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.
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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.