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Involutions of Lie algebras in positive characteristic

Levy, Paul (2007) Involutions of Lie algebras in positive characteristic. Advances in Mathematics, 210 (2). pp. 505-559.

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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.

Item Type: Article
Journal or Publication Title: Advances in Mathematics
Uncontrolled Keywords: Symmetric spaces ; Lie algebras in positive characteristic
Subjects:
Departments: Faculty of Science and Technology > Mathematics and Statistics
ID Code: 51956
Deposited By: ep_importer_pure
Deposited On: 09 Dec 2011 13:20
Refereed?: Yes
Published?: Published
Last Modified: 09 Apr 2014 22:56
Identification Number:
URI: http://eprints.lancs.ac.uk/id/eprint/51956

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