Seminare und Vorträge im SS 2020

am Dienstag, 09. Juni

Oberseminar

Christian Steinert (Köln), Fano Varieties and Fano Polytopes

16:00

online

am Dienstag, 16. Juni

Oberseminar

tba

16:00

online

am Dienstag, 23. Juni

Oberseminar

Ilia Smilga (Marseille), Action of the Weyl group on the space of L-invariant vectors
Let G be a semisimple real Lie group, A its Cartan subspace (or maximal split torus), W = N_G(A)/Z_G(A) its restricted Weyl group. Consider a finite-dimensional irreducible representation rho of G (acting on a space V).
Then W has a well-defined action on the subspace V^L comprising the vectors of V fixed by the subgroup, often denoted by MA or L, which is the centralizer of A. We are especially interested in the action on V^L of the longest word w_0 of the Weyl group (the unique element that maps all positive restricted roots to negative restricted roots). More specifically, we want to classify the cases in which this action is nontrivial. (This question arose in connection with a certain question about groups of affine transformations acting properly on the affine space.)
This question naturally splits into two parts: first, which are the representations for which at least the space V^L is itself nontrivial? and then, among these, which are the ones where, additionally, w_0 acts nontrivially? In the particular case where G is split, the first question is very easy, and we have found an answer to the second question, which is: "almost all". In the general case, I have recently obtained an answer to the first question, and for the second question I have experimental results. I will present all of this work.

16:00

online