Theoretical Statistics and Mathematics Unit, ISI Delhi

March 11, 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.
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\noindent Problem 1: Let $G$ be a group. Consider the action of $G$ on itself by conjugation. For $g\in G$, let $\mathcal{O}_g$ denote the orbit of $g$ under this action. Two elements $x,y\in G$ are said to have the same orbit type if $\mathcal{O}_x$ and $\mathcal{O}_y$ 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.
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\noindent 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 $k$.