connection with theoretical physics had a stimulating effect on the development of the general theory of representations of Lie groups.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> I.M. Gel'fand, M.A. Naimark, "Unitäre Darstellungen der klassischen Gruppen" , Akademie Verlag (1957) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) {{MR|0583191}} {{ZBL|1116.28002}} {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}} {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}} {{ZBL|0156.06001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Borel, "Repŕesentations de groupes localement compacts" , Springer (1972) {{MR|0414779}} {{ZBL|0242.22007}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> E. Nelson, "Analytic vectors" ''Ann. of Math.'' , '''70''' (1959) pp. 572–615 {{MR|0107176}} {{ZBL|0091.10704}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> G.W. Mackey, "Infinite-dimensional group representations" ''Bull. Amer. Math. Soc.'' , '''69''' (1963) pp. 628–686 {{MR|0153784}} {{ZBL|0136.11502}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M.A. Naimark, "Infinite-dimensional representations of groups and related problems" ''Itogi Nauk. Ser. Mat.'' : 2 (1964) pp. 38–82 (In Russian)</TD></TR><TR><TD valign="top">[7]</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">[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"> G. Warner, "Harmonic analysis on semi-simple Lie groups" , '''1''' , Springer (1972) {{MR|0499000}} {{MR|0498999}} {{ZBL|0265.22021}} {{ZBL|0265.22020}} </TD></TR><TR><TD valign="top">[10]</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></table>



====Comments====
The notions of a differentiable or analytic representation are commonly related to the strong topology [[#References|[9]]].

The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840357.png" /> (of generalized functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840358.png" /> with compact support) is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840359.png" /> in the West. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840360.png" />, if used, is then a synonym for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840361.png" />.

Recently (1986), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840362.png" /> has been determined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840363.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840364.png" /> is the field of real or complex numbers or the skew-field of quaternions (D.A. Vogan), for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840365.png" /> a complex simple Lie group of real rank 2 (M. Duflo) and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840366.png" /> a split-rank or semi-simple real Lie group (Baldoni–Silva–Barbasch). For a survey of the current state-of-affairs see [[#References|[a3]]], [[#References|[a5]]].

An analogue of the Paley–Wiener theorem is also known for real reductive Lie groups (cf. [[#References|[a7]]], [[#References|[a10]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Dixmier, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050840/i050840367.png" /> algebras" , North-Holland (1977) (Translated from French) {{MR|0498740}} {{MR|0458185}} {{ZBL|0372.46058}} {{ZBL|0346.17010}} {{ZBL|0339.17007}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> Harish-Chandra, "Collected papers" , '''1–4''' , Springer (1984) {{MR|}} {{ZBL|0652.01036}} {{ZBL|0561.01030}} {{ZBL|0561.01029}} {{ZBL|0546.01015}} {{ZBL|0546.01014}} {{ZBL|0546.01013}} {{ZBL|0541.01013}} {{ZBL|0527.01020}} {{ZBL|0527.01019}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D.A. Vogan, "Representations of real reductive Lie groups" , Birkhäuser (1981) {{MR|0632407}} {{ZBL|0469.22012}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> W. Casselman, D. Miličić, "Asymptotic behaviour of matrix coefficients of admissible representations" ''Duke. Math. J.'' , '''49''' (1982) pp. 869–930</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A.W. Knapp, B. Speh, "Status of classification of irreducible unitary representations" F. Ricci (ed.) G. Weiss (ed.) , ''Harmonic analysis'' , ''Lect. notes in math.'' , '''908''' , Springer (1982) pp. 1–38 {{MR|0654177}} {{ZBL|0496.22018}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Duflo, "Construction de représentations unitaires d'un groupe de Lie" , ''Harmonic analysis and group representations'' , C.I.M.E. &amp; Liguousi (1982) {{MR|0777341}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> J. Arthur, "A Paley–Wiener theorem for real reductive groups" ''Acta. Math.'' , '''150''' (1983) pp. 1–89 {{MR|0697608}} {{MR|0733803}} {{ZBL|0533.43005}} {{ZBL|0514.22006}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> W. Rossman, "Kirillov's character formula for reductive Lie groups" ''Invent. Math.'' , '''48''' (1978) pp. 207–220</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Duflo, G. Heckman, M. Vergne, "Projection d'orbites, formule de Kirillov et formule de Blattner" ''Mém. Soc. Math. France Nouvelle Série'' , '''15''' (1985) pp. 65–128 {{MR|0789081}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> P. Delorme, "Théorème de type Paley–Wiener pour les groupes de Lie semi-simple réels avec une seule classe de conjugaison de sous-groupes de Cartan" ''J. Funct. Anal.'' , '''47''' (1982) pp. 26–63</TD></TR></table>