Sheets of Symmetric Lie Algebras and Slodowy Slices

Details Sheets of Symmetric Lie Algebras and Slodowy Slices Sheets of Symmetric Lie Algebras and Slodowy Slices

2009-06-21

arxiv.org/abs/0906.3881v3

M. Bulois, Sheets of Symmetric Lie Algebras and Slodowy Slices, Journal of Lie Theory 21 (2001), 1--54

Mathematics

Representation Theory

38 pages, revised version, more synthetic presentation

Scientific paper

Let T be an involution of the finite dimensional complex reductive Lie algebra g and g=k+p be the associated Cartan decomposition. Denote by K the adjoint group of k. The K-module p is the union of the subsets p^{(m)}={x | dim K.x =m}, indexed by integers m, and the K-sheets of (g,T) are the irreducible components of the p^{(m)}. The sheets can be, in turn, written as a union of so-called Jordan K-classes. We introduce conditions in order to describe the sheets and Jordan K-classes in terms of Slodowy slices. When g is of classical type, the K-sheets are shown to be smooth; if g=gl_N a complete description of sheets and Jordan K-classes is then obtained.

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