# Gelfand pairs admit an Iwasawa decomposition

## TL;DR

This paper proves that all Gelfand pairs can be decomposed into a product involving an amenable subgroup, leading to new classifications and a canonical family of spherical functions.

## Contribution

It establishes that every Gelfand pair admits an Iwasawa decomposition with an amenable subgroup, providing new insights into their structure and applications.

## Key findings

- Gelfand pairs admit an Iwasawa decomposition G=KP.
- Complete classification of non-positively curved Gelfand pairs.
- Introduction of a canonical family of spherical functions.

## Abstract

Every Gelfand pair (G,K) admits a decomposition G=KP, where P<G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous. Applications include the complete classification of non-positively curved Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand--Godement for general Gelfand pairs.