Noncompact Semisimple Lie Group Research Articles

Fundamental groups of negatively curved manifolds and actions of semisimple groups

THIS PAPFR is part of the investigation of the fundamental groups of manifolds on which a non-compact semisimple Lie group may act. Earlier work in this direction has dealt largely with the linear representation theory of the fundumcntal group. In this paper we consider homomorphisms into isomctry groups of negatively curved manifolds. and in particular the question as to when a scmisimplc Lie group may act on a manifold whose fundamcnt;ll group is isomorphic to th;tt of a ncgativcly curved manifold.

Invariant Differential Operators on Hermitian Symmetric Spaces

Our object of study is the ring of invariant differential operators on a hermitian symmetric space G/K of classical and noncompact type. Here, as usual, G is a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup of G. It is well-known that the ring, denoted by ?(G/K), is isomorphic to a polynomial ring of 1 variables, 1 being the rank of G/K. This is true even in the nonhermitian case. If 1 = 1, it is generated by the Laplace-Beltrami operator z/ which is essentially self-adjoint; moreover -Iz2 is nonnegative. In the general case, an easily posable problem is to find an explicitly defined set of generators. We can go one step further by focusing our attention on the nonnegativity, which necessarily limits the range of eigenvalues under a suitable integrability condition on functions. It is thus natural to ask whether there exist some canonically defined operators which generate -9(G/K) and have the property of nonnegativity. The main purpose of the present paper is to give an affirmative answer to this question. In fact, our answer applies to a somewhat more general type of rings of differential operators which includes 9(G/K) as a special case. To be explicit, we take an irreducible representation p: K -* GL(V) with a complex vector space V of finite dimension, and consider the set C'(p) of all V-valued Cx functions f on G such that f(xk -1) = p(k)f(x) for every k e K. We then denote by ?(p) the ring of left-invariant difrrential operators on G which map C'(p) into itself. Now the complexification g of the Lie algebra of G has abelian subalgebras p+ and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on G/K at the origin. For any complex vector space W let Sr(W) denote the ring of all complex-valued homogeneous polynomial functions on W of degree r, and let S(W) = Er=0Sr(W). Through the adjoint representation of G on g, K acts naturally on Sr( ?+). It is a known fact, due to Hua and Schmid, that each irreducible constituent of this representation of K has multiplicity one. Now our principal

Lyapunov exponents, symmetric spaces, and a multiplicative ergodic theorem for semisimple Lie groups

The classical theory of Lyapunov characteristic exponents is reformulated in invariant geometric terms and carried over to arbitrary noncompact semisimple Lie groups with finite center. A multiplicative ergodic theorem (a generalization of a theorem of Oseledets) and the global law of large numbers are proved for semisimple Lie groups, as well as a criterion for Lyapunov regularity of linear systems of ordinary differential equations with subexponential growth of coefficients.

Local Uncertainty Inequalities for Locally Compact Groups

Let $G$ be a locally compact unimodular group equipped with Haar measure $m$, $\hat G$ its unitary dual and $\mu$ the Plancherel measure (or something closely akin to it) on $\hat G$. When $G$ is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given $\theta \in \left [ {0,\tfrac {1} {2}} \right .)$ there exists a constant ${K_\theta }$ such that for all $f$ in a certain class of functions on $G$ and all measurable $E \subseteq \hat G$, \[ {\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f)) d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2}\] where ${\phi _\theta }$ is a certain weight function on $G$ (for which an explicit formula is given). When $G = {{\mathbf {R}}^k}$ the inequality has been established with ${\phi _\theta }(x) = |x{|^{k\theta }}$.

