University of Birmingham > Talks@bham > Algebra Seminar > Periodic automorphisms of simple Lie algebras

Periodic automorphisms of simple Lie algebras

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If you have a question about this talk, please contact David Craven.

Let θ be an automorphism of order m of the complex simple algebraic group G, and let g be the Lie algebra of G. Then there is a direct sum decomposition g = g(0)+g(1)+···+ g(m-1), where g(j) is the eij/m-eigenspace for the action of the differential dθ on g. In fact, this is a Z/mZ-grading: [g(i),g(j)]< g(i+j) (where i and j should be considered as integers modulo m). Let G(0) be the connected component of the fixed point subgroup for the action of θ on G; then G(0) is reductive, Lie(G(0)) = g(0) and G(0) stabilizes each of the subspaces g(i).

The first example to consider is the case m=2, that is, where θ is an involution. In this case G(0) is commonly denoted K, g(0) = k and g(1) = p. Then it is well known that the action of K on p shares many invariant-theoretic properties with the adjoint representation: closed orbits are orbits of semisimple elements, the invariants are polynomial, and so on. It is rather less well-known that (due to the seminal work of Vinberg) most of these properties also hold for the action of G(0) on g(1) for arbitrary m. In this talk I will give an overview of Vinberg’s results, explain how they can be extended to positive characteristic and discuss a long-standing conjecture of Popov concerning the existence of an analogue of Kostant’s slice to the regular orbits in the adjoint representation.

This talk is part of the Algebra Seminar series.

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