SMRI Algebra and Geometry Online: He -- Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras

SMRI Algebra and Geometry Online 

’Tits groups of Iwahori-Weyl groups and presentations of Hecke algebras’ 
Xuhua He (Chinese University of Hong Kong) 

Thursday 5th August 3:30pm - 5:00pm (AEST) 
Register:
https://uni-sydney.zoom.us/meeting/register/tZYtcu-przouH9zYrbDw_tL7yC9JirzOWzZH 

Abstract: Let $G(\mathbb C)$ be a complex reductive group and $W$ be its Weyl group. 
In 1966, Tits introduced a certain subgroup of $G(\mathbb C)$, which is an extension 
of $W$ by an elementary abelian $2$-group. This group is called the Tits group and 
provides a nice lifting of $W$. 

In this talk, I will discuss a generalization of the notion of the Tits group 
$\mathcal T$ to a reductive $p$-adic group $G$. Such $\mathcal T$, if exists, gives a 
nice lifting of the Iwahori-Weyl group of $G$. I will show that the Tits group exists 
when the reductive group splits over an unramified extension of the $p$-adic field and 
will provide an example in the ramified case that such a Tits group does not exist. 
Finally, as an application, we will provide a nice presentation of the Hecke algebra 
of the $p$-adic group $G$ with $I_n$-level structure. 

This talk is based on the recent joint work with Ganapathy arXiv:2107.01768. 

Biography: Xuhua He received his PhD from MIT in 2005.  After postdoctoral positions at
the IAS and Stony Brook University, He was an A/Professor at HKUST from 2008 and a Full
Professor at the University of Maryland from 2014 before joining CUHK in 2019 as
Choh-Ming Professor of Mathematics.  He received the Morningside Gold Medal of
Mathematics in 2013 and the Xplorer Prize in 2020, and was an invited sectional speaker
of the International Congress of Mathematicians in 2018.