Analysis on the crown domain

Abstract

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.

Revised version. 103 pages. To appear in GAFA