Seminar at SMU Delhi

August 5, 2014 (Tuesday) , 3:30 PM at Webinar
Speaker: Anirban Bose, Indian Statistical Institute, Delhi
Title: Some Conjugacy Problems in Algebraic Groups
Abstract of Talk
In this talk we discuss two problems related to the study of algebraic groups and Lie groups. \vskip0.1in Problem 1: Let $G$ be a group. Consider the action of $G$ on itself by conjugation. For $g ∈ G,$ let $O_g$ denote the orbit of $g$ under this action. Two elements $x,$ $y \in G$ are said to have the same orbit type if $O_x$ and $O_y$ are isomorphic as G-sets. We describe a method of computing the number of orbit types of simply connected algebraic groups over an algebraically closed field and that of compact simply connected Lie groups. \vskip0.1in Problem 2: Let $G$ be a group (resp. an algebraic group defined over a field $k$). For the latter case, let $G(k)$ denote the group $k$-rational points of $G.$ An element $g \in G$ (resp. $G(k)$) is called real (resp. $k$-real) if there exists $h \in G$ (resp. $G(k)$) such that $hgh^{−1} = g^{−1} .$ An element $g \in G$ (resp. $G(k)$) is said to be strongly real (resp. strongly $k$-real) if there exists $h \in G$ (resp. $G(k)$) such that $hgh^{−1} = g^{−1}$ and $h^2 = 1.$ We prove that in a compact connected Lie group of type $F_4 ,$ every element is strongly real. We also describe the structure of k-real elements in algebraic groups of type $F_4$ defined over a field $k.$