Connected Semisimple Lie Group Research Articles

The Schwartz space of a general semisimple Lie group. II. Wave packets associated to Schwartz functions

Let G G be a connected semisimple Lie group. If G G has finite center, Harish-Chandra used Eisenstein integrals to construct Schwartz class wave packets of matrix coefficients and showed that every K K -finite function in the Schwartz space is a finite sum of such wave packets. This paper is the second in a series which generalizes these results of Harish-Chandra to include the case that G G has infinite center. In this paper, the Plancherel theorem is used to decompose K K -compact Schwartz class functions (those with K K -types in a compact set) as finite sums of wave packets. A new feature of the infinite center case is that the individual wave packets occurring in the decomposition of a Schwartz class function need not be Schwartz class. These wave packets are studied to obtain necessary conditions for a wave packet of Eisenstein integrals to occur in the decomposition of a Schwartz class function. Applied to the case that f f itself is a single wave packet, the results of this paper yield a complete characterization of Schwartz class wave packets.

Deformations of the representations of the fundamental groups of three-dimensional manifolds

The Well theorem [I] on local rigidity is one of the fundamental results of the theory of discrete subgroups of Lie groups. The theorem asserts that for any connected semisimple Lie group G without compact components whose Lie algebra does not have any sl(2: R) factors, the orbit of any uniform lattice F c G (with respect to the adjoint action adG) is open in Hom(F, G). The assertion follows from the fact that the cohomology group HI(F, Ad) is trivial (for the definitions, see e.g. [2]). The above result of Weil can often be generalized ([3-6], etc.). Garland and Raghunathan [7] proved the "disappearance theorem," according to which for any lattice F in a simple connected Lie group G of real rank 1 that is not locally isomorphic with SL2, the cohomology group HI(F, Ad) is equal to zero. Thurston [8] proved that the corresponding "disappearance theorem" is no longer valid for nonuniform lattices in the case G----SL~(C). Namely, if F is a lattice in SL2(C) that has no finite-order elements and n is the number of conjugacy classes of the maximal parabolic subgroups of F, then the complex dimension of Horn(F, SL2(C))/adSL2(C) at any point corresponding to an irreducible representation p is not less than n. Alternative proofs of this fact were presented in [9, i0]. However, each of the proofs rests upon some algebraic (or geometric) properties of the group SLy(C) , the representations in which were considered in these articles. ]in particular~ Thurston's proof was based on the fact that for any a, b~SL~(C) and for any word w(a, b) = i, the word w(a -l, b -I) is also equal to i.

The multiplicities of certain K-types in spherical representations

Let G be a connected semisimple Lie group with finite center, G 0 its Lie algebra. G 0 = K 0 ⊕ P 0 a Cartan decomposition of G 0 and let K be the corresponding maximal compact subgroup. We denote the complexification of P 0 by P . For an irreducible spherical representation, X, of G we compute dim Hom K ( P X). The result is given explicitly in terms of the Langlands parameters of X.

Geometric realization of discrete series for semisimple symmetric spaces

Let G be a connected semisimple Lie group, a an involution of G and H the connected component of the fixed point set of a containing the identity. Then G/H is a semisimple symmetric space. We assume G is a real form of a complex Lie group and rank(G/H)= rank(K/K c~ H) where K is a a-stable maximal compact subgroup of G. Flensted-Jensen, using his duality principle and Poisson transform, constructed countably many discrete series in U(G/H) [F-J i]. Then using deep results of hyperfunction theory and microlocal analysis, Oshima and Matsuki gave a detailed characterization of these discrete series, including a proof of Flensted-Jensen's conjecture "c = 0" [OM]. Based on these results we give the discrete series a geometric interpretation in locally symmetric spaces F\c~ where ~=G/K and F cG is a cocompact discrete group such that F c~ H c H is cocompact. Our geometric interpretation is the following: first of all to a Flensted-Jensen function ~ . which is the generating function of a discrete series ~V] we associate a harmonic form cb z on G with values in a finite dimensional G module V. Right G translates of (coefficients of) (5z realizes the contragredient ,:t~* of r and tba is left invariant under H, and under right K action it transforms according to the minimal K type with highest weight /~ of ~/~*. (3~ is the pull back of a V valued from coz on ~ , and V determines a locally constant bundle E on F \ @ so that if convergence holds

