Representations Of Semisimple Lie Groups Research Articles

Positive operators and automorphism groups of bounded symmetric domains

The spectrum of self-adjoint operators arising from unitary representations of semi-simple Lie groups is investigated. A series of irreducible unitary representations in which certain generators of non-compact one-parameter subgroups are realized by positive operators is described. These representations occur only for groups of automorphisms of bounded symmetric domains.

Classification of irreducible tempered representations of semisimple Lie groups

For each connected real semisimple matrix group, one obtains a constructive list of the irreducible tempered unitary representations and their characters. These irreducible representations all turn out to be instances of a more general kind of representation, here called basic. The result completes Langland's classification of all irreducible admissible representations for such groups. Since not all basic representations are irreducible, a study is made of character identities relating different basic representations and of the commuting algebra for each basic representation.

Some Properties of Square-Integrable Representations of Semisimple Lie Groups

In the theory of irreducible representations of a compact Lie group, the formula for the multiplicity of a weight and the so-called theorem of the highest weight are among the important results. At least conjecturally, both of these statements have analogues for the discrete series of representations of a semisimple Lie group. Let G be a connected, semisimple Lie group, K c G a maximal compact subgroup, and suppose that rk K = rk G. Exactly in this situation, G has a non-empty discrete series [8]. Blattner's conjecture predicts how a given discrete series representation should break up under the action of K; precise statements can be found in [10], [15], [16]. Formally, the conjectured multiplicity formula looks just like the formula for the multiplicity of a weight. Partial results toward the conjecture have been proved in [10], [15]. More recently, the full conjecture was established for those linear groups G, whose quotient G/K admits a Hermitian symmetric structure [16]. As this paper was being completed, H. Hecht and I succeeded in proving Blattner's conjecture for all linear groups, by extending the arguments of [16]. According to Blattner's conjecture, any particular discrete series representation 7t contains a distinguished irreducible K-module V, with multiplicity one; moreover, w contains no irreducible K-module with a highest weight which is lower, in the appropriate sense, than that of V,:. For most discrete series representations, it was known that these two properties characterize at, up to infinitesimal equivalence, among all irreducible representations of G [10], [15]. In this paper, I shall give an infinitesimal characterization, by lowest K-type, for all discrete series representations. The result, which is stated as Theorem (1.3) below, closely resembles the theorem of the highest weight. I shall also draw a number of conclusions from it. The methods of this paper have some further, less immediate consequences, which will be taken up elsewhere. For the remainder of the introduction, I assume that G is a linear group.

Supplements and corrections to my paper : The characters of some induced representations of semisimple Lie groups

Supplements and corrections to my paper : The characters of some induced representations of semisimple Lie groups

On the tensor product of representations of semisimple Lie groups

The problem of the decomposition of the tensor product of finite and infinite representations of a complex semigroup of a Lie group is examined by using the theory of characters of completely irreducible representations. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite and elementary representation.

Square Integrable Representations of Semisimple Lie Groups

Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component ${A_0}(D)$ of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle $E \to D$ admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle $E \to D$ must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of $E \to D$ by defining a map $\Phi :G \to {\text {GL}}(V)$ (V being the fiber of the vector bundle) which extends the classical “universal factor of automorphy” for the action of ${A_0}(D)$ on D. Then, we study the space H of all square integrable holomorphic sections of $E \to D$. The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over $G/Z$ (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in ${L_2}(G/Z)$.

Square integrable representations of semisimple Lie groups

Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component A 0 ( D ) {A_0}(D) of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle E → D E \to D admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle E → D E \to D must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of E → D E \to D by defining a map Φ : G → GL ( V ) \Phi :G \to {\text {GL}}(V) (V being the fiber of the vector bundle) which extends the classical “universal factor of automorphy” for the action of A 0 ( D ) {A_0}(D) on D. Then, we study the space H of all square integrable holomorphic sections of E → D E \to D . The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over G / Z G/Z (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in L 2 ( G / Z ) {L_2}(G/Z) .

Spectra in Representations of Semisimple Lie Groups

Spectra in Representations of Semisimple Lie Groups

Algebraic Results on Representations of Semisimple Lie Groups

Let G be a noncompact connected real semisimple Lie group with finite center, and let K be a maximal compact subgroup of G. Let $\mathfrak {g}$ and $\mathfrak {k}$ denote the respective complexified Lie algebras. Then every irreducible representation $\pi$ of $\mathfrak {g}$ which is semisimple under $\mathfrak {k}$ and whose irreducible $\mathfrak {k}$-components integrate to finite-dimensional irreducible representations of K is shown to be equivalent to a subquotient of a representation of $\mathfrak {g}$ belonging to the infinitesimal nonunitary principal series. It follows that $\pi$ integrates to a continuous irreducible Hilbert space representation of G, and the best possible estimate for the multiplicity of any finite-dimensional irreducible representation of $\mathfrak {k}$ in $\pi$ is determined. These results generalize similar results of Harish-Chandra, R. Godement and J. Dixmier. The representations of $\mathfrak {g}$ in the infinitesimal nonunitary principal series, as well as certain more general representations of $\mathfrak {g}$ on which the center of the universal enveloping algebra of $\mathfrak {g}$ acts as scalars, are shown to have (finite) composition series. A general module-theoretic result is used to prove that the distribution character of an admissible Hilbert space representation of G determines the existence and equivalence class of an infinitesimal composition series for the representation, generalizing a theorem of N. Wallach. The composition series of Weylgroup-related members of the infinitesimal nonunitary principal series are shown to be equivalent. An expression is given for the infinitesimal spherical functions associated with the nonunitary principal series. In several instances, the proofs of the above results and related results yield simplifications as well as generalizations of certain results of Harish-Chandra.

