of a Lie group
A representation of a Lie group (cf. 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), 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; Topological algebra).
Let be a Lie group. A representation of in a general sense is any homomorphism , where GL is the group of all invertible linear transformations of the vector space . If is a topological vector space, the homomorphisms which are usually considered are those with values in the algebra of all continuous linear transformations of or in the algebra of all weakly-continuous transformations of . The algebras and have one of the standard topologies (for example, the weak or the strong). A representation is said to be continuous (separately continuous) if the vector function is continuous (separately continuous) on . If is a quasi-complete barrelled space, any separately continuous representation is continuous. A continuous representation is called differentiable (analytic) if the operator function is differentiable (analytic) on . The dimension of a representation is the dimension of . The most important example of a representation of a group is its regular representation , , which can be defined on some class of functions on . If is a Lie group, its regular representation is continuous in and in (where is defined with respect to the Haar measure on ), and is differentiable in (with respect to the standard topology in : the topology of compact convergence). Every continuous finite-dimensional representation of a group is analytic. If 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 is compact, all its irreducible (continuous) representations are finite-dimensional. Similarly, if 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 ; the algebra , which is the completion of in the smallest regular norm (cf. Algebra of functions); — the algebra of all infinitely-differentiable functions on with compact support; — the algebra of all complex Radon measures with compact support on ; — the algebra of all generalized functions (Schwarz distributions) on G with compact support; and also, for a complex Lie group, the algebra of all analytic functionals over . The linear spaces , , are dual to, respectively, , , , where is the set of all holomorphic functions on (with the topology of compact convergence). All these algebras have a natural topology. In particular, is a Banach algebra. The product (convolution) of two elements , where is one of the group algebras indicated above, is defined by the equality
with respect to a right-invariant measure on , with a natural extension of this operation to the class of generalized functions. The integral formula
establishes a natural connection between the representations of the group and the representations of the algebra (if the integral is correctly defined): If the integral is weakly convergent and defines an operator for each , then the mapping is a homomorphism. One then says that the representation of the group is extended to the representation of the algebra , or that it is an -representation. Conversely, all weakly-continuous non-degenerate representations of the algebra are determined, in accordance with the formula above, by some representation of the group (weakly continuous for , weakly differentiable for , weakly analytic for ). This correspondence preserves all natural relations between the representations, such as topological irreducibility or equivalence. If is a unimodular group, its unitary representations (in Hilbert spaces, cf. Unitary representation) correspond to symmetric representations of the algebra with respect to the involution in (cf. Group algebra; Involution representation). If is a sequentially complete, locally convex Hausdorff space, any continuous representation of a group in is an -representation. If, moreover, the representation of the group is differentiable, it is a -representation. In particular, if is a reflexive or a quasi-complete barrelled space, any separately-continuous representation is an -representation, and for all .
The infinitesimal method.
If a representation is differentiable, it is infinitely often differentiable, and the space has the structure of a -module, where is the Lie algebra of the group , by considering the Lie infinitesimal operators:
The operators form a representation of the algebra , called the differential representation . A vector is said to be differentiable (with respect to ) if the vector function is differentiable on . A vector is said to be analytic if is an analytic function in a neighbourhood of the unit . If is a -representation, the space of all infinitely-differentiable vectors is everywhere-dense in . In particular, this is true for all continuous representations in a Banach space; moreover, in this case  the space of analytic vectors is everywhere-dense in . The differential representation in may be reducible, even if is topologically irreducible in . To two equivalent representations of correspond equivalent differential representations in (); the converse is, generally speaking, not true. For unitary representations in Hilbert spaces , it follows from the equivalence of differential representations in , that the representations are equivalent . 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 is said to be integrable (-integrable) if it coincides with a differential representation of the group in a subspace which is everywhere-dense in the representation space. Integrability criteria are now (1988) known only in isolated cases . If is simply connected, all finite-dimensional representations of the algebra are -integrable.
