Semisimple Symmetric Space Research Articles

Fundamental Groups of Semisimple Symmetric Spaces, II

The aim of this report is to determine the fundamental group of an arbitrary irreducible semisimple symmetric space $G/H$ when $G$ is a connected semisimple Lie group with trivial center. The fundamental group $\pi_1(G/H)$ is well-known if $G/H$ is Riemannian. Therefore, we restrict our attention to the case where $G/H$ is non-Riemannian so both $G$ and $H$ are not compact. The result is summarized in Table 4.

Harmonic analysis on compact polar homogeneous spaces

In this paper we are concerned with the existence of discrete series for semisimple homogeneous spaces. This leads to the definition of polar homogeneous spaces. We first study the harmonic analysis on compact polar homogeneous spaces. The proof of the existence of disrete series and the Plancherel formula for non-compact and non-symmetric polar homogeneous spaces will be given in a consequent paper. One of the greatest achievements of mathematics in twentieth century is Harish-Chandra's work on discrete series for semisimple Lie groups. Suppose G is a semisimple Lie group with a maximal compact Lie group K. Harish-Chandra proved that G has discrete series if and only if the real rank of G is equal to the rank of K ([HC]). In other words, G has discrete series if and only if G has a compact Cartan subgroup. Harish-Chandra also obtained a classification of the discrete series for semisimple Lie groups. Harish-Chandra's results on discrete series have been generalized to semisimple symmetric spaces. Flensted-Jensen ([FJ]) proved that a semisimple symmetric space G/H has discrete series if

Correction to: “A convexity theorem for semisimple symmetric spaces”

Correction to: “A convexity theorem for semisimple symmetric spaces”

A convexity theorem for semisimple symmetric spaces

In this paper we prove a convexity theorem for semisimple symmetric spaces which generalizes Kostant's convexity theorem for Riemannian symmetric spaces. Let τ be an involution on the semisimple connected Lie group G and H = GQ the 1-component of the group of fixed points. We choose a Cartan involution θ of G which commutes with τ and write K = Gθ for the group of fixed points. Then there exists an abelian subgroup A of G, a subgroup M of K commuting with A, and a nilpotent subgroup N such that HMAN is an open subset of G and there exists an analytic mapping L: HMAN -* α = h(A) with L(hman) = log a. The set of all elements in A for which aH C HMAN is a closed convex cone. Our main result is the description of the projections L(aH) C α for these elements as the sum of the convex hull of the Weyl group orbit of log a and a certain convex cone in α. 0. Introduction. If G is a connected semisimple Lie group and G — KAN an Iwasawa decomposition, then the convexity theorem of Kostant describes the image of the sets aK under the projection G = KA'N -> α' = L(A') ,kexpXn*-+X as the convex hull of the Weyl group orbit through log a. Recently van den Ban proved a generalization of this theorem to the following situation. Let τ be an involution on the semisimple Lie group G with finite center, G = KA'N a compatible Iwasawa decomposition, i.e., K is τ-invariant, and o! = α^ + αq the corresponding decomposition of a' = L(^4/) into 1 and — 1 eigenspaces for τ. Suppose that H c Gτ is an essentially connected subgroup (see §1 for the definition). Then he describes the image of the sets aH, a £ exp αq under the projection F: G —• α q defined by g e K exp(a^) exp F(g)N. This set is the sum of the convex hull of the orbit of log a under a certain Weyl group and a convex cone in α q. We generalize Kostant's theorem into another direction. We consider the projection L: HMAN -> α defined by g e HMexpL(g)N, where H c Gτ is essentially connected and M, A, and N are defined in §1. This makes sense because the ^4-component in a product hman is unique and HMAN is open in G. So the main new difficulties are the non-compactness of H and the fact that the projection L

On the Irreducibility of Flensted-Jensen's Fundamental Series Representations for Semisimple Symmetric Spaces.

On the Irreducibility of Flensted-Jensen's Fundamental Series Representations for Semisimple Symmetric Spaces.

Symmetric spaces of hermitian type

Let M = G/ H be a semisimple symmetric space,τ the corresponding involution and D = G/ K the Riemannian symmetric space. Then we show that the followingare equivalent: M is of Hermitian type; τ induces a conjugation on D; thereexists an open regular H-invariant cone Ω in q = h [bottom] such that k ∩ Ω ≠ 0. We relate the spaces of Hermitian type to the regular and parahermitian symmetric spaces, analyze the fine structure of D under τ and construct an equivariant Cayley transform. We collect also some results on the classification of invariant cones in q. Finally we point out some applications in representations theory.

