Peter Slodowy

Simple singularities and simple algebraic groups

Regular group actions.- Deformation theory.- The quotient of the adjoint action.- The resolution of the adjoint quotient.- Subregular singularities.- Simple singularities.- Nilpotent elements in simple lie algebras.- Deformations of simple singularities.

Elliptic genera, involutions, and homogeneous spin manifolds

We study the normalized elliptic genera Φ(X)=ϕ(X)/εk/2 for 4k-dimensional homogeneous spin manifolds X and show that they are constant as modular functions. The basic tool is a reduction formula relating Φ(X) to that of the self-intersection of the fixed point set of an involution γ on X. When Φ(X) is a constant it equals the signature of X. We derive a general formula for sign(G/H), G⊃H compact Lie groups, and determine its value in some cases by making use of the theory of involutions in compact Lie groups.

On Kleinian Singularities and Quivers

Starting from McKay’s observation on the description of (an essential part of) the representation theory of binary polyhedral groups Γ in terms of extended Coxeter-Dynkin-Witt diagrams \(\tilde \Delta (\Gamma )\) and working in the differential geometric framework of Hyper-Kahler-quotients P.B. Kronheimer was able to give a new construction of the semiuniversal deformations of the Kleinian singularities X = ℂ2/Γ as well as of their simultaneous resolutions ([24], [25], [26]). As far as the deformations were concerned, he already gave a purely algebraic geometric formulation of his results in terms of representations of certain quivers naturally attached to the diagrams \(\tilde \Delta (\Gamma )\). By making use of the invariant-theoretic notion of “linear modification” (cf. Section 6, below) and applying it to Kronheimer’s quiver construction we show here how to obtain a purely algebraic geometric simultaneous resolution as well (Section 7). On the way, we shall take the opportunity to remind the reader of various facts about Kleinian singularities (Section 1), McKay’s observation (Section 2), Symplectic geometry (Section 3), Kronheimer’s work (Section 4), and Quivers (Section 5).

The hydrogen algebra as centerless twisted Kac-Moody algebra

Abstract We show that the dynamical symmetry of the hydrogen atom leads in a natural way to the infinite-dimensional algebra, which we identify as the positive subalgebras of twisted Kac-Moody algebras of so(4) and so(3,1), which are isomorphic to each other over R . The relevant automorphism can be associated with parity.

An Adjoint Quotient for Certain Groups Attached to Kac-Moody Algebras

In this article we want to give a survey of that part of our Habilitationsschrift [16] which deals with conjugacy classes in certain groups G attached to Kac-Moody Lie algebras. These investigations were motivated on one side by the result of Brieskorn relating simple singularities and simple algebraic groups (see for instance [14]) and on the other side by recent results of Looijenga on the deformation theory of simply elliptic and cusp singularities ([9], [10]). The results in [16] show that at least to some extent there is a similar relationship between these singularities and associated Kac-Moody Lie groups as there is between simple singularities and simple algebraic groups. Here, we shall limit ourselves to the group-theoretical aspects, i.e. we give a definition (due to E. Looijenga) of an adjoint quotient for an arbitrary Kac-Moody Lie group G and we analyze the structure of its fibers. A large part of the notes will be dedicated to an explanation of Looijenga’s “partial compactification” T/W of the quotient of a maximal torus T of G by the Weyl group W since this space will figure as the base of the adjoint quotient of G. Its stratification into boundary components induces a partition of G which can be described in terms of the building associated to G. We conjecture a representation-theoretic interpretation of this partition which seems to be relevant when dealing with a character-theoretic construction of the adjoint quotient. Some open problems in that direction are mentioned at the end. Detailed proofs may be found in [16]. There also the relations to singularities are explained.

On the signature of homogeneous spaces

We compute the signature of real and quaternionic Grassmannians, thereby completing the table of signatures of symmetric spaces given in a previous paper [4]. In addition, all homogeneous spaces of exceptional Lie groups with non-zero signature are listed.

Simple Singularities and Complex Reflections

The Weyl groups of type A k , D k , E k play an important role in the study of the corresponding simple singularities, e.g. they are realised as the full monodromy groups of these singularities (in dimension 2), and the base of the semiuniversal deformation of such a singularity is naturally identified with the complex orbit space of the Weyl group, cf. [1, 3], [5], [34, 35].

Zur Geometrie der Bahnen Reeller Reduktiver Gruppen

Sei G eine uber R definierte reduktive algebraische Gruppe und G R die Gruppe ihrer reellen Punkte. In diesem Aufsatz betrachten wir einige geometrische Eigenschaften der Bahnen von G R , unter einer linearen algebraischen Darstellung G R , → GL(V) auf einem reellen Vektorraum V. Sei K ⊂ G R , eine maximal kompakte Untergruppe. Gegebenenfalls nach Mittelung uber K konnen wir dann die Existenz eines K-invarianten Skalarproduktes \(\langle ,\rangle :\) V × V → R auf V annehmen. Unser Hauptinteresse gilt dem Verhalten der Langenfunktion \(v \mapsto\langle v,v\rangle={\left\|v\right\|^2}\) bei Einschrankung auf eine G R -Bahn. Diese Situation wurde bereits von Kempf und Ness [KN], [Ne] in dem Fall untersucht, das G eine komplexe reduktive Gruppe und G C → GL(W) eine komplexe Darstellung ist. Dabei erhalt man eine Langenfunktion \(w \mapsto \langle w,w\rangle\) mittels eines K-invarianten hermiteschen Skalarproduktes auf W. Da nur der Realteil dieses Skalarproduktes in die Untersuchung eingeht, last sich dieser Fall unter die von uns anvisierte Situation subsummieren. Unser Hauptziel ist es, die Resultate von Kempf und Ness in unserer allgemeineren Situation herzuleiten. Dabei werden wir auch von den Vereinfachungen Gebrauch machen, die deren Theorie in den Arbeiten [DK] und [PS] erhalten hat.