Ш-rigidity of Chevalley groups over local rings

Abstract We prove that every locally inner endomorphism of a Chevalley group (or its elementary subgroup) over a local ring with an irreducible root system of rank <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi/> <m:mo>&gt;</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> &gt;1 (with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> 1/2 for the systems <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">A</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathbf{A}_{2} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">F</m:mi> <m:mn>4</m:mn> </m:msub> </m:math> \mathbf{F}_{4} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">B</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msub> </m:math> \mathbf{B}_{\ell} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">C</m:mi> <m:mi mathvariant="normal">ℓ</m:mi> </m:msub> </m:math> \mathbf{C}_{\ell} and with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mn>3</m:mn> </m:mrow> </m:math> 1/3 for the system <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="bold">G</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathbf{G}_{2} ) is inner, so that all these groups are <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Ш</m:mi> </m:math> \Sha -rigid.