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In this article we give an overview of the Plancherel theory for Riemannian symmetric spaces Z = G/K. In particular we illustrate recently developed methods in Plancherel theory for real spherical spaces by explicating them for Riemannian symmetric spaces, and we explain how Harish-Chandra's Plancherel theorem for Z can be proven from these methods.

In this paper we give an overview on $L^p$-factorizations of Lie group representations and introduce the notion of smooth $L^p$-factorization., Comment: This article is dedicated to the fond memories of Gerrit van Dijk

Our concern is with Riemannian symmetric spaces $Z=G/K$ of the non-compact type and more precisely with the Poisson transform $\mathcal{P}_\lambda$ which maps generalized functions on the boundary $\partial Z$ to $\lambda$-eigenfunctions on $Z$. Special emphasis is given to a maximal unipotent group $N0})^r

We give an example of a semisimple symmetric space $G/H$ and an irreducible representation of $G$ which has multiplicity 1 in $L^2(G/H)$ and multiplicity 2 in $C^\infty(G/H)$.

We formulate and prove a Paley-Wiener theorem for Harish-Chandra modules for a real reductive group. As a corollary we obtain a new and elementary proof of the Helgason conjecture., Comment: Submitted version; with two appendices on the Helgason conjecture and an application

We explain by elementary means why the existence of a discrete series representation of a real reductive group $G$ implies the existence of a compact Cartan subgroup of $G$. The presented approach has the potential to generalize to real spherical spaces.

We define Sobolev norms of arbitrary real order for a Banach representation $(\pi, E)$ of a Lie group, with regard to a single differential operator $D=d\pi(R^2+\Delta)$. Here, $\Delta$ is a Laplace element in the universal enveloping algebra, and $R>0$ depends explicitly on the growth rate of the representation. In particular, we obtain a spectral gap for $D$ on the space of smooth vectors of $E$. The main tool is a novel factorization of the delta distribution on a Lie group., Comment: 10 pages, to appear in Journal of Functional Analysis

Given a unimodular real spherical space $Z=G/H$ we construct for each boundary degeneration $Z_I=G/H_I$ of $Z$ a Bernstein morphism $B_I: L^2(Z_I)_{\rm disc }\to L^2(Z)$. We show that $B:=\bigoplus_I B_I$ provides an isospectral $G$-equivariant morphism onto $L^2(Z)$. Further, the maps $B_I$ are finite linear combinations of orthogonal projections which translates in the known cases where $Z$ is a group or a symmetric space into the familiar Maass-Selberg relations. As a corollary we obtain that $L^2(Z)_{\rm disc }\neq \emptyset$ provided that ${\mathfrak h}^\perp$ contains elliptic elements in its interior., Comment: 101 pages. Final version. Accepted to J. Amer. Math. Soc

Let $Z=G/H$ be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of $G$ on $L^2(Z)$. It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of $L^2(Z)$, have infinitesimal characters which are real and belong to a lattice. Moreover, let $K$ be a maximal compact subgroup of $G$. Then each irreducible representation of $K$ occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of $H$., Comment: To appear in GAFA

We provide $L^p$-versus $L^\infty$-bounds for eigenfunctions on a real spherical space $Z$ of wavefront type. It is shown that these bounds imply a non-trivial error term estimate for lattice counting on $Z$. The paper also serves as an introduction to geometric counting on spaces of the mentioned type. Section 7 on higher rank is new and extends the result from v1 to higher rank. Final version. To appear in Acta Math. Sinica., Comment: 46 pages

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple (see arXiv:1609.00963) and generalizes the results of Brion and Mikityuk in the (complex) spherical case., Comment: Extended revised version. Section 6 and Appendix B are new. To appear in Transformation Groups. 40p

Let $Z$ be a unimodular real spherical space. We develop a theory of constant terms for tempered functions on $Z$ which parallels the work of Harish-Chandra. The constant terms $f_I$ of an eigenfunction $f$ are parametrized by subsets $I$ of the set $S$ of spherical roots which determine the fine geometry of $Z$ at infinity. Constant terms are transitive i.e. $(f_J)_I=f_I$ for $I\subset J$, and our main result is a quantitative bound of the difference $f-f_I$, which is uniform in the parameter of the eigenfunction., Comment: Final version. 72 pages. Accepted to IMRN

