Reality Properties of Conjugacy Classes in algebraic Groups
| dc.creator | Singh, Anupam | |
| dc.creator | Thakur, Maneesh | |
| dc.date | 2008-04-08 | |
| dc.date.accessioned | 2026-07-07T09:47:48Z | |
| dc.date.available | 2026-07-07T09:47:48Z | |
| dc.description | Let $G$ be an algebraic group defined over a field $k$. We call $g\in G$ {\bf real} if $g$ is conjugate to $g^{-1}$ and $g\in G(k)$ as {\bf $k$-real} if $g$ is real in $G(k)$. An element $g\in G$ is {\bf strongly real} if $\exists h\in G$, $h^{2}=1$ (i.e. $h$ is an {\bf involution}) such that $hgh^{-1}=g^{-1}$. Clearly, strongly real elements are real and are product of two involutions. Let $G$ be a connected adjoint semisimple group over a perfect field $k$, with -1 in the Weyl group. We prove that any strongly regular $k$-real element in $G(k)$ is strongly $k$-real (i.e. is a product of two involutions in $G(k)$). For classical groups, with some mild exceptions, over an arbitrary field $k$ of characteristic not 2, we prove that $k$-real semisimple elements are strongly $k$-real. We compute an obstruction to reality and prove some results on reality specific to fields $k$ with $cd(k)\leq 1$. Finally, we prove that in a group $G$ of type $G_2$ over $k$, characteristic of $k$ different from 2 and 3, any real element in $G(k)$ is strongly $k$-real. This extends our results in \cite{st}, on reality for semisimple and unipotent real elements in groups of type $G_2$. | |
| dc.description | 26 pages | |
| dc.identifier | https://arxiv.org/abs/0804.1245 | |
| dc.identifier | http://arxiv.org/abs/0804.1245 | |
| dc.identifier | Israel Journal of Mathematics, 165 (2008), 1--27 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/163980 | |
| dc.subject | Group Theory | |
| dc.subject | 20G15, 20G10 | |
| dc.title | Reality Properties of Conjugacy Classes in algebraic Groups | |
| dc.type | text |