One of the main tasks of the theory of representations is the classification of all irreducible representations (cf. Irreducible representation) of a given group , 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 of all unitary equivalence classes of irreducible unitary representations of a group ; and 2) the description of the set of all Fell equivalence classes  of totally-irreducible representations (also called completely-irreducible representations) of a group . For semi-simple Lie groups with a finite centre, Fell equivalence is equivalent to Naimark equivalence , and the natural imbedding holds. The sets , have a natural topology, and their topologies are not necessarily Hausdorff . If is a compact Lie group, then is a discrete space. The description of the set in such a case is due to E. Cartan and H. Weyl. The linear envelope of matrix entries of the group (i.e. of matrix entries of the representations ) here forms a subalgebra in (the algebra of spherical functions) which is everywhere-dense in and in . The matrix entries form a basis in . If the matrices of all representations are defined in a basis with respect to which they are unitary, the corresponding matrix entries form an orthogonal basis in ( the Peter–Weyl theorem). If the group 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 , and became the starting point of an intensive development of the theory of unitary infinite-dimensional representations. G.W. Mackey's  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 () 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 be a semi-simple Lie group with a finite centre, let be an -representation in the space and let be a compact subgroup in . A vector is said to be -finite if its cyclic envelope is finite-dimensional with respect to . The subspace of all -finite vectors is everywhere-dense in and is the direct (algebraic) sum of subspaces , , where is the maximal subspace in in which the representation of is a multiple of . A representation is said to be -finite if for all . A subgroup is said to be massive (large or rich) if every totally-irreducible representation of is -finite. The following fact is of paramount importance in the theory of representations: If is a maximal compact subgroup in , then is massive. If the vectors of are differentiable, is invariant with respect to the differential of the representation . The representation is said to be normal if it is -finite and if the vectors of are weakly analytic. If is normal, there exists a one-to-one mapping (defined by restriction to ) between closed submodules of the -module and submodules of the -module , where is the Lie algebra of the group . Thus, the study of normal representations can be algebraized by the infinitesimal method. An example of a normal representation of the group is its principal series representation . This representation is totally irreducible for points in general position. In the general case can be decomposed into a finite composition series the factors of which are totally irreducible. Any quasi-simple irreducible representation of the group in a Banach space is infinitesimally equivalent to one of the factors of for a given . This is also true for totally-irreducible representations of in quasi-complete locally convex spaces. If is real or complex, it is sufficient to consider subrepresentations of instead of its factors . In the simplest case of , the representation is defined by a pair of complex numbers with integral difference , and operates in accordance with the right-shift formula , , , on the space of all functions which satisfy the homogeneity condition . If and are positive integers, contains the irreducible finite-dimensional subrepresentation (in the class of polynomials in ), the factors of which are totally irreducible. If and are negative integers, has a dual structure. In all other cases the module is totally irreducible. In such a case is in one-to-one correspondence with the set of pairs , where is an integer, factorized with respect to the relation . The subset consists of the representations of the basis series ( is purely imaginary) (cf. Series of representations), the complementary series and the trivial (unique) representation , which results if . Let be a semi-simple connected complex Lie group, let be its maximal solvable (Borel) subgroup, let be a maximal torus, let be a Cartan subgroup, and let be a character of the group (extended to ). Then is in one-to-one correspondence with , where is the set of all characters and is the Weyl group of the complex algebra . For characters in "general position" the representation is totally irreducible. The description of the set 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) (direct sums in ). All irreducible representations of the discrete series are classified  by describing the characters of these representations.
For nilpotent connected Lie groups  the set is equivalent to , where is the linear space dual to , and the action of in is conjugate with the adjoint representation on . The correspondence is established by the orbit method . A subalgebra is called the polarization of an element if annihilates and if
where is the orbit of with respect to (all orbits are even-dimensional). If is the corresponding analytic subgroup in and is a character of , the representation corresponding to is induced by . Here, is equivalent to if and only if the corresponding functionals lie on the same orbit . In the simple case of the group of all unipotent matrices with respect to a fixed basis in , the orbits of general position in are the two-dimensional planes and the points in the plane . To each orbit in general position corresponds an irreducible representation of the group , determined by the formula
in the Hilbert space . The infinitesimal operators of this representation coincide with the operators , , , where is the identity operator on . This result is equivalent to the Stone–von Neumann theorem on self-adjoint operators , with the commutator relationship . To each point corresponds a one-dimensional representation (a character) of . The set is then described in an analogous way, with values of the parameters 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 (where is the complexification of ), which satisfy certain integer conditions .