We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces $Z=G/H$ attached to a real reductive Lie group $G$. A special emphasis is made to the case where $Z$ is real spherical., Comment: Shortened title, typos fixed, more details on dual smooth Frobenius reciprocity (now Lemma 6.6). 38 pages, lecture notes for the Sanya meeting on spherical varieties. To appear in Acta Math. Sinica

This paper gives a classification of all pairs $(\mathfrak g, \mathfrak h)$ with $\mathfrak g$ a simple real Lie algebra and $\mathfrak h < \mathfrak g$ a reductive subalgebra for which there exists a minimal parabolic subalgebra $\mathfrak p < \mathfrak g$ such that $\mathfrak g = \mathfrak h + \mathfrak p$ as vector sum., Comment: 44 pages, 8 tables. Minor corrections in Table 8. Published version

In this paper, we study rationality properties of reductive group actions which are defined over an arbitrary field of characteristic zero. Thereby, we unify Luna's theory of spherical systems and Borel-Tits' theory of reductive groups. In particular, we define for any reductive group action a generalized Tits index whose main constituents are a root system and a generalization of the anisotropic kernel. The index controls to a large extent the behavior at infinity (i.e., embeddings). For k-spherical varieties (i.e., where a minimal parabolic has an open orbit) we obtain explicit (wonderful) completions of the set of rational points. For local fields this means honest compactifications generalizing the maximal Satake compactification of a symmetric space. Our main tool is a k-version of the local structure theorem., Comment: v1: 62 pages; v2: 66 pages, some material added (especially 13.5-13.8), minor corrections; v3: 69 pages, section 12 on boundary degenerations rewritten, mistake in Prop. 13.1 fixed, Ex. 13.9 and Prop. 14.7 added

Let G/H be a unimodular real spherical space which is either absolutely spherical or wave-front. It is shown that every tempered representation of G/H embeds into a relative discrete series of a boundary degeneration of G/H. If in addition G/H is of wave-front type it follows that the tempered representation is parabolically induced from a discrete series representation of a lower dimensional real spherical space. Final version. To appear in Acta Math., Comment: 64 pages

By the collective name of {\it lattice counting} we refer to a setup introduced in Duke-Rudnick-Sarnak that aim to establish a relationship between arithmetic and randomness in the context of affine symmetric spaces. In this paper we extend the geometric setup from symmetric to real spherical spaces and continue to develop the approach with harmonic analysis which was initiated in Duke-Rudnick-Sarnak., Comment: Extended and revised version with 4 additional pages. To appear in Documenta math

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure $L^p$-integrability of matrix coefficients on Z., Comment: Additional material of 4 pages added. To appear in J. Funct. Analysis

Let $H$ be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold $X$ and a real algebraic bundle $\mathcal{E}$ on $X$. Let $\mathfrak{h}$ be the Lie algebra of $H$. Let $\mathcal{S}(X,\mathcal{E})$ be the space of Schwartz sections of $\mathcal{E}$. We prove that $\mathfrak{h}\mathcal{S}(X,\mathcal{E})$ is a closed subspace of $\mathcal{S}(X,\mathcal{E})$ of finite codimension. We give an application of this result in the case when $H$ is a real spherical subgroup of a real reductive group $G$. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $\pi$ be a Casselman-Wallach representation of $G$ and $V$ be the corresponding Harish-Chandra module. Then the natural morphism of coinvariants $V_{\mathfrak{h}}\to \pi_{\mathfrak{h}}$ is an isomorphism if and only if any linear $\mathfrak{h}$-invariant functional on $V$ is continuous in the topology induced from $\pi$. The latter statement is known to hold in two important special cases: if $H$ includes a symmetric subgroup, and if $H$ includes the nilradical of a minimal parabolic subgroup of $G$., Comment: v4: version appearing in Math. Z + erratum added in the end

Let Z be an algebraic homogeneous space Z=G/H attached to real reductive Lie group G. We assume that Z is real spherical, i.e., minimal parabolic subgroups have open orbits on Z. For such spaces we investigate their large scale geometry and provide a polar decomposition. This is obtained from the existence of simple compactifications of Z which is established in this paper., Comment: Extended revised version. Section 5 with a systematic study of the compression (or valuation) cone is new

