Theoretical Statistics and Mathematics Unit, ISI 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.
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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.
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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 $k.$