On an analogue of the Wiener Tauberian theorem for symmetric spaces of the noncompact type

Let X be a symmetric space and G the connected component of the group of isometries of X. If / G Lι (X), we consider conditions under which Sp{*/: g e G} is dense in Lι(X) in terms of the of /. This continues earlier work on this kind of problem by L. Ehrenpreis and F. I. Mautner, R. Krier and the author. 1. Introduction. Let /be an integrable function on R. Then we have the famous Wiener-Tauberian theorem: If the Fourier transform / is a nowhere vanishing function on i?, then the ideal generated by / is dense in Lι(R). In [EMI] Ehrenpreis and Mautner observed that the exact analogue of the above theorem is no longer true if one considers the commutative Banach algebra of ^-bi-invarian t Lι-functions on a non-compact semi-simple Lie group G, where K is a maximal compact subgroup of G—i.e. Let I (G) denote the commutative Banach algebra of Λ^-bi-invaria nt Lι-functions on G. For / e I\(G) let / denote its spherical Fourier transform (see §2). Then there exist functions / G I\(G) such that / is nowhere vanishing on the maximal ideal space M of I (G) and yet the algebra generated by / is not dense in h{G). However when G = SL(2, i?) they were able to show that a modified version of Wiener's theorem is true i.e. / nowhere vanishing on M together with the condition that it does not go to zero too fast at oo would indeed imply that the ideal generated by / is dense in I\(G). (Theorems 6 and 7 of [EMI].) This kind of result has been generalized by R. Krier [Kl] when G is a non-compact connected semisimple Lie group of real rank 1 and by the author for G of arbitrary rank [Si].

Local uncertainty inequalities for locally compact groups

Let G G be a locally compact unimodular group equipped with Haar measure m m , G ^ \hat G its unitary dual and μ \mu the Plancherel measure (or something closely akin to it) on G ^ \hat G . When G G is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given θ ∈ [ 0 , 1 2 ) \theta \in \left [ {0,\tfrac {1} {2}} \right .) there exists a constant K θ {K_\theta } such that for all f f in a certain class of functions on G G and all measurable E ⊆ G ^ E \subseteq \hat G , \[ ( ∫ E Tr ⁡ ( π ( f ) ∗ π ( f ) ) d μ ( π ) ) 1 / 2 ⩽ K θ μ ( E ) θ | | ϕ θ f | | 2 {\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f))\,d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2} \] where ϕ θ {\phi _\theta } is a certain weight function on G G (for which an explicit formula is given). When G = R k G = {{\mathbf {R}}^k} the inequality has been established with ϕ θ ( x ) = | x | k θ {\phi _\theta }(x) = |x{|^{k\theta }} .

Some remarks on measures on noncompact semisimple Lie groups

This paper answers a question posed by K. R. Parthasarathy: Let X be a symmetric space of non-compact type and G the connected component of the group of isometries of X. Let m be the canonical G-invariant measure on X and E a Borel set in X such that E is compact and 0 < m(E) < ∞. If μ,ν are probability measures on X such that μ(g ⋅ E) = ν(g ⋅ E) for all g ∈ G, then is μ = ν? We answer the question in the affirmative (Theorem A) and also find that the condition “E is compact” is unnecessary. A special case of this problem (under the condition that μ and ν are K-invariant probabilities on X, where K is a maximal compact subgroup of G) was settled by I. K. Rana.

Discrete Series for Semisimple Symmetric Spaces

We give a sufficient condition for the existence of minimal closed G-invariant subspaces of L2(G/H) for a semisimple symmetric space G/H. As a semisimple Lie group with finite center may always be considered as a symmetric space, this provides, in particular, a new and elementary proof of Harish-Chandra's result that G has a discrete series if rank (G) = rank (K), where K is a maximal compact subgroup. Let G be a connected noncompact semisimple Lie group, let z be an involution on G, and let H be the connected component of the fixed-point group Gr containing the identity. Then G/H is a semisimple symmetric space, and the group G acts by left translation on C*(G/H) and L2(G/H). In the introduction we will, for simplicity, assume that G has a finite center. By the discrete series for G/H we shall mean the set of equivalence classes of the representations of G on minimal closed invariant subspaces of L2(G/H). Let a be a Cartan involution commuting with z. The fixed-point group K for a is a maximal compact subgroup. Our main result is THEOREM 1.1. The discrete series for G/H is nonempty and infinite if

Global solvability of the laplacians on pseudo-Riemannian symmetric spaces

Let G be a noncompact semisimple Lie group with finite center and H an open subgroup of the fixed point group of an involution of G. G H becomes a pseudo-Riemannian manifold. We prove that the Laplacian P on G H is globally solvable in the sense that PC ∞( G H ) = C ∞( G H ) . This generalizes the global solvability of the Casimir operators on non-compact semisimple Lie groups with finite center due to J. Rauch and D. Wigner.