Representations of fundamental groups of manifolds with a semisimple transformation group

group ? 1 (M) , and the Lie groups that can act on M. More precisely, let G be a connected semisimple Lie group of higher real rank, and suppose G acts continuously on a (topological) manifold M, preserving a finite measure. The main theme of this paper is that the representation theory of 7r I(M) in low dimensions is to a large extent controlled by that of G (the latter of course being well understood). In particular, under natural hypotheses (e.g., that the action of G on M is engaging, i.e., there is no loss of ergodicity in passing to finite covers; see Definition 3.1 below), we prove that if G has no nontrivial

The Orders of Invariant Eigendistributions

In a series of papers, Harish-Chandra studied the invariant eigendistributions and in particular established a fundamental theorem which states that any invariant eigendistribution on a connected semisimple Lie group is locally L (cf. [H]). It may be available to reconsider this result from the view point of microlocal analysis. On the other hand, recently Hotta and Kashiwara [HK] have shown that the system of differential equations which governs an invariant eigendistribution on a semisimple Lie algebra is regular holonomic (=a holonomic system with regular singularities in the sense of [KK]). Among other things they showed, by using this result, that the holonomic system in question corresponds to the intersection cohomology complex defining Springer's representations of the Weyl group through the Riemann-Hilbert correspondence. In this paper we examine a microlocal property of the invariant eigendistributions. The results of this paper is quite unsatisfactory in comparison with those mentioned above. But the author hopes that our attempts will be developed in future. We now explain the contents shortly. In the first half we consider the homonomic system M^ which governs an invariant eigendistribution on a connected linear semisimple Lie group. An invariant of a holonomic system is the set ordA(u) of the orders along each irreducible component A of the characteristic variety of the system in question. Here u is a section of the system on the generic points of A. We attempt to determine ord^(w) for the system MI. Unfortunately, we cannot do it for every irreducible component A of the characteristic Ch(*SKx) but if an irreducible component A of Ch(^) satisfies the condition (A) in (3.1), we can calculate the orders along A. In this case, though c5^z is not a simple holonomic system in the sense of Sato-Kashiwara, ord^w) along such an irreducible component A consists of only one element 0. This is the main result of the first half (Theorem (3.4)). It rarely occurs that

A convexity theorem for semisimple symmetric spaces

In this paper we prove a generalization of the convexity theorem in [3] for a symmetric space G/H. Here G is a connected semisimple Lie group with finite center, σ an involution of G and H an open subgroup of the group G of fixed points for σ. The generalization involves Iwasawa decompositions related to minimal parabolic subgroups of G of arbitrary type instead of the particular type of parabolic subgroup considered in [3]. From now on we assume more generally that G is a real reductive group of the HarishChandra class; this will allow an inductive argument relative to the real rank of G. Let θ : G → G be a Cartan involution of G that commutes with σ; for its existence, see [20, Thm. 6.16]. The associated group K := G of fixed points is a maximal compact subgroup of G. For the infinitesimal involutions determined by σ and θ we use the same symbols: θ, σ : g → g; here g denotes the Lie algebra of G. With respect to the infinitesimal involutions, g decomposes as

On orbits of unipotent flows on homogeneous spaces, II

AbstractWe show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

Knapp-Wallach Szegö integrals and generalized principal series representations: The parabolic rank one case

To each discrete series representation of a connected semisimple Lie group G with finite center, a G-equivariant embedding into a generalized principal series representation is given. This representation is induced from specified parameters on a maximal parabolic subgroup of G and the mapping is defined by an integral formula, analogous to the Szegö integral introduced by Knapp and Wallach for a minimal parabolic subgroup. In a limiting case, embeddings of limits of discrete series representations are obtained and used to exhibit a reducibility theorem.

