On the Langlands retraction

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.