Transversal slices to conjugacy classes in algebraic groups and an analogue of the Kostant cross-section theorem for them

The set of conjugacy classes in a group is one of the most fundamental objects in group theory. E.g. In the case of finite groups the elements of this set are in one-to-one correspondence with irreducible representations of the group over an algebraically closed field of characteristic zero. 

The structure of algebraic groups is much more reach, and  conjugacy classes do not completely control the representation theory. However, they naturally appear as some parts of the classification data when we study certain categories of representations. 

For the lecturer, the motivation for the study of conjugacy classes in algebraic groups comes from the representation theory of quantum groups at roots of unity, where they play the key role in the proof of the De Concini-Kac-Procesi conjecture, from the study of the generalized Gelfand-Graev representations for quantum groups, and from the relevant theory of q-W-algebras. However, the results that will be presented in these lectures are quite general and may be interesting to all experts in group theory. 

The main purpose of these lectures is to construct algebraic group analogues of the Slodowy transversal slices and to study their properties. The definition is based on a deep generalisation of the results by Coxeter and Steinberg on the properties of Coxeter elements in Weyl groups.

The three  90 min lectures will be mainly based on Chapter 1 of arXiv:2102.03208 [4].

Plan of the first three lectures:

1. Generalized Coxeter hyperplanes and positive root systems associated to conjugacy classes of Weyl group elements;

2. Transversal slices to conjugacy classes in algebraic groups and an analogue of the Kostant cross-section theorem for them;

3. The Lusztig partition and the strict transversality condition.

If there will be enough interest, the lectures will be followed by 2-3 talks by the participants based on 

F. A. Garside. The braid group and other groups. Quart. J. Math. Oxford Ser. (2), 20:235–254, 1969;

arXiv:2110.09266 [5] 

and

arXiv:2111.01313 [6]