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Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Part of: (Co)homology theory Linear algebraic groups and related topics Lie algebras and Lie superalgebras Algebraic groups Representation theory of groups Birational geometry

Published online by Cambridge University Press:  01 February 2011

Jean-Louis Colliot-Thélène ,
Boris Kunyavskiĭ ,
Vladimir L. Popov  and
Zinovy Reichstein
Jean-Louis Colliot-Thélène
Affiliation:
C.N.R.S., UMR 8628, Mathématiques, Bâtiment 425, Université Paris-Sud, F-91405, Orsay, France (email: jlct@math.u-psud.fr)
Boris Kunyavskiĭ
Affiliation:
Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel (email: kunyav@macs.biu.ac.il)
Vladimir L. Popov
Affiliation:
Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991, Russia (email: popovvl@mi.ras.ru)
Zinovy Reichstein
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada BC V6T 1Z2 (email: reichst@math.ubc.ca)
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Abstract

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Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let 𝔤 be its Lie algebra. Let k(G), respectively, k(𝔤), be the field of k-rational functions on G, respectively, 𝔤. The conjugation action of G on itself induces the adjoint action of G on 𝔤. We investigate the question whether or not the field extensions k(G)/k(G)G and k(𝔤)/k(𝔤)G are purely transcendental. We show that the answer is the same for k(G)/k(G)G and k(𝔤)/k(𝔤)G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type An or Cn, and negative for groups of other types, except possibly G2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Keywords

algebraic groupsimple Lie algebrarationality problemintegral representationalgebraic torusunramified Brauer group

MSC classification

Secondary: 14E08: Rationality questions 17B45: Lie algebras of linear algebraic groups 14L30: Group actions on varieties or schemes (quotients) 20C10: Integral representations of finite groups 20G15: Linear algebraic groups over arbitrary fields 14F22: Brauer groups of schemes
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 428 - 466
Copyright
Copyright © Foundation Compositio Mathematica 2011

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