Continuous Equivariant Images of Lattice-Actions on Boundaries

Let G be a connected semisimple Lie group with finite center and R-rank at least 2. For any parabolic subgroup P of G the quotient space G/P (equipped with the G-quasi-invariant measure class) is called a boundary of G. In [9] G. A. Margulis proved that if F is any irreducible lattice in G then any measurable factor (equivariant image) of the non-singular F-action on any boundary G/P is again of the same form-viz., it is measurably isomorphic to the F-action on a boundary G/Q, where Q is a parabolic subgroup containing P. This result happens to be a major ingredient of his proof of finiteness of either the cardinality or the index of any normal subgroup of F as above. Here and in the sequel, by an irreducible lattice, we mean a lattice F such that FF is dense for any non-central normal subgroup (in particular, G can have no compact factors). The purpose of this note is to establish the following topological analogue of the above result. (See also Remark 3.2 for a generalisation.)

Characteristic classes and representations of discrete subgroups of Lie groups

A volume invariant is used to characterize those representations of a countable group into a connected semisimple Lie group G which are injective and whose image is a discrete cocompact subgroup of G. Let IT be a discrete cocompact subgroup of G and consider the analytic variety Hom(7r, G) consisting of homomorphisms 0: TT —> G. Denote by K a maximal compact subgroup of G and X = K\G the associated symmetric space. Let M be the orbit space X/TT. (For convenience we shall henceforth assume that n is torsionfree: by Selberg's lemma [12] this may be accomplished by replacing TT by a subgroup of finite index. This insures that M is a compact smooth manifold having TT as its fundamental group. The case when n has torsion follows from the torsionfree case with minor modifications but these modifications need not concern us here.) To every representation 0 G Hom(7r, G) we associate a foliated bundle E^ over M with fibre X and structure group G (see e.g. [6] ). If co is a closed Ginvariant differential fc-form on X then we may spread co over the fibres of E^ (copies of X) to obtain a closed fc-form co^ on E^. We define co(0) = fj^f*^^ where ƒ is any section of E^. Moreover co(0) is independent of the choice of section. For example taking co to be the G-invariant volume form on X we obtain a real number u(0) which depends on 0. When X is even-dimensional the Chern-Gauss-Bonnet theorem implies that u(0) may be described as an Euler number, i.e. the self-intersection number of any section, which is a topological invariant of E^. When X is odd-dimensional this volume invariant is related to a more recent kind of topological invariant (based on bounded cohomology and due to Gromov [3] ) and is constant on the connected components of Hom(7r, G). The volume invariant satisfies an inequality

A duality for symmetric spaces with applications to group representations. III. Tangent space analysis

1. IN~ODUCTI~N Let X = G/K be a symmetric space of the noncompact type, G being a connected semisimple Lie group with finite center and K a maximal compact subgroup. Let X0 be the tangent space to X at the fixed pont o of K and let G, denote the group of af%ne transformations of X,, generated by the translations and the natural action of K on X,, . The group GO which is the semi-direct product K x S X,, is often called the Car-tan motion group. While the irreducible unitary representations of GO are described by Mackey’s general semi-direct product theorem [13, p. 1311, and their characters given by Gindikm [4], our purpose here is to develop GO-invariant analysis on the space X0 . More specifically, let D(G,,/K) denote the algebra of G,,-invariant differential operators on Go/K = X,,. We are interested in the joint eigenfunctions of D(G,,/K) and in the representations of GO on the joint eigenspaces, ([8(d)] Ch. IV). As tools for this study we prove Paley-Wiener type theorems for the spherical transform and for the generalized Bessel transform on X0. The first of these comes from Theorem 2.1 which is an analog for entire functions of exponential type of Chevalley’s restriction theorem for polynomials. An important property of the generalized Bessel transform comes from Theorem 4.6 which shows that the generalized Bessel function is divisible by the Jg-matrix. Another useful tool is Theorem 4.8 which represents the generalized Bessel function as a certain derivative of the zonal spherical function. These results are used in Section 6 to derive an integral formula for the K-finite joint eigenfunctions of D(G,/K). This in turn is used to give (in Theorem 6.6) an explicit irreducibility criterion for the associated eigenspace representation of G,, . This is the principal application of our results. The results for the flat space X, = GO/K are to a large extent analogous to those proved in [8(g)] for the curved space X = G/K, most of the proofs are

