Real-valued characters of finite reductive groups
Abstract: Let $F_q$ be a finite field with $q$ elements, and let $G=G(F_q)$ be the group of $F_q$-points of a connected reductive group with connected center defined over $F_q$. The main topic of this talk is to address the following problem: Given an irreducible complex character $\chi$ of $G$, how does one tell from the parameters of $\chi$ whether $\chi$ is real-valued? I will begin by giving some motivation for this problem, and then give a description of the Jordan decomposition (or Lusztig parametrization) of the irreducible characters of $G$. In particular, this parametrization consists of a pair $(s, \psi)$, where $s$ is a semisimple character of the dual group $G^*$, and $\psi$ is a unipotent character of the centralizer of $s$ in $G^*$. I will then motivate the following conjecture on the classification of real-valued characters of $G$: the character corresponding to the pair $(s, \psi)$ is real-valued if and only if $s$ is a real element, and if $h$ is an element of $G^*$ which conjugates $s$ to $s^{-1}$, then $h$ also conjugates $\psi$ to its complex conjugate. I will then outline a proof of this conjecture in the case that the centralizer of $s$ in $G^*$ is a Levi subgroup. This material in this talk is all ongoing joint work with Bhama Srinivasan from University of Illinois at Chicago.