Reductive Symmetric Space Research Articles

Relative Non-cuspidality of Representations Induced from Split Parabolic Subgroups

Let $G/H$ be a reductive symmetric space over a $p$-adic field, defined by an involution $\sigma$ on $G$. We study the induced representations ${\rm Ind}_P^G(\rho)$ for any proper $\sigma$-split parabolic subgroup $P=MU$ and any $M\cap H$-distinguished representation $\rho$ of $M$. Let ${\mathcal X}$ denote the complex torus of unramified characters of $M$ which are trivial on $M\cap H$ and $\rho_{\chi}$ the twist of $\rho$ by $\chi\in {\mathcal X}$. We show that ${\rm Ind}_P^G(\rho_{\chi})$ is not $H$-relatively cuspidal {\it for generic} $\chi$, that is, for $\chi$ in a Zariski-open subset of ${\mathcal X}$.

Branching laws for the Steinberg representation: the rank 1 case

Let G/H be a reductive symmetric space over a p-adic field F, the algebraic groups G and H being assumed semisimple of relative rank 1. One of the branching problems for the Steinberg representation StG of G is the determination of the dimension of the intertwining space , for any irreducible representation π of H. We show how this dimension is related to the dimensions of some other intertwining spaces , for a certain finite family Ki, , of anisotropic subgroups of H (here denote the contragredient representation, and 1 the trivial character).

K-invariant cusp forms for reductive symmetric spaces of split rank one

Abstract Let G / H {G/H} be a reductive symmetric space of split rank one and let K be a maximal compact subgroup of G. In a previous article the first two authors introduced a notion of cusp forms for G / H {G/H} . We show that the space of cusp forms coincides with the closure of the space of K-finite generalized matrix coefficients of discrete series representations if and only if there exist no K-spherical discrete series representations. Moreover, we prove that every K-spherical discrete series representation occurs with multiplicity one in the Plancherel decomposition of G / H {G/H} .

Normalizations of Eisenstein integrals for reductive symmetric spaces

We construct minimal Eisenstein integrals for a reductive symmetric space G/H as matrix coefficients of the minimal principal series of G. The Eisenstein integrals thus obtained include those from the σ-minimal principal series. In addition, we obtain related Eisenstein integrals, but with different normalizations. Specialized to the case of the group, this wider class includes Harish-Chandra's minimal Eisenstein integrals.

Poincaré series for non-Riemannian locally symmetric spaces

We initiate the spectral analysis of pseudo-Riemannian locally symmetric spaces Γ\\G/H, beyond the classical cases where H is compact (automorphic forms) or Γ is trivial (analysis on symmetric spaces).For any non-Riemannian reductive symmetric space X=G/H on which the discrete spectrum of the Laplacian is nonempty, and for any discrete group of isometries Γ whose action on X is “sufficiently proper”, we construct L2-eigenfunctions of the Laplacian on XΓ:=Γ\\X for an infinite set of eigenvalues. These eigenfunctions are obtained as generalized Poincaré series, i.e. as projections to XΓ of sums, over the Γ-orbits, of eigenfunctions of the Laplacian on X.We prove that the Poincaré series we construct still converge, and define nonzero L2-functions, after any small deformation of Γ inside G, for a large class of groups Γ. Thus the infinite set of eigenvalues we construct is stable under small deformations. This contrasts with the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface.We actually construct joint L2-eigenfunctions for the whole commutative algebra of invariant differential operators on XΓ.

Radon transformation on reductive symmetric spaces: Support theorems

We introduce a class of Radon transforms for reductive symmetric spaces, including the horospherical transforms, and derive support theorems for these transforms.A reductive symmetric space is a homogeneous space G/H for a reductive Lie group G of the Harish-Chandra class, where H is an open subgroup of the fixed-point subgroup for an involution σ on G. Let P be a parabolic subgroup such that σ(P) is opposite to P and let NP be the unipotent radical of P. For a compactly supported smooth function ϕ on G/H, we define RP(ϕ)(g) to be the integral of NP∋n↦ϕ(gn⋅H) over NP. The Radon transform RP thus obtained can be extended to a large class of distributions containing the rapidly decreasing smooth functions and the compactly supported distributions.For these transforms we derive support theorems in which the support of ϕ is (partially) characterized in terms of the support of RPϕ. The proof is based on the relation between the Radon transform and the Fourier transform on G/H, and a Paley–Wiener-shift type argument. Our results generalize the support theorem of Helgason for the Radon transform on a Riemannian symmetric space.

