Frobenius-twisted conjugacy classes of loop groups and Demazure product of Iwhaori-Weyl groups

Abstract: The affine Deligne-Lusztig varieties, roughly speaking, describe the intersection of Iwahori-double cosets and Frobenius-twisted conjugacy classes in a loop group. For each fixed Iwahori-double coset $I w I$, there exists a unique Frobenius-twisted conjugacy class whose intersection with $I w I$ is open dense in $I w I$. Such Frobenius-twisted conjugacy class $[b_w]$ is called the generic Frobenius-twisted conjugacy class with respect to the element $w$. Understanding $[b_w]$ leads to some important consequences in the study of affine Deligne-Lusztig varieties. In this talk, I will give an explicit description of $[b_w]$ in terms of Demazure product of the Iwahori-Weyl groups. It is worth pointing out that a priori, $[b_w]$ is related to the conjugation action on $I w I$, and it is interesting that $[b_w]$ can be described using Demazure product instead of conjugation action. This is based on my preprint arXiv:2107.14461.

If time allows, I will also discuss an interesting application. Lusztig and Vogan recently introduced a map from the set of translations to the set of dominant translations in the Iwahori-Weyl group. As an application of the connection between $[b_w]$ and Demazure product, we will give an explicit formula for the map of Lusztig and Vogan.

representation theory

Audience: researchers in the topic

MIT Lie groups seminar

Series comments: Description: Research seminar on Lie groups

This seminar will take place entirely online: Zoom Meeting Link. You should be able to watch live video at this link. Your microphone will be muted, but you are welcome to unmute it (microphone icon on the lower left of the Zoom window, perhaps visible only when you put your mouse near there) to ask a question.