Domains of holomorphy for irreducible unitary representations of simple Lie groups

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 page

The complex crown for homogeneous harmonic spaces

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

Harmonic analysis for real spherical spaces

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. Sinic

The tempered spectrum of a real spherical space

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 page

The harmonic analysis of lattice counting on real spherical spaces

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 mat