MIT Lie Groups Seminar
Wednesday, December 11, 2019 at 4:30pm to 5:30pm
Room : 2 - 142 182 memorial Drive, Cambridge, MA 02142
Title : On the sections of the Weyl group
Let G be a connected reductive group over an algebraically closed field and W the Weyl group of a maximal torus. In 1966, Tits defined a representative nα, in G, for each simple reflection sα in the Weyl group. These representatives satisfy the braid relations, and therefore define a section of W . This section has been used extensively ever since, in many works. On the other hand, there exist sections of the Weyl group that are not Tits section. For example, in some cases (for example type Bn adjoint and even simpler, GL2), W embeds into G, but the Tits section does not realize such an embedding. Motivated by a question about the Kottwitz homomorphism in p-adic groups, we have computed and classified all sections of the Weyl group, in types A through G. It turns out that these sections can be partially ordered by “how homomorphic they are,” and this partially ordered set of sections has a unique maximal (i.e. “most homomorphic”) element. We will describe this poset, which seems to be interesting in its own right. Moreover, we explain that the “most homomorphic” section answers our question from p-adic groups: it splits the Kottwitz homomorphism. Finally, essentially by definition, this most homomorphic section always realizes an embedding of W in G, whenever such an embedding exists.
Followed by dinner