On the Unitarized Adjoint Representation of a Semisimple Lie Group II

Let G be a connected semisimple Lie group with Lie algebra . Lebesgue measure on is invariant under the adjoint action of G; and so there is a natural unitary representation TG of G on L2 given by

Tensor Products of Finite and Infinite Dimensional Representations of Semisimple Lie Groups

study of the category of Harish-Chandra modules over a real semisimple Lie algebra equipped with a Cartan involution. In this paper we prove some very general theorems about tensor products of finite dimensional modules with Harish-Chandra modules. These theorems yield new structural information about the category of Harish-Chandra modules, as well as new information on the classification, construction, and properties of the infinite dimensional irreducible Harish-Chandra modules. We study the Harish-Chandra modules associated to a connected semisimple Lie group G having finite center. We let q, be the Lie algebra of G,

Spherical Functions on a Simply Connected Semisimple Lie Group

Spherical Functions on a Simply Connected Semisimple Lie Group

Composition series and intertwining operators for the spherical principal series. I

Let G be a connected semisimple Lie group with finite center and let K be a maximal compact subgroup. Let π \pi be a not necessarily unitary principal series representation of G on the Hilbert space H π {H^\pi } . If X π {X^\pi } denotes the space of K-finite vectors of H π , π {H^\pi },\pi induces a representation π 0 {\pi _0} of U ( g ) U(g) , the enveloping algebra of the Lie algebra of G, on X π {X^\pi } . In this paper, we determine when π 0 {\pi _0} is irreducible, and if π 0 {\pi _0} is not irreducible we determine the composition series of X π {X^\pi } and the structure of the induced representations on the subquotients. Explicit computation of the intertwining operators for the different principal series representations are obtained and their relationship to polynomials defined by B. Kostant are obtained.

The spherical Bochner theorem on semisimple Lie groups

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by C p( G)(0< p⩽2), (ii) the K-biinvariant elements in C p(G) by I p(G), (iii) the positive definite (zonal) spherical functions by P , and (iv) the spherical transform on C p( G) by ϑ → \\ ̂ gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto C l( G), (ii) there exists a unique measure μ of polynomial growth on P such that T[ ψ]=∫ pψdμ for all ψϵI 1( G) (iii) all measures μ of polynomial growth on P are obtained in this way, and (iv) T may be extended to a particular I p(G) space (1 ⩽ p ⩽ 2) if and only if the support of μ lies in a certain easily defined subset of P . These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.

Lie algebra cohomology of certain infinite-dimensional representations

Let G denote a connected semisimple Lie group, g its Lie algebra, gc the complexification of g, and G the conjugation of gc with respect to g. The Lie algebra cohomology of certain nilpotent subalgebras of gc plays an important role in the Bott-Borel-Weil theorem and its generalizations, notably in the works of Kostant [1], Schmid [3], and others. An important consideration is the regular behavior of the group actions associated to the Lie algebra cohomology complex. In this note we announce some results concerning the case where the nilpotent subalgebra mentioned above comes from an Iwasawa decomposition of g. This case is virtually the opposite of that considered by Schmid, and the aforementioned regularity is only partial. I would like to thank my thesis advisor, Robert Langlands, for his encouragement and generous advice.

Central Measures on Semisimple Lie Groups have Essentially Compact Support

In this paper it is shown that for a connected semisimple Lie group with no nontrivial compact quotient any finite central measure is a discrete measure concentrated on the center of the group. More generally, the largest possible support set for a central measure on any semisimple Lie group is determined. From these results it follows that the center of the algebra ${L_1}(H)$ is trivial for any locally compact group H which has a noncompact connected simple Lie group as a homomorphic image.

Central measures on semisimple Lie groups have essentially compact support

In this paper it is shown that for a connected semisimple Lie group with no nontrivial compact quotient any finite central measure is a discrete measure concentrated on the center of the group. More generally, the largest possible support set for a central measure on any semisimple Lie group is determined. From these results it follows that the center of the algebra L 1 ( H ) {L_1}(H) is trivial for any locally compact group H which has a noncompact connected simple Lie group as a homomorphic image.