Alladi Ramakrishnan Hall

Real elements in groups of type $F_4$

Anirban Bose

IMSc, Chennai

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 $h^2=1$.

An exceptional algebraic group of type $F_4$ over a field $k$, is defined as the automorphism group of an Albert algebra over $k$. In this talk 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 an arbitrary field $k$.