Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten

Simple singularities are classified by simply-laced Dynkin diagrams (type ADE) which in turn define certain simple algebraic groups. Grothendieck conjectured and Brieskorn proved a direct connection between these structures.

 In this paper, a completely different connection between simple singularities and simple algebraic groups is given. Let G be a simple algebraic group with simply-laced Dynkin diagram. Let P be the projective space attached to the adjoint representation g. Then P contains exactly one closed orbit, a certain flag variety X. Using the killing form, every non-zero x in g defines a hyperplane in P, hence a hyperplane section X_x of X.

Theorem: Assume x is regular nilpotent. Then X_x has exactly one singular point. This singularity is simple with the same Dynkin diagram as G. Furthermore, "perturbing x" is a versal deformation of the singularity.

One of the advantages of this construction is that it holds, properly modified, also in positive characteristic, even if the characteristic is bad. This predicted certain degenerations of simple singularities which are impossible in characteristic zero, e.g., E_7-->A_7, E_8-->A_8 (p=2), and E_8-->A_8 (p=3).

Appeared in: Inventiones Mathematicae 90 (1987) 579-604

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Remark: There is a paper dealing with hyperplane sections of X when V is an arbitrary simple G-module.

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Last updated: August 25, 2006