University of Birmingham > Talks@bham > Birmingham and Warwick 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)+ยทยทยท+ 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 Birmingham and Warwick Algebra Seminar series.

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