Finite Maximal Tori

Finite Loop Space with Maximal Tori have Finite Weyl Groups

Connected Finite Loop Spaces with Maximal Tori

Finite loop spaces are a generalization of compact Lie groups. However, they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connected finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence...

Cosets and genericity

In a connected group of finite Morley rank in which the generic element belongs to a connected nilpotent subgroup, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite Morley rank. The only known infinite simple groups of finite Morley rank are the simple algebraic g...

Lecture 9: Finite Groups of Lie Type and Hecke Algebras

We continue to study the representation theory of finite groups of Lie type and its connection to Hecke algebras, now in a more general setting. We start by defining Hecke algebras of arbitrary Weyl groups. Then we introduce finite groups G of Lie type and relate the coinduced representations C[B \G] to Hecke algebras, similarly to what was done for GLn(Fq). In the second part of this lecture w...

Maximal Hamiltonian tori for polygon spaces

We study the poset of Hamiltonian tori for polygon spaces. We determine some maximal elements and give examples where maximal Hamiltonian tori are not all of the same dimension.