Local boundary data of eigenfunctions on a Riemannian symmetric space

Let X = G/K be a Riemannian symmetric space of non-compact type, and let D(X) be the algebra of invariant differential operators on X. In a previous paper [1] we developed a theory of asymptotic expansions for joint eigenfunctions of D(X) of at most exponential growth. In the present paper we show that local asymptotic data determine the eigenfunctions completely. We also develop a theory of asymptotic expansions "along walls". Both results are of importance for the theory of the discrete series of a semisimple symmetric space. Let a be a maximal abelian split subspace of the Lie algebra 9 of G, and denote by a* its complexified dual. Let E be the restricted root system of a in g, and W the associated Weyl group. Then Harish-Chandra's isomorphism D ( X ) ~ S(a) w determines a bijection 2~--~Z~ from a*/W to the set D(X) ^ of algebra homomorphisms D ( X ) ~ C . Given 2ca* , let ga(X) denote the joint eigenspace of functions f e C o~ (X) satisfying:

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

Algebraic groups with a commuting pair of involutions and semisimple symmetric spaces

Let G be a connected reductive algebraic group defined over an algebraically closed field F of characteristic not 2. Denote the Lie algebra of G by 9. In this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of G. This is shown to be independent of the characteristic of F and can be applied to describe all semisimple locally symmetric spaces together with their line structure. Involutorial automorphisms of g occur in several places in the literature. Cartan has already shown that for F= C, the isomorphism classes of involutorial automorphisms of g correspond bijectively to the isomorphism classes of real semisimple Lie algebras, which correspond in their turn to the isomorphism classes of Riemannian symmetric spaces (see Helgason [ 111). If one lifts this involution to the group G, then the present work gives a characteristic free description of these isomorphism classes. In a similar manner we can show that semisimple locally symmetric spaces correspond to pairs of commuting involutorial automorphisms of g. Namely let (go, a) be a semisimple locally symmetric pair; i.e., go is a real semisimple Lie algebra and rr E Aut(g,) an involution. Then by a result of Berger [2], there exists a Cartan involution 8 of go, such that 00 = ea. If we denote the complexilication of go by g, then o and 8 induce a pair of commuting involutions of g. Conversely, if c, 8 E Aut(g) are commuting involutions, then c and 8 determine two locally semisimple symmetric pairs. For if u is a O- and O-stable compact real form with conjugation r, then (ger, (~1 ge,) and (g,,, f3 I ger) are semisimple locally symmetric pairs where

Isotropy representations of semisimple symmetric spaces and homogeneous hypersurfaces

Isotropy representations of semisimple symmetric spaces and homogeneous hypersurfaces

On the support of Plancherel measure

Let G be a real reductive group. As follows from Plancherel formula for G, proved by Harish-Chandra, only tempered representations of G contribute to the decomposition of the regular representation in L 2(G). We give a simple direct proof of this result, based on Gelfand-Kostyuchenko method. We also prove similar results for representations, which appear in the decomposition of L 2(X), where X is a homogeneous G-space of polynomial growth. (See precise definition in 3.5). Important examples of such space X are semisimple symmetric spaces and quotient of G by arithmetic subgroups.

A Plancherel formula for the isotropic cone

In this short note we propose a definition of the isotropic cone related to a semisimple symmetric space and derive a Plancherel formula for this cone.

Invariant differential operators on a semisimple symmetric space and finite multiplicities in a Plancherel formula

Let G be a connected real semisimple Lie group with finite centre, and let z be an involutive automorphism of G. Put G'={x~G: z(x)=x}, and let H be a closed subgroup of G with (G'),cHcG'; here (G'), denotes the identity component of GL In this paper we investigate some properties of the algebra D(X) of invariant differential operators on the semisimple symmetric space X=G/H. Our main results are that the action of D(X) diagonalizes over the discrete part of L2(X) (Theorem 1.5), and that the irreducible constituents of an abstract Plancherel formula for X occur with finite multiplicities (Theorem 3.1). Both results are proved by using techniques of Harish--Chandra adapted to the situation at hand.

Fourier and Poisson transformation associated to a semisimple symmetric space

Soit G un groupe de Lie semisimple connexe contenu dans un groupe de Lie semisimple simplement connexe dont l'algebre de Lie est la complexification de celle de G. Soit τ une involution non triviale de G et θ une involution de Cartan commutant avec τ. Soit H un groupe point fixe de τ. On etudie les representations H-spheriques de dimension finie de G contenant un vecteur K-fixe, ou K est un sous-groupe compact maximal τ-stable de G. Sous-representations de dimension finie de Πσ,ν. Le noyau de Poisson. La distribution uσ , ν i . La transformation de Fourier et Poisson. Formules integrales

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

An explicit construction of the K-finite vectors in the discrete series for an isotropic semisimple symmetric space

An explicit construction of the K-finite vectors in the discrete series for an isotropic semisimple symmetric space

A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group

LetG/H be a semisimple symmetric space. Generalizing results of Flensted-Jensen we give a sufficient condition for the existence of irreducible closed invariant subspaces of the unitary representations ofG induced from unitary finite dimensional representations ofH. This provides a method of constructing unitary irreducible representations ofG, and we show by examples that for some irreducible admissible representations ofG, this method exhibits not previously known unitarity.

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

The Structure of Semisimple Symmetric Spaces

A semisimple symmetric space can be defined as a homogeneous space G/H, where G is a semisimple Lie group, H an open subgroup of the fixed point group of an involutive automorphism of G. These spaces can also be characterized as the affine symmetric spaces or pseudo-Riemannian symmetric spaces or symmetric spaces in the sense of Loos [4] with semisimple automorphism groups [3, 4]. The connected semisimple symmetric spaces are all known: they have been classified by Berger [2] on the basis of Cartan's classification of the Riemannian symmetric spaces. However, the list of these spaces is much too long to make a detailed case by case study feasible. In order to do analysis on semisimple symmetric spaces, for example, one needs a general structure theory, just as in the case of Riemannian symmetric spaces and semisimple Lie groups.