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q \times_L S$. Here $Q$ is a parabolic subgroup with Levi decomposition $LU$, and $S$ is a homogeneous space for a quotient $D=L/L_n$ of $L$, where $L_n$ is normal in $L$, such that $D$ is compact modulo center., Comment: v1: 18 pages, no figures; v2: 19 pages, revised, final version

Let G be a real semi-simple Lie group and H a closed subgroup which admits an open orbit on the flag manifold of a minimal parabolic subgroup. Let V be a Harish-Chandra module. A sharp finite bound is given for the dimension of the space of H-fixed distribution vectors for V and a related subrepresentation theorem is derived. Extended final version. To appear in Trans. Amer. Math. Soc., Comment: 18 pages

Let G be a connected real semi-simple Lie group and H a closed connected subgroup. Let P be a minimal parabolic subgroup of G. It is shown that H has an open orbit on the flag manifold G/P if and only if it has finitely many orbits on G/P. This confirms a conjecture by T. Matsuki.

Let Z be a homogeneous space Z=G/H of a real reductive Lie group G with a reductive subgroup H. The investigation concerns the quantitative decay of matrix coefficients on $Z$ under the assumption that Z is of spherical type, that is, minimal parabolic subgroups have open orbits on Z.

Let G be a real reductive group and Z=G/H a unimodular homogeneous G-space. The space Z is said to satisfy VAI if all smooth vectors in the Banach representations L^{p}(Z) vanish at infinity, 1 <=p. For H connected we show that Z satisfies VAI if and only if it is of reductive type., Comment: An assumption that G/H is algebraic is added to deduce the volume estimate needed for the if direction of the main result. A shorter proof and improved exposition of the "only if direction" of the main result

A theory of holomorphic extension of eigenfunctions on homogeneous harmonic spaces is developed., Comment: 17 pages, 1 figure.

Let G be a real reductive group and G/H a unimodular homogeneous G space with a closed connected subgroup H. We establish estimates for the invariant measure on G/H. Using these, we prove that all smooth vectors in the Banach representation L^p(G/H) of G are functions that vanish at infinity if and only if G/H is of reductive type. An application to lattice counting on G/H is presented., Comment: An expanded version entitled "Decay on homogeneous spaces of reductive type" was posted as article 1106.1331

In this article a general framework for studying analytic representations of a real Lie group G is introduced. Fundamental topological properties of the representations are analyzed. A notion of temperedness for analytic representations is introduced, which indicates the existence of an action of a certain natural algebra A(G) of analytic functions of rapid decay. For reductive groups every Harish-Chandra module V is shown to admit a unique tempered analytic globalization, which is generated by V and A(G) and which embeds as the space of analytic vectors in all Banach globalizations of V., Comment: Main file unchanged. Erratum added at the end

For every moderate growth representation of a real Lie group G on a Frechet space E, we prove a factorization theorem of Dixmier--Malliavin type for the space of analytic vectors E^{\omega}. There exists a natural algebra of superexponentially decreasing analytic functions A(G), such that E^{\omega} = A(G) * E^{\omega}. As a corollary we obtain that E^\omega coincides with the space of analytic vectors for the Laplace--Beltrami operator on G., Comment: 14 pages

In this paper we determine the Bochner measure for spherical functions on a semi-simple Lie group of rank one.

An elementary and self-contained approach to smooth Frechet globalizations of Harish-Chandra modules is presented. We provide applications to the theory of Eisenstein series and Automatic Continuity., Comment: 65 p., no figures. revised version

Let X be a Riemannian symmetric space of non-compact type. We prove a theorem of holomorphic extension for eigenfunctions of the Laplace-Beltrami operator on X, by techniques from the theory of partial differential equations., Comment: 8p, no figures

We investigate the non-diagonal normal forms of a quadratic form on R^n, in particular for n=3. For this case it is shown that the set of normal forms is the closure of a 5-dimensional submanifold in the 6-dimensional Grassmannian of 2-dimensional subspaces of \R^5.