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 analogue of the Cartan decomposition forp-adic symmetric spaces of splitp-adic reductive groups

Let k be a nonarchimedean locally compact field of residue characteristic p, let G be a connected reductive group defined over k, let σ be an involutive k-automorphism of G, and H an open k-subgroup of the fixed points group of σ. We denote by G k and H k the groups of k-points of G and H. We obtain an analogue of the Cartan decomposition for the reductive symmetric space H k \Gk in the case where G is k-split and p is odd. More precisely, we obtain a decomposition of G k as a union of (H k , K)-double cosets, where K is the stabilizer of a special point in the Bruhat-Tits building of G over k. This decomposition is related to the H k -conjugacy classes of maximal σ-antiinvariant k-split tori in G. In a more general context, Benoist and Oh obtained a polar decomposition for any p-adic reductive symmetric space. In the case where G is k-split and p is odd, our decomposition makes more precise that of Benoist and Oh, and generalizes results of Offen for GL n .

Polynomial Estimates for c-functions on Reductive Symmetric Spaces

The c-functions, related to a reductive symmetric space G/H and a fixed representation τ of a maximal compact subgroup K of G, are shown to satisfy polynomial bounds in imaginary directions.

Stable spectrum for pseudo-Riemannian locally symmetric spaces

Let X = G / H be a reductive symmetric space with rank G / H = rank K / K ∩ H , where K (resp. K ∩ H ) is a maximal compact subgroup of G (resp. of H). We investigate the discrete spectrum of certain Clifford–Klein forms Γ \\ X , where Γ is a discrete subgroup of G acting properly discontinuously and freely on X: we construct an infinite set of joint eigenvalues for “intrisic” differential operators on Γ \\ X , and this set is stable under small deformations of Γ in G.

A PALEY–WIENER THEOREM FOR DISTRIBUTIONS ON REDUCTIVE SYMMETRIC SPACES

Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. We study Fourier transforms of compactly supported K-finite distributions on X and characterize the image of the space of such distributions.

A Paley–Wiener theorem for reductive symmetric spaces

Let X = G/H be a reductive symmetric space and K a maximal compact subgroup of G. The image under the Fourier transform of the space of K-finite compactly supported smooth functions on X is characterized.

The Plancherel decomposition for a reductive symmetric space. I. Spherical functions

We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space. Our starting point is an inversion formula for spherical smooth compactly supported functions. The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus. In the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized Eisenstein integrals and of the Maass–Selberg relations satisfied by the associated C-functions.

The Plancherel decomposition for a reductive symmetric space. II. Representation theory

We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I. The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized principal series.

Filtration de certains espaces de fonctions sur un espace symétrique réductif

We introduce a filtration of a ( g,K) -module of some space of functions on a reductive symmetric space G/ H, and compute the associated grading as a direct sum of induced representations. As an application of this result to the reductive groups viewed as symmetric spaces, we are able to realize any Harish-Chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups, the inducing representations being trivial on the unipotent radical.

