×

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

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

Citations:

Zbl 0577.00010