Cartan-Iwahori-Matsumoto decompositions for reductive groups

Abstract: We provide a short and self-contained proof of the existence of Cartan--Iwahori--Matsumoto decompositions for reductive groups.   This is the key algebraic input in the proof of the Hilbert--Mumford criterion in geometric invariant theory.  This is joint work with Daniel Halper-Leistner and Jochen Heinloth.