TY - JOUR

T1 - Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

AU - Colliot-Thélène, Jean Louis

AU - Kunyavskiǐ, Boris

AU - Popov, Vladimir L.

AU - Reichstein, Zinovy

PY - 2011/3

Y1 - 2011/3

N2 - Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let fraktur small g sign be its Lie algebra. Let k(G), respectively, k(fraktur small g sign), be the field of k-rational functions on G, respectively, fraktur small g sign. The conjugation action of G on itself induces the adjoint action of G on fraktur small g sign. We investigate the question whether or not the field extensions k(G)/k(G) G and k(fraktur small g sign)/k(fraktur small g sign)G are purely transcendental. We show that the answer is the same for k(G)/k(G) G and k(fraktur small g sign)/k(fraktur small g sign)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 C n, and negative for groups of other types, except possibly G 2. 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.

AB - Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let fraktur small g sign be its Lie algebra. Let k(G), respectively, k(fraktur small g sign), be the field of k-rational functions on G, respectively, fraktur small g sign. The conjugation action of G on itself induces the adjoint action of G on fraktur small g sign. We investigate the question whether or not the field extensions k(G)/k(G) G and k(fraktur small g sign)/k(fraktur small g sign)G are purely transcendental. We show that the answer is the same for k(G)/k(G) G and k(fraktur small g sign)/k(fraktur small g sign)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 C n, and negative for groups of other types, except possibly G 2. 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.

KW - algebraic group

KW - algebraic torus

KW - integral representation

KW - rationality problem

KW - simple Lie algebra

KW - unramified Brauer group

UR - http://www.scopus.com/inward/record.url?scp=80052832291&partnerID=8YFLogxK

U2 - 10.1112/S0010437X10005002

DO - 10.1112/S0010437X10005002

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AN - SCOPUS:80052832291

SN - 0010-437X

VL - 147

SP - 428

EP - 466

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 2

ER -