Let Y=G/H be a semisimple symmetric space. It is shown that the smooth vectors for the regular representation of G on L^p(Y) vanish at infinity., Comment: 8 pages

We define and study the (minimal) Radon transform on a real symmetric variety., Comment: Final, fully revised version. 22 pages

It is about the uniqueness of the Iwasawa decomposition., Comment: To appear in an appendix of a physics lecture notes volume

I exemplify part of my recent work on the upper halfplane., Comment: Minor stylistic changes

In this paper I raise a question on the structure of the boundary of the crown domain., Comment: revised final version, 9 p

We classify the domains of holomorphy of all Harish-Chandra modules of irreducible unitary representations of simple non-compact Lie groups., Comment: revised version, to appear in Invent. math., 14 pages

In this paper one finds:1) A simple combinatorical description of the distinguished boundary of the crown domain in terms of the affine Weyl group; 2) Optimal upper and lower bounds for holomorphically extended spherical functions; 3) First progress on how to attach complex invariants to irreducible representations; 4) A new unipotent model for the crown domain; 5) Optimal bounds for the exponential decay of cuspidal automorphic forms., Comment: Revised version. 103 pages. To appear in GAFA

We develop integral geometry for non-compactly causal symmetric spaces. We define a complex horospherical transform and, for some cases, identify it with a Cauchy type integral., Comment: Revised, final version; to appear in IMRN, 38 p

We continue our discussion from part I., Comment: 46 pages, to appear in GAFA

We define a complex horospherical transform for affine symmetric spaces of Hermitian type and show that it has no kernel on the holomorphic discrete series., Comment: 24 pages

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., Comment: Expanded version. Two new sections added in order to clarify two remarks in the previous verson. 15 p

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., Comment: 13 pages

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L^2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown $\Xi$. In this article we construct a minimal G-invariant subdomain $\Xi_H$ of $\Xi$ with G/H as Shilov boundary. Let $\pi$ be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of $\pi$, which admit a holomorphic extension to $\Xi_H$, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L^2(G/H)_mc in a space of holomorphic functions., Comment: Due to a very careful and professional referee's work we could strongly improve on both exposition and detail. To appear in Invent. math., 35 pages, 1 figure

We give an overview on the c-function of a non-compactly causal symmetric space G/H and explain its interplay with harmonic analysis and representation theory., Comment: 24 pages. Minor errors corrected, new format. To appear in the Karpelevic memorial volume (AMS series Translations 2)

Let G be a connected, real, semisimple Lie group contained in its complexification G_C, and let K be a maximal compact subgroup of G. We construct a K_C-G double coset domain in G_C, and we show that the action of G on the K-finite vectors of any irreducible unitary representation of G has a holomorphic extension to this domain. For the resultant holomorphic extension of K-finite matrix coefficients we obtain estimates of the singularities at the boundary, as well as majorant/minorant estimates along the boundary. We obtain L^\infty bounds on holomorphically extended automorphic functions on G/K in terms of Sobolev norms, and we use these to estimate the Fourier coefficients of combinations of automorphic functions in a number of cases, e.g. of triple products of Maass forms., Comment: 84 pages, published version

We make a fine study of the SO(2,C)-double coset decomposition in SL(2,C) and give a full description of the intersection of the various cells with the complex crown Xi of SL(2,R)/SO(2). A non-linear convexity theorem is proved and applications to analytically continued spherical functions are given., Comment: 15 pages, 2 figures. Simpler geometric proofs of the results in Section 3 are given. To appear in Math. Z

Let $G/H$ be a semisimple symmetric space. Then the space $L^2(G/H)$ can be decomposed into a finite sum of series representations induced from parabolic subgroups of $G$. The most continuous part of the spectrum of $L^2(G/H)$ is the part induced from the smallest possible parabolic subgroup. In this paper we introduce Hardy spaces canonically related to this part of the spectrum for a class of non-compactly causal symmetric spaces. The Hardy space is a reproducing Hilbert space of holomorphic functions living on a tube type bounded symmetric domain, containing $G/H$ as a boundary component. A boundary value map is constructed and we show that it induces an $G$-isomorphism onto a multiplicity free subspace of full spectrum in the most continuous part $L_{\rm mc}^2(G/H)$ of $L^2(G/H)$. We also relate our Hardy space with the classical Hardy space on the tube domain., Comment: Revised version: readability improved, minor errors corrected. To appear in Math. Annalen, 36 pages