Slodowy, Peter An adjoint quotient for certain groups attached to Kac-Moody algebras. (English) Zbl 0613.22006 Infinite dimensional groups with applications, Publ., Math. Sci. Res. Inst. 4, 307-333 (1985). [For the entire collection see Zbl 0577.00010.] In this article, the author gives a survey on the subject stated in the title developed by the author [Habilitationsschrift, Universität Bonn, 1984]. The author gives a definition (due to E. Looijenga) of an adjoint quotient for an arbitrary Kac-Moody Lie group G and analyses the structure of fibers. Due to Brieskorn, there is a relationship between simple singularities and simple algebraic groups. The author shows that at least to some extent there is a similar relationship between the deformation theory of simple elliptic and cusp singularities due to Looijenga and associated Kac-Moody Lie groups. In the finite dimensional case, let G be a simply connected semi-simple algebraic group over \({\mathbb{C}}\), B be a Borel subgroup and \(T\subset B\) a maximal torus of G, N be the normalizer of T in G. Then, \(N/T=W\) is the finite Weyl group. The adjoint quotient of G is the quotient of G by its adjoint actions which is canonically isomorphic to T/W. In the infinite-dimensional case, to obtain a reasonable adjoint quotient of G, the author uses the Tits cone attached to the root basis and its Weyl group and Looijenga’s partial compactification of T/W. Its stratification into boundary components induces a partition of G which can be described in terms of the building associated with G. Some open problems on a representation theoretic interpretation of this partition are mentioned at the end. Detailed proofs and relations to singularities may be found in the authors notes indicated above. Reviewer: E.Abe Cited in 3 Documents MSC: 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 14H20 Singularities of curves, local rings 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Keywords:Kac-Moody algebras; Kac-Moody group; deformation theory of singularities; adjoint quotient; Kac-Moody Lie group; semi-simple algebraic group; Borel subgroup; Weyl group; Tits cone Citations:Zbl 0577.00010 PDFBibTeX XML