It was proved by Huckleberry that the Akhiezer-Gindikin domain is included in the ``Iwasawa domain'' using complex analysis. But we can see that we need no complex analysis to prove it. In this paper, we generalize the notions of the Akhiezer-Gindikin domain and the Iwasawa domain for two associated symmetric subgroups in real Lie groups and prove the inclusion. Moreover, by the symmetry of two associated symmetric subgroups, we also give a direct proof of the known fact that the Akhiezer-Gindikin domain is included in all cycle spaces.
We prove a complex version of Kostant's non-linear convexity theorem. Applications to the construction of G-invariant Grauert tubes of non-compact Riemannian symmetric G/K spaces are given.
Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon--Vonk can be computed in terms of $\mathcal{L}$-invariants of the associated Galois representation. The same argument applies …
We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic deco…
For a symmetric pair $(G,H)$ of reductive groups we extend to a large class of generalized principal series representations our previous construction of meromorphic families of symmetry breaking operators. These operators intertwine between a possibly vector-valued principal series of $G$ and one for $H$ and are given explicitly in terms of their integral kernels. As an application we give a complete classification of symmetry breaking operators from spinors on a Euclidean space to spinors on a hyperplane, intertwining for a double cover of the conformal group of the hyperplane.
For a symmetric pair $(G,H)$ of reductive groups we construct a family of intertwining operators between spherical principal series representations of $G$ and $H$ that are induced from parabolic subgroups satisfying certain compatibility conditions. The operators are given explicitly in terms of their integral kernels and we prove convergence of the integrals for an open set of parameters and meromorphic continuation. We further discuss uniqueness of intertwining operators, and for the rank one cases $$ (G,H)=(SU(1,n;\mathbb{F}),S(U(1,m;\mathbb{F})\times U(n-m;\mathbb{F}))), \qquad \mathbb{F}=…
We prove Dipendra Prasad's conjecture on the distinction of the Steinberg representation for symmetric spaces of the form G(E)/G(F), where G is a split reductive group defined over F and E/F an unramified quadratic extension of non-archimedean local fields.
We use Littlewood-Paley-Stein type g-functions (also called generalized square functions) associated to symmetric diffusion semigroups to obtain a characterization of inhomogeneous abstract Besov spaces on the abstract Hilbert spaces. Then we apply our results for the abstract Besov spaces defined through the Poisson and Gauss-Weierstrass semigroups.
A convexity theorem for certain G-orbits in a complexified Riemannian symmetric space G_C/K_C is proved. Applications to analytically continued spherical functions will be given.
Consider the (Helgason-) Fourier transform on a Riemannian symmetric space G/K. We give a simple proof of the L^p-Schwartz space isomorphism theorem (0 <p \le 2) for K-finite functions. The proof is a generalization of J.-Ph. Anker's proof for K-invariant functions.
We investigate the structure of invariant distributions on a non-isotropic non-Riemannian symmetric space of rank one. Especially, the $J$-criterion related to the generalized Gelfand pair is shown for this space without imposing the condition on the eigenfuction of the Laplace-Bertrami operator.
Let X=G/K be a Riemannian symmetric space of the noncompact type. We give a short exposition of the representation theory related to X, and discuss its holomorphic extension to the complex crown, a G-invariant subdomain in the complexified symmetric space X_\C=G_\C/K_\C. Applications to the heat transform and the Radon transform for X are given.
We characterize the image of the Poisson transform on any distinguished boundary of a Riemannian symmetric space of the noncompact type by a system of differential equations. The system corresponds to a generator system of a two sided ideals of an universal enveloping algebra, which are explicitly given by analogues of minimal polynomials of matrices.
In this paper we define a distinguished boundary for the complex crowns $Ξ\subeq G_\C /K_\C$ of non-compact Riemannian symmetric spaces $G/K$. The basic result is that affine symmetric spaces of $G$ can appear as a component of this boundary if and only if they are non-compactly causal symmetric spaces.
Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of Riemannian symmetric spaces of compact and noncompact type inside a common complexification, we prove an analogue of Ramanujan's Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extend the results proven by Bertram for Riemannian symmetric spaces of rank-one.
This paper deals with some simple results about spherical functions of type $δ$, namely new integral formulas, new results about behavior at infinity and some facts about the related $C_σ$ functions.
A basic exact sequence by Harish-Chandra related to the invariant differential operators on a Riemannian symmetric space G/K is generalized for each K-type in a certain class which we call `single-petaled'. The argument also includes a further generalization of Broer's generalization of the Chevalley restriction theorem.
We summarize recent results initiating spectral analysis on pseudo-Riemannian locally symmetric spaces $Γ\backslash G/H$, beyond the classical setting where $H$ is compact (e.g. theory of automorphic forms for arithmetic $Γ$) or $Γ$ is trivial (e.g. Plancherel-type formula for semisimple symmetric spaces).
The heat kernel or Bargmann-Segal transform on a noncompact Riemannian symmetric space X=G/K maps a square integrable function on X to a holomorphic function on the complex crown. In this article we determine the range of this transform.
Following our approach to metric Lie algebras developed in math.DG/0312243 we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a triple consisting of (i) a Lie algebra with involution (of dimension much smaller than the dimension of the transvection group of M), (ii) a semi-simple orthogonal module of the Lie algebra with involution…
Riemannian and pseudo-Riemannian symmetric spaces with semisimple transvection group are known and classified for a long time. Contrary to that the description of pseudo-Riemannian symmetric spaces with non-semisimple transvection group is an open problem. In the last years some progress on this problem was achieved. In this survey article we want to explain these results and some of their applications. Among other things, the material developed in our previous papers math.DG/0312243, math.DG/0408249, and math.DG/0503220 is presented in a unified way.
We deduce a proof of the isomorphism theorem for certain closed subspace $\mc S^p_Γ(X)$ of the $L^p$-Schwartz class functions $(0< p \leq 2) $ on a Riemannian symmetric space $X$ where $Γ$ is a finite subset of $\what{K}_M$. The Fourier transform considered is the Helgason Fourier transform. Our proof relies only on the Paley-Wiener theorem for the corresponding class of functions and hence it does not use the complicated higher asymptotics of the elementary spherical functions.