Expansions for spherical functions on noncompact symmetric spaces

Section 0 Lot G be a connected noncompact semisimple Lie group with finite center and real rank one; fix a maximal compact subgroup K. Our concern in this paper is Fourier analysis on the Riemannian symmetric space G]K. We shall analyze the local and global behavior of spherical functions, the boundedness of multiplier operators, and the inversion of differential operators. The core of the paper, however, is an analysis of how the size of a function is controlled by the size of its Fourier transform. There is an extensive literature on such topics, centered about the Paley-Wiener and Plancherel theorems. Our work relies heavily on these earlier ideas and techniques, to which detailed reference will be made in the body of the paper. The problems we wish to solve, however, require estimates more precise and of a different nature than are necessary for the Plancherel or Paley-Wienor theorem. Thus the first two sections of this paper are devoted to the construction of various asymptotic expansions for spherical functions; in later sections we show how these expansions may be applied to the Fourier analysis of G/K.

An algebraic construction of a class of representations of a semi-simple Lie algebra

It is weltknown that the discrete class representations of a semisimple Lie group form the building blocks for the representation theory of semisimple Lie groups. Several attempts have been made to realize these representations by proving analogoues of the Borel-Weil-Bott theorem for noncompact symmetric spaces. We note that in any such analogue there is no "ab initio" proof that the space of "LZ-harmonic forms" concerned is nonzero. In this paper we give a straightforward algebraic construction of a class of irreducible, infinitesimally unitarizable representations of a semisimple Lie algebra. This class contains a special subseries of the discrete series. Our method is by explicitly constructing (through algebraic results about existence and uniqueness) certain operators on the direct sum of some cohomology spaces (of bundles on a compact flag manifold); the operators so defined will represent the given Lie algebra. Going into the details of this paper one can see that our construction has some applications (among them, for example, is a proof of Blattner's conjecture for the special subseries of the discrete series). In [1] Enright and Varadarajan obtained some modules which include all discrete classes. Recently (i.e. at the time of writing up this paper) Schmid has also obtained some modules which include all discrete series. We now begin to describe our results in detail. Let G be a connected noncompact semisimple Lie group with finite center. Let K be a maximal compact subgroup of G. Assume rank of of K = rank of G. Let T C K C G be a Cartan subgroup of G. Let t c k c g denote the Lie algebras of T, K, and G respectively, Let tCc kCc gc denote the complexifications of t C k C g. Let S be the set of roots of( t c, gC) and P C S a positive system of roots. Let Pk and P, denote the set of compact and noncompact roots in P respectively so that P = P k u P , . Let b c k c be the Borel subalgebra of k c, defined by

A proof of Blattner's conjecture

Let G be a connected, semisimple Lie group. In Harish-Chandra's work on the Plancherel formula, the discrete series of irreducible unitary representations [5] plays a crucial role. Roughly speaking, besides the discrete series itself, the representations which enter the Plancherel decomposition of L2(G) can all be built up from discrete series representations of subgroups of G. Among the various irreducible unitary representations, the discrete series representations are therefore of particular importance. The main technique in the representation theory of compact Lie groups is to investigate the restriction of any given representation to a maximal torus. Similarl~r to understand the structure of an irreducible representation of a noncompact semisimple Lie group, one may try to determine its restriction to a maximal compact subgroup. Blattner's conjecture predicts how the discrete series representations should break up under the action of a maximal compact subgroup. It formally resembles the formula for the multiplicity of a weight for finite-dimensional representations. A companion statement to Blattner's conjecture, analogous to the theorem of the highest weight, characterizes discrete series representations, up to infinitesimal equivalence, in terms of their restriction to a maximal compact subgroup [13]. Once and for all, we make the assumption that G contains a compact Cartan subgroup. Exactly in this situation G has a non-empty discrete series [5]. Let K be a maximal compact subgroup, with HcK. The Lie algebras of G, K, H will be written as go, ~0, bo, and their complexifications as g, [, b. A non-zero root of (g, b) is called compact or noncompact, depending on whether or not its root space lies in L We denote the sets of compact and noncompact roots by, respectively, q~ and 4"; ~ = ~c w 4" is thus the setof all non-zero roots of (g, b). The Killing form of g induces an inner product ( , ) on ib*, the space of linear functions on b which assume imaginary values on b0An element ~teib* is said to be singular if it is perpendicular to at least one ct~q~, and nonsingular otherwise. The differentials of the characters of H form a lattice A cib*, which contains ~b. Since

