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.
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 |