Spectra in representations of semisimple Lie group

The spectrum of the infinitesimal generator of a one-parameter group of unitary operators arising from a representation of a semisimple Lie group is determined. The support of the spectral measure depends only on whether the group is a group of automorphisms of a bounded symmetric domain. 1. Let [I be a (strongly) continuous unitary representation of a semisimple Lie group G on a separable Hilbert space H. Let C' (ll) denote the set of all vectors v EC H such that HI()v is a C' function on G. Then C'(T) is a dense linear subspace of H, and contains the dense set of analytic vectors for the representation [6]. We denote the Lie algebra of G by g0. If x E g0, (II(exp(tx))) is a oneparameter group of unitary operators and has a skew-adjoint infinitesimal generator dll(x). Thus IH(exp(tx))=exp(t dJl(x)), t E R. Also C'(1I) is contained in the domain of dHl(x), and is invariant under dJl(x) for all x e g0. Moreover, x-dr1(x)IC'([J) defines a representation of g0 by essentially skew-adjoint operators. We shall examine the spectrum of the selfadjoint operator -idf(x) whenever exp(tx) satisfies a certain noncompactness condition explained below. The information obtained will depend on G but not on HI or x. More precisely, if f A dE, is the spectral resolution of i dJJ(x), C. C. Moore [5] has shown that the projection valued measure E is absolutely continuous with respect to Lebesgue measure, and its support is either (-oo, oo), (0, xo) or (o, 0). In the first instance, we shall say -i dll(x) has two-sided spectrum, while in the second and third instances we shall say idfl (x) has one-sided spectrum. One's feeling, and this is borne out by our investigations, is that most of the time we can expect a two-sided spectrum, and that a one-sided Received by the editors February 6, 1973. AMS (MOS) subject classifications (1970). Primary 22D10, 22E45, 47B1 5; Secondary 32M15, 81A78.

Algebraic results on representations of semisimple Lie groups

Algebraic results on representations of semisimple Lie groups

On the boundedness and unboundedness of certain convolution operators on nilpotent Lie groups

One method of proving irreducibility of the “principal series” representations of semisimple Lie groups involves showing that a certain nonintegrable function on a nilpotent subgroup X X cannot be regularized to give a bounded convolution operator on L 2 ( X ) {L_2}(X) . This note gives an elementary proof of this unboundedness property for the groups X X which occur in real-rank one semisimple groups.

Cyclic vectors and irreducibility for principal series representations.

Canonical sets of cyclic vectors for principal series representations of semisimple Lie groups having faithful representations are found. These cyclic vectors are used to obtain estimates for the number of irreducible subrepresentations of a principal series representations. The results are used to prove irreducibility for the full principal series of complex semisimple Lie groups and for S L ( 2 n + 1 , R ) , n ≧ 1 SL(2n + 1,R),n \geqq 1 .

Representations of Noncompact Semisimple Lie Groups

A method is presented for the construction of unitary representations of semisimple Lie groups (or, more precisely, of the corresponding algebras), proceeding directly from the commutation relations among the canonical generators e±α and hα. In the case of the orthogonal groups, the correspondence between the canonical generators and the more usual tensor generators is written down explicitly. It is shown how this method can be used to construct a certain class of representations (bounded above or below) of the Lie algebras of the noncompact groups SU(p, q) and SO(p, q), which arise in the dynamical group treatment of certain physical systems.

APPLICATIONS OF THE RADON TRANSFORM TO REPRESENTATIONS OF SEMISIMPLE LIE GROUPS

From the point of view of the duality between points and horocycles in a symmetric space, the counterparts to the spherical functions on the symmetric space are the conical distributions on the manifold of horocycles. While the conical functions are closely related to certain finite-dimensional representations of semisimple Lie groups, in the present work the conical distributions are found to play various roles in the principal series of infinitedimensional representations of these groups.

The characters of some induced representations of semisimple Lie groups

The characters of some induced representations of semisimple Lie groups

The characters of some induced representations of semisimple Lie groups

The characters of some induced representations of semisimple Lie groups

Representation of a Group in Relativistic Quantum Mechanics

ESSENTIALLY all infinite unitary representations of semi-simple Lie groups are known1, they are of Murray–von Neumann2 type I. Further, while it has been known for a long time that non-relativistic quantum mechanics is such a representation for the rotation group of space3, the same does not carry over to relativistic quantum mechanics. Infinite representations of the Lorentz group4 do not correspond to anything physically identifiable. This, among other things, suggests that the underlying group of relativistic quantum theory is not the Lorentz group but rather one for which non-unitary representations have a special significance.

Representations of Semisimple Lie Groups VI: Integrable and Square-Integrable Representations

Representations of Semisimple Lie Groups VI: Integrable and Square-Integrable Representations

Representations of Semisimple Lie Groups, V

Representations of Semisimple Lie Groups, V