Global Solvability of the Casimir Operator

Let G be a semisimple Lie group, g its Lie algebra. The Killing form on g is a nondegenerate symmetric bilinear form which induces an isomorphism a: g* -. g. The Casimir operator is the unique biinvariant differential operator on G which annihilates constants and whose principal symbol is the pullback of the Killing form under the isomorphism a. Let C denote the Casimir operator and C-(G) the space of infinitely differentiable functions on G. THEOREM. If G is a non-compact semisimple Lie group with finite center, f e C-(G), then there is a u e C-(G) such that Cu = f. Since G is not compact, C is not elliptic but ultrahyperbolic. Analogous questions have been considered by S. Helgason [3] and A. Cerezo and F. Rouviere [1]. Helgason proves local solvability of all biinvariant differential operators and Cerezo and Rouviere prove global solvability of the Casimir operator on complex semisimple Lie groups. The latter authors prove that the imaginary part of the complex Casimir operator on SL(2, C) is not globally solvable. This shows that not every biinvariant operator is globally solvable. Henceforward G will denote a group satisfying the hypotheses of the theorem and g will denote the Lie algebra of left invariant vector fields on G. Proof. The remainder of the paper is devoted to a proof of this theorem. In spite of its length the proof is simple in outline. A result from functional analysis and Hormander's propagation of singularities theorem can be combined to show that it suffices to demonstrate three things.

Matrix elements of tensor operators

The Wigner-Eckart theorem for matrix elements of tensor operators transforming according to a finite-dimensional representation of a non-compact semisimple Lie group is proved. The applications of this theorem are discussed.

L p -multipliers for Noncompact Symmetric Spaces

Let G be a real noncompact semi-simple Lie group with finite center and K a maximal compact sub-group. The symmetric space M = G/K carries a measure invariant under the action of G. The operators which map L(p)(M) continuously into itself and commute with the action of G, can be easily characterized when p = 2 or p = 1. This note gives some results on "singular integrals" which map L(p) into itself (1 < p < + infinity).

The tail of a positivity-preserving semigroup

With certain assumptions a representation theorem is proved for the elements of $$ \cap _{\sigma \in \Sigma } $$ σS, where Σ is an abelian semigroup of, endomorphisms of a real vector space, andS is a convex antisymmetric cone. Application is made to chacterization of nonnegative harmonic functions on bounded Lipschitz domains, of Hausdorff-Stieltjes moment sequences, and of “bilateral Laplace transforms” on locally compact abelian groups, Euclidean motion groups, and noncompact semi-simple Lie groups. Uniqueness of the representation is proved in both the Euclidean motion and the semi-simple cases.

An Analogue of the Borel-Weil-Bott Theorem for Hermitian Symmetric Pairs of Non-Compact Type

