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Involutions of Lie algebras in positive characteristic

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Involutions of Lie algebras in positive characteristic. / Levy, Paul.
In: Advances in Mathematics, Vol. 210, No. 2, 01.04.2007, p. 505-559.

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Levy P. Involutions of Lie algebras in positive characteristic. Advances in Mathematics. 2007 Apr 1;210(2):505-559. doi: 10.1016/j.aim.2006.07.002

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Levy, Paul. / Involutions of Lie algebras in positive characteristic. In: Advances in Mathematics. 2007 ; Vol. 210, No. 2. pp. 505-559.

Bibtex

@article{819c9724965441f4b1be0f067815fa54,
title = "Involutions of Lie algebras in positive characteristic",
abstract = "Let G be a reductive group over a field k of characteristic ≠2, let g=Lie(G), let θ be an involutive automorphism of G and let g=k⊕p be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G^θ on p is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety N of nilpotent elements of p has a dense open orbit, and that the same is true for every fibre of the quotient map p→p//G^θ. However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of N, extending a result of Sekiguchi for k=C. Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants k[p]^k.",
keywords = "Symmetric spaces, Lie algebras in positive characteristic",
author = "Paul Levy",
year = "2007",
month = apr,
day = "1",
doi = "10.1016/j.aim.2006.07.002",
language = "English",
volume = "210",
pages = "505--559",
journal = "Advances in Mathematics",
publisher = "Academic Press Inc.",
number = "2",

}

RIS

TY - JOUR

T1 - Involutions of Lie algebras in positive characteristic

AU - Levy, Paul

PY - 2007/4/1

Y1 - 2007/4/1

N2 - Let G be a reductive group over a field k of characteristic ≠2, let g=Lie(G), let θ be an involutive automorphism of G and let g=k⊕p be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G^θ on p is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety N of nilpotent elements of p has a dense open orbit, and that the same is true for every fibre of the quotient map p→p//G^θ. However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of N, extending a result of Sekiguchi for k=C. Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants k[p]^k.

AB - Let G be a reductive group over a field k of characteristic ≠2, let g=Lie(G), let θ be an involutive automorphism of G and let g=k⊕p be the associated symmetric space decomposition. For the case of a ground field of characteristic zero, the action of the isotropy group G^θ on p is well understood, since the well-known paper of Kostant and Rallis [B. Kostant, S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971) 753–809]. Such a theory in positive characteristic has proved more difficult to develop. Here we use an approach based on some tools from geometric invariant theory to establish corresponding results in (good) positive characteristic.Among other results, we prove that the variety N of nilpotent elements of p has a dense open orbit, and that the same is true for every fibre of the quotient map p→p//G^θ. However, we show that the corresponding statement for G, conjectured by Richardson, is not true. We provide a new, (mostly) calculation-free proof of the number of irreducible components of N, extending a result of Sekiguchi for k=C. Finally, we apply a theorem of Skryabin to describe the infinitesimal invariants k[p]^k.

KW - Symmetric spaces

KW - Lie algebras in positive characteristic

U2 - 10.1016/j.aim.2006.07.002

DO - 10.1016/j.aim.2006.07.002

M3 - Journal article

VL - 210

SP - 505

EP - 559

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 2

ER -