Ideal triangles in Euclidean buildings and branching to Levi subgroups

TL;DR

This paper introduces ideal triangles in Euclidean buildings, establishing a correspondence with genuine triangles and deriving saturation theorems for branching to Levi subgroups, advancing geometric and algebraic understanding.

Contribution

It defines ideal triangles in Bruhat-Tits buildings and links their algebraic varieties to genuine triangles, leading to new saturation theorems for Levi subgroup branching.

Findings

Isomorphism between varieties of ideal and genuine triangles

Saturation theorems for branching to Levi subgroups

Applications to constant term homomorphisms

Abstract

We introduce the notion of ideal triangle in the Bruhat-Tits building associated to a split group -- it is analogous to the usual notion of triangle, but one vertex is "at infinity" in a certain direction. We prove that the algebraic variety of based ideal triangles with prescribed side-lengths is naturally isomorphic to a suitable variety of genuine triangles. From theorems pertaining to genuine triangles, we deduce saturation theorems related to branching to Levi subgroups and to the constant term homomorphisms.