Let G be a connected non-compact semi-simple Lie group admitting a finite dimensional faithful representation. Let K be a maximal compact subgroup of G. We suppose that GIK is hermitian symmetric. When G -SL(2, R), V. Bargmann [31 constructed the discrete series for G on spaces of square-integrable holomorphic functions on GIK (which is isomorphic to the unit disc) and in the general case Harish-Chandra (cf. [5]) constructed part of the discrete series for G in a similar way. (See also [6].) However, in the general case, this method does not yield all the discrete series. In analogy with the Borel-Weil-Bott theorem [4, 8(a)], it was suggested in [91 that all the discrete classes (which were obtained by Harish-Chandra in [7(f)]) might be realized on spaces of square-integrable harmonic forms of type (0, q) (square-integrable D-cohomology spaces) with coefficients in holomorphic vector bundles on GIK arising from finite dimensional irreducible unitary representations of K. (When q = 0, we have the case which was considered in [5].) We prove in this paper that most of the discrete classes are obtained in this way. We proceed to describe the results of this paper in more detail. Let g and f be the Lie algebras of G and K respectively. Let t be a Cartan subalgebra of t. Then t is also a Cartan subalgebra of g. Choose an ordering on the roots of (gC, tc) compatible with the complex structure on GIK. For an irreducible unitary representation z-A of K with the highest weight A, we denote by the holomorphic vector bundle on GIK associated to the contragredient representation. Let H,,q(EA) denote the Hilbert space of square-integrable harmonic forms of type (0, q) with coefficients in (these are the squareintegrable a-cohomology spaces attached to EA defined in [11]). The unitary representation wr of G on H,,q(EA) decomposes into a finite number of irreducible representations each of which belongs to the discrete series for G (Proposition 3.1). Let p denote the half sum of positive roots of (gC, tc) and let qA

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.

Remarks on Differentiable Actions of Non-Compact Semi-Simple Lie Groups on Euclidean Spaces

Introduction. It is well known that differentiable actions of compact Lie groups are far more regular than those of the non-compact Lie groups and the behavior of compact Lie group actions are comparatively much more well understood than those of the non-compact Lie groups. Of course, one of the most basic and important cases of non-compact Lie group actions are those Rl-actions which are naturally associated with ordinary differential equations [6 ]. So far, much important research has been done in that direction. However, knowledge about R'-actions still seems rather inadequate. Since the group R1 is a simply connected one-dimensional Lie group and it contains no non-trivial compact subgroups, neither the theory of Lie groups nor knowledge about compact Lie group actions are of any help in the study of R'-actions. In this note, we try to look at the other extreme case of actions of large semi-simple non-compact Lie groups. The situation here is quite different from the study of RI-actions. Roughly speaking, the problem is more complicated on the one hand, and has much more built in structures on the other hand. For example, one naturally expects that the profound theory of semisimple Lie groups and the knowledge of compact Lie grfoup actions might be useful at least for some special but natural cases. Due to the fact that one knows so little about the behavior of differentiable actions of non-compact Lie groups in general, it seems to be worthwhile to investigate some special but natural cases such as semi-simple Lie group actions on euclidean spaces. Let G be a semi-simple non-compact Lie group. It is well known that there exists a maximal compact subgroup H which is unique up to conjugacy. For a given differentiable manifold M'4, let r(GMm]Ii4) and r (H, Mm) be the set of equivalence classes of differentiable actions of G and H on Ml', respectively. To restrict those G-actions on MIlm to H induces a natural luap i !: r (G, Mm) r (H, Mm) which is independent of the choice of the maximal compact subgroup H. A reasonable way to study G-actions on Mm is to investigate the image and the kernel of the above map. To be more precise,

Fatou's Theorem for Symmetric Spaces: I

and Fatou's theorem is the assertion that, as r tends to 1, h tends to f almost everywhere. If the disc and the circle are viewed as homogeneous spaces of the group SL(2, R), the setting for this theorem may be generalized as follows. Let G be any connected non-compact semi-simple Lie group with finite center, and let K be a maximal compact subgroup. G/K is the symmetric space of G, and a complex-valued function on GIK is harmonic if it is annihilated by every G-invariant differential operator on G/K. In [3] Furstenberg exhibited a Poisson integral formula for the bounded harmonic functions on GIK, and he generalized his results to positive harmonic functions as part of [4]. Furstenberg knew that the boundary (analogous to the circle) was a homogeneous space of G (and actually of K), and he wrote the Poisson kernel as Radon-Nikodym derivatives of the action of G on the K-invariant measure on the boundary. Moore [9] identified the boundary explicitly, and a concrete formula for the kernel followed from calculations of Harish-Chandra in [5]. Now a symmetric space admits polar coordinates, in which the radial direction is indexed by a cone in a euclidean space and the other coordinate is indexed by the boundary. A theorem of Fatou type would say that, as the radial coordinate tends to co in some fashion, the Poisson integral of an integrable function on the boundary tends to the function at almost every point of the boundary. Theorems of this sort are known in several special cases. In addition to