Semisimple Symmetric Space Research Articles

Distribution vectors on SU(p, q)/SO(p, q)

In this paper, we consider Flensted-Jensen's discrete series on semisimple symmetric space SU(p,q)/SO(p,q). The main idea is to construct the (g+,K∩SO(p,q))-module, where K=S(U(p)×U(q)), g+ is the Lie algebra of G+ and G+ denotes a connected linear reductive subgroup of SU(p,q) with the maximal compact subgroup K∩SO(p,q).

Totally geodesic submanifolds in exceptional symmetric spaces

We classify maximal totally geodesic submanifolds in exceptional symmetric spaces up to isometry. Moreover, we introduce an invariant for certain totally geodesic embeddings of semisimple symmetric spaces, which we call the Dynkin index. We prove a result analogous to the index conjecture: for every irreducible symmetric space of non-compact type, there exists a totally geodesic submanifold which is of minimal codimension and whose non-flat irreducible factors have Dynkin index equal to one.

Discrete series representations with non-tempered embedding

We give an example of a semisimple symmetric space G/H and an irreducible representation of G which has multiplicity 1 in L2(G/H) and multiplicity 2 in C∞(G/H).

Obstructions to the existence of compact Clifford–Klein forms for tangential symmetric spaces

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

The Radon transform for double fibrations of semisimple symmetric spaces

The Radon transform for double fibrations of semisimple symmetric spaces

Relative Trace Formula for Compact Quotient and Pseudo-Coefficients for Relative Discrete Series

Abstract We introduce the notion of relative pseudo-coefficient for relative discrete series representations of real spherical homogeneous spaces of reductive groups. We prove that $K$-finite relative pseudo-coefficient does not exist for semisimple symmetric spaces of type $G_{\mathbb{C}}/G_{\mathbb{R}}$, where $K$ is a maximal compact subgroup of $G_{\mathbb{C}}$, and construct strong relative pseudo-coefficients for some hyperbolic spaces. We establish a toy model for the relative trace formula of H. Jacquet for compact discrete quotient $\Gamma \backslash G$. This allows us to prove that a relative discrete series representation, which admits strong pseudo-coefficients with sufficiently small support, occurs in the spectral decomposition of $L^2(\Gamma \backslash G)$ with a nonzero period.

Spectral analysis on pseudo-Riemannian locally symmetric spaces

We summarize recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces $\Gamma \backslash G/H$, beyond the classical setting where $H$ is compact (e.g. theory of automorphic forms for arithmetic $\Gamma$) or $\Gamma$ is trivial (e.g. Plancherel-type formula for semisimple symmetric spaces).

Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

<pre>Let X = G/H be a semisimple symmetric space of non-compact style. Our purpose is to construct a compact real analytic manifold in which the semisimple symmetric space X = G/H is realized as an open subset and that $G$ acts analytically on it.</pre><pre> By the <span>Cartan</span> decomposition <span>G = KAH,</span> we must <span>compacify</span> the <span>vectorial</span> part <span>A.$</span></pre><pre> In [6], by using the action of the Weyl group, we constructed a compact real analytic manifold in which the semisimple symmetric space G/H is realized as an open subset and that G acts analytically on it.</pre><pre>Our construction is a motivation of the <span>Oshima's</span> construction and it is similar to those in N. <span>Shimeno</span>, J. <span>Sekiguchi</span> for <span>semismple</span> symmetric spaces.</pre><pre>In this note, first we will <span>inllustrate</span> the construction via the case of <span>SL (n, </span>R)/SO_e (1, n-1) and then show that the system of invariant differential operators on X = G/H extends analytically on the corresponding compactification. </pre>

Homogeneous space with non-virtually abelian discontinuous groups but without any proper SL(2, ℝ)-action

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of “continuous analogue” gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space [Formula: see text] admits a discontinuous group which is not virtually abelian if and only if [Formula: see text] admits a proper [Formula: see text]-action (T. Okuda [J. Differ. Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces [Formula: see text] of reductive type. We give a counterexample in the case [Formula: see text].

Convexity theorems for semisimple symmetric spaces

Abstract We prove a convexity theorem for semisimple symmetric spaces G / H ${G/H}$ which generalizes an earlier theorem of the second named author to a setting without restrictions on the minimal parabolic subgroup involved. The new more general result specializes to Kostant’s non-linear convexity theorem for a real semisimple Lie group G in two ways, firstly by taking H maximal compact and secondly by viewing G as a symmetric space for G × G ${G\times G}$ .