The algebra of K-invariant vector fields on a symmetric space G/K

When $G$ is a complex reductive algebraic group and $G/K$ is a reductive symmetric space, the decomposition of $\C[G/K]$ as a $K$-module was obtained (in a non-constructive way) by Richardson, generalizing the celebrated result of Kostant-Rallis for the linearized problem (the harmonic decomposition of the isotropy representation). To obtain a constructive version of Richardson's results, this paper studies the infinite dimensional Lie algebra $\X(G/K)^K$ of $K$-invariant regular algebraic vector fields using the geometry of $G/K$ and the $K$-spherical representations of $G$. Assume $G$ is semisimple and simply-connected and let $\J$ be the algebra of $K$ biinvariant functions on $G$. An explicit set of free generators for the localization $ \X(G/K)^K_{\psi}$ is constructed for a suitable $\psi \in \J$. A commutator formula is obtained for $K$-invariant vector fields in terms of the corresponding $K$-covariant maps from $G$ to the isotropy representation of $G/K$. Vector fields on $G/K$ whose horizontal lifts to $G$ are tangent to the Cartan embedding of $G/K$ into $G$ are called \emph{flat}. When $G$ is simple and simply connected, it is shown that every element of $\X(G/K)^K$ is flat if and only if $K$ is semisimple. The gradients of the fundamental characters of $G$ are shown to generate all conjugation-invariant vector fields on $G$. These results are applied in the case of the adjoint representation of $G = \SL(2,\C)$ to construct a conjugation invariant differential operator whose kernel furnishes a harmonic decomposition of $\C[G]$.

Analytic families of eigenfunctions on a reductive symmetric space

Let X = G / H X=G/H be a reductive symmetric space, and let D ( X ) \mathbb D(X) denote the algebra of G G -invariant differential operators on X X . The asymptotic behavior of certain families f λ f_\lambda of generalized eigenfunctions for D ( X ) \mathbb D(X) is studied. The family parameter λ \lambda is a complex character on the split component of a parabolic subgroup. It is shown that the family is uniquely determined by the coefficient of a particular exponent in the expansion. This property is used to obtain a method by means of which linear relations among partial Eisenstein integrals can be deduced from similar relations on parabolic subgroups. In the special case of a semisimple Lie group considered as a symmetric space, this result is closely related to a lifting principle introduced by Casselman. The induction of relations will be applied in forthcoming work on the Plancherel and the Paley-Wiener theorem.

Invariant fundamental solutions and solvability for $\mathrm{GL} (n, \mathsf{C})/U(p,q)$.

where is the Dirac measure at the origin of G=H. Consider now the reductive symmetric space G=H ˆ GL n;C†=U p; q†. Let aq be a fundamental Cartan subspace for G=H (the `most compact' Cartan subspace) and let Aq be the associated Cartan subset of G=H, identified with a real abelian subgroup of G. For every non-trivial G-invariant differential operator D we let q D† be the differential operator with constant coefficients on Aq defined via the Harish-Chandra isomorphism. We use the Plancherel formula for GL n;C†=U p; q†, obtained by Bopp and Harinck in [4], to construct invariant fundamental solutions for G-invariant differential operators D on G=H for which the differential operator q D† has a fundamental solution, i.e. a distribution Tq on Aq solving the differential equation: q D†Tq ˆ q; where q is the Dirac measure at the origin of Aq. This result is similar to the results obtained by Benabdallah and Rouvie© re for semisimple Lie groups, see [2, Theore© me 1]. Their and our approach can be seen as a generalization of the method used by Ho« rmander to find fundamental solutions for non-zero differential operators with constant coefficients on R, see [7, p.189f].

Propriétés de certaines fonctions tempérées sur un espace symétrique réductif

We introduce a seminorm on a space of C ∞ functions on a reductive symmetric space G/H. We show that the C ∞, K-finite eigenfunctions of the algebra of all G-invariant differential operators on G/H for which this seminorm is finite as well as for their derivatives from the left, are tempered. Reciprocally, we explicite this seminorm for the K-finite tempered eigenfunctions with a regular character. Our work generalises and specifies the results of H. Midorikawa [7] in the group case.

Fourier inversion on a reductive symmetric space

Let X be a semisimple symmetric space. In previous papers, [8] and [9], we have dened an explicit Fourier transform for X and shown that this transform is injective on the space C 1 c (X) ofcompactly supported smooth functions on X. In the present paper, which is a continuation of these papers, we establish an inversion formula for this transform.