On the Langlands retraction

TL;DR

This paper reviews Langlands' canonical retraction of a root system onto the dominant chamber, highlighting its role in geometric lemmas and its application in stratification of G-bundles.

Contribution

It provides a concise overview of Langlands' retraction, serving as a reference for its use in stratification of G-bundles in algebraic geometry.

Findings

Clarifies the construction of Langlands' retraction.

Connects the retraction to geometric lemmas.

Demonstrates application in stratification of G-bundles.

Abstract

Given a root system in a vector space V, Robert Langlands defined in 1973 a canonical retraction of V onto the dominant chamber. In this note we give a short review of the material on this retraction (which is well known under the name of "Langlands' geometric lemmas"). The main purpose of this review is to provide a convenient reference for e-print arXiv:1112.2402 by D.Gaitsgory and me, in which the Langlands retraction is used to define a certain coarsening of the Harder-Narasimhan-Shatz stratification of the stack of G-bundles on a smooth projective curve.