The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra

TL;DR

This paper explores the geometry of polygons in symmetric spaces and buildings to address algebraic problems related to eigenvalues, singular values, and structure constants, providing new proofs and generalizations in algebraic group theory.

Contribution

It introduces a unified geometric approach to generalized triangle inequalities and applies it to solve algebraic problems, including a new proof of the Saturation Conjecture for GL(m).

Findings

Generalized triangle inequalities in symmetric spaces and buildings.

Solution to the saturation problem for Hecke structure constants.

New proof of the Saturation Conjecture for GL(m).

Abstract

In this paper we apply our results on the geometry of polygons in Cartan subspaces, symmetric spaces and buildings to four problems in algebraic group theory. Two of these problems are generalizations of the problems of finding the constraints on the eigenvalues (resp. singular values) of a sum (resp. product) when the eigenvalues (singular values) of each summand (factor) are fixed. The other two problems are related to the computation of the structure constants of the Hecke and representation rings associated with a split reductive algebraic group over Q and its complex Langlands' dual. We give a new proof of the ``Saturation Conjecture'' for GL(m) as a consequence of our solution of the corresponding ``saturation problem'' for the Hecke structure constants for all split reductive algebraic groups over Q.