Physik Sommersemester 2007
06062 Topics in representation theory [V]
Dozent Wendland K., Mencattini I.
Dauer 2 SWS
Studiensemester 6
Schein Ja (Vortrag)
Termin Do, 10:00-11:30, 2001 T
Inhalt The goal of this course is to provide an introduction to some geometric constructions in representation theory. During the first part of the lectures, I will review the necessary background from the theory of algebraic groups and I will discuss some geometry and representation theory related to flag varieties and other varieties associated to semisipmle groups. During the second part we will study the geometry of the nilpotent cone and of the Steinberg's variety. Applications to the Springer's representation of Weyl groups and to the geometry of the kleinian singularities will be discussed as well.
Tentative plan:
1. Symplectic structures, moment maps and coadjoint orbits.
2. Review of the theory of algebraic groups: Borel subgroups, maximal tori, Weyl groups, Bruhat's decomposition, etc.
3. Universal resolution of $g$, semisimple and nilpotent elements, regular elements. Chevalley theorem.
4. Nilpotent cone and Steinberg's variety. Lagrangian construction of $U (g)$ and applications to the representation theory of the Weyl groups.
5. Kleinian singularities.
Begleitend 06062
Vorkenntnisse algebraic topology and (some) algebraic geometry; theory of Lie groups and Lie algebras.
Literatur J.E. Humphreys: Linear algebraic groups; N. Chriss and V. Ginzburg: Representation theory and complex geometry; P. Slodowy: Four lectures on simple groups and singularities; J.P. Serre: Algebres de Lie semi-simples complexes
Weitere Informationen There will be no exam; each student who needs a grade will have to make a presentation on some topic related to the subject of the course.