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 , which permits one to describe the irreducible unitary representations of a semi-direct product with normal subgroup in terms of irreducible representations of and of certain subgroups of the group (in view of the Levi–Mal'tsev theorem, cf. Levi–Mal'tsev decomposition). In practice, this method is only effective if the radical is commutative. Another method for studying (and also ) is the description of the characters of the irreducible unitary representations of ; the set of such characters is in one-to-one correspondence with . The validity of the general formula for characters, proposed by A.A. Kirillov , has been verified (1988) only for a few special classes of Lie groups.
Harmonic analysis of functions on .
For a compact Lie group, the harmonic analysis is reduced to the expansion of functions , , into generalized Fourier series by the matrix entries of the group (the Peter–Weyl theorem for and its analogues for other function classes). For non-compact Lie groups the foundations of harmonic analysis were laid in  by the introduction of the generalized Fourier transform
where is the operator of the elementary representation and is the Haar measure on , and by the introduction of the inversion formula (in analogy to the Plancherel formula) for for the case of classical matrix groups . This result was generalized to locally compact unimodular groups (the abstract Plancherel theorem). The Fourier transform converts convolution of functions on the group to multiplication of their (operator) Fourier images and is accordingly a very important tool in the study of group algebras. If is a semi-simple Lie group, the operators satisfy structure relations of the form
, , where are intertwining operators, is the Weyl group of the symmetric space ( is a maximal compact subgroup in ), and is the Weyl group of the algebra , where is the complexification of the Lie algebra of the group . If the functions have compact support, the operator functions are entire functions of the complex parameter . For the group algebras , , where is a semi-simple connected complex Lie group, analogues of the classical Paley–Wiener theorem  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 . 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 . 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 of a group to a subgroup (cf. 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 itself is considered, but rather one of its group algebras , and the problem of spectral analysis is treated as the study of two-sided ideals of the algebra . The problem of spectral analysis (and spectral synthesis) is also closely connected with the problem of approximation of functions on the group or on the homogeneous space , where is a subgroup, by linear combinations of matrix entries of the group .
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 . 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 . 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.
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The notions of a differentiable or analytic representation are commonly related to the strong topology .
The algebra (of generalized functions on with compact support) is usually denoted by in the West. The notation , if used, is then a synonym for .
Recently (1986), has been determined for , where is the field of real or complex numbers or the skew-field of quaternions (D.A. Vogan), for a complex simple Lie group of real rank 2 (M. Duflo) and for a split-rank or semi-simple real Lie group (Baldoni–Silva–Barbasch). For a survey of the current state-of-affairs see [a3], [a5].
|[a1]||J. Dixmier, " algebras" , North-Holland (1977) (Translated from French) MR0498740 MR0458185 Zbl 0372.46058 Zbl 0346.17010 Zbl 0339.17007|
|[a2]||Harish-Chandra, "Collected papers" , 1–4 , Springer (1984) 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|
|[a3]||D.A. Vogan, "Representations of real reductive Lie groups" , Birkhäuser (1981) MR0632407 Zbl 0469.22012|
|[a4]||W. Casselman, D. Miličić, "Asymptotic behaviour of matrix coefficients of admissible representations" Duke. Math. J. , 49 (1982) pp. 869–930|
|[a5]||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 MR0654177 Zbl 0496.22018|
|[a6]||M. Duflo, "Construction de représentations unitaires d'un groupe de Lie" , Harmonic analysis and group representations , C.I.M.E. & Liguousi (1982) MR0777341|
|[a7]||J. Arthur, "A Paley–Wiener theorem for real reductive groups" Acta. Math. , 150 (1983) pp. 1–89 MR0697608 MR0733803 Zbl 0533.43005 Zbl 0514.22006|
|[a8]||W. Rossman, "Kirillov's character formula for reductive Lie groups" Invent. Math. , 48 (1978) pp. 207–220|
|[a9]||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 MR0789081|
|[a10]||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|
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