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Involutions of Lie algebras in positive characteristic

Research output: Contribution to journalJournal article

Published

Journal publication date1/04/2007
JournalAdvances in Mathematics
Journal number2
Volume210
Number of pages55
Pages505-559
Original languageEnglish

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.