Ein neuer Zusammenhang zwischen einfachen Gruppen und einfachen Singularitäten. (A new connection between simple groups and simple singularities)
Let G be a simple complex Lie group of type A, D, E. The author gives a construction of the hypersurface singularity of the same type in terms of conjugacy classes of G which is different from the one found by Grothendieck-Brieskorn [see \textit{F. Brieskorn}, ``Singular elements of semisimple algebraic groups'' in Actes Congr. Internat. Math. 1970, vol. 2, 279-284 (1971; Zbl 0223.22012)]. Namely, let X be the closed orbit of G under the adjoint representation in the projective space attached to the Lie algebra \({\mathfrak g}\) of G. Consider a regular nilpotent element \(y\in {\mathfrak g}\) and let H(y) be the hyperplane orthogonal to y with respect to the Killing form. Then \(X\cap H(y)\) has exactly one singularity which is simple of the desired type. Its dimension is quite high, for example equal to 57 in the case of \(E_ 8\). By varying y in \({\mathfrak g}\) one can realize a versal deformation of this singularity. These results are generalized to algebraic groups over fields of any characteristic and applied to get interesting deformations in characteristic two and three.
scientific article; zbMATH DE number 4057656
scientific article