On Lie groups and hyperbolic symmetry—From Kunze–Stein phenomena to Riesz potentials

Sharp forms of Kunze–Stein phenomena on SL(2,R) are obtained by using symmetrization and Stein–Weiss potentials. A new structural proof with remarkable simplicity can be given on SL(2,R) which effectively transfers the analysis from the group to the symmetric space corresponding to a manifold with negative curvature. Our methods extend to include the Lorentz groups and n-dimensional hyperbolic space through application of the Riesz–Sobolev rearrangement inequality. A new framework is developed for Riesz potentials on semisimple symmetric spaces and the semi-direct product of groups analogous to the Iwasawa decomposition for semisimple Lie groups. Extensions to higher-rank Lie groups and analysis on multidimensional connected hyperboloids including anti de Sitter space are suggested by the analysis outlined here.

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

Let Gℂbe a complex simple Lie group, GUa compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂis either an empty set or a single orbit of GUif [Formula: see text] is isomorphic to a complex symmetric space.

Semisimple symmetric spaces without compact manifolds locally modelled thereon

Let $G$ be a real reductive Lie group and $H$ a closed subgroup of $G$ which is reductive in $G$. In our earlier work it was shown that, if the homomorphism $i : H^{\bullet}(\mathfrak{g}_{\mathbf{C}}, \mathfrak{h}_{\mathbf{C}}; \mathbf{C}) \to H^{\bullet}(\mathfrak{g}_{\mathbf{C}},(\mathfrak{k}_{H})_{\mathbf{C}}; \mathbf{C})$ is not injective, there does not exist a compact manifold locally modelled on $G/H$. In this paper, we give a classification of the semisimple symmetric spaces $G/H$ for which $i$ is not injective. We also study the case when $G$ cannot be realised as a linear group.

Classification of Semisimple Symmetric Spaces with Proper-Actions

We give a complete classification of irreducible symmetric spaces for which there exist proper $SL(2,\mathbb{R})$-actions as isometries, using the criterion for proper actions by T. Kobayashi and combinatorial techniques of nilpotent orbits. In particular, we classify irreducible symmetric spaces that admit surface groups as discontinuous groups, combining this with Benoist’s theorem.

Cuspidal discrete series for semisimple symmetric spaces

We propose a notion of cusp forms on semisimple symmetric spaces. We then study the real hyperbolic spaces in detail, and show that there exists both cuspidal and non-cuspidal discrete series. In particular, we show that all the spherical discrete series are non-cuspidal.

A remark on the orbit structure of the complexification of a semisimple symmetric space

We consider the action of a real semisimple Lie group G on the complexification GC/HC of a semisimple symmetric space G/H and we present a refinement of Matsukiʼs results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in GC/HC, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G-orbit structure of GC/HC. Every such point p¯ lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair (g,h). The slice representation at p¯ is equivalent to the isotropy representation of a real reductive symmetric space, namely ZG(p4)/Gp¯. In principle, this gives the possibility to explicitly parametrize all G-orbits in GC/HC.

An exact formula relating lattice points in symmetric spaces to the automorphic spectrum

We extract an exact formula relating the number of lattice points in an expanding region of a complex semi-simple symmetric space and the automorphic spectrum from a spectral identity, which is obtained by producing two expressions for the automorphic fundamental solution of the invariant differential operator (∆ − λz) . On one hand, we form a Poincare series from the solution to the corresponding differential equation on the free space G/K, which is obtained using the harmonic analysis of bi-K-invariant functions. On the other hand, a suitable global automorphic Sobolev theory, developed in this paper, enables us to use the harmonic analysis of automorphic forms to produce a solution in terms of the automorphic spectrum.

Proper actions of $SL(2,\mathbf{R})$ on semisimple symmetric spaces

We classify semisimple symmetric spaces $G/H$ for which there exist proper $SL(2,\mathbf{R})$-actions via $G$. This leads us to the classification of semisimple symmetric spaces that admit surface groups as discontinuous groups.

Horospherical transform on real symmetric varieties: Kernel and cokernel

In this paper, we define a horospherical transform for a semisimple symmetric space Y. A natural double fibration is used to assign a more geometrical space Ξ of horospheres to Y. The horospherical transform relates certain integrable analytic functions on Y to analytic functions on Ξ by fiber integration. We determine the kernel of the horospherical transform and establish that the transform is injective on functions belonging to the most continuous spectrum of Y.

Local orbit types of s-representations for exceptional semisimple symmetric spaces

We investigate the local orbit types of hyperbolic or elliptic orbits of the linear isotropy representations of exceptional semisimple (pseudo-Riemannian) symmetric spaces in terms of the restricted root systems with respect to maximal split abelian subspace.