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Consider the real algebraic group $SO(p,q)$, this is the automorphism group of the vector space $\mathbb{R}^n$ of dimension $n=p+q$ over $\mathbb{R}$, endowed with the diagonal quadratic form with $p …

I prefer to use the language of algebraic groups. All algebraic groups and Lie algebras are defined over $\Bbb C$. 1. Let ${\mathfrak g}$ be a semisimple Lie algebra. Consider the automorphism group $ …

The abstract subgroup generated by $H$ and $K$ is closed. We may assume that $G$ is connected. The groups $G$, $H$, $K$ are the groups of real points of real algebraic groups $\mathbf{G}$, $\mathbf{H …

Let us write the Frobenius-Schur indicator of the representation $V$ with highest weight $\lambda$ as $\mathrm{FS}(\lambda)$. Let $R$ denote the root system, and let $\Pi=\{\alpha_1,\dots,\alpha_l\}$ …

Let $G$ be a compact (anisotropic) real algebraic group. Let $\rho\colon G\to {\rm GL}(n,{\mathbb{C}})$ be a complex linear (polynomial) representation of $G$. Following OP, we say that $\rho$ is pse …

The answer is YES. It suffices to assume that $H_1$ and $H_2$ are conjugate over $\mathbb{C}$ or, what is the same, that they are conjugate over $\overline{\mathbb{Q}}$. Theorem 1. Let $G$ be a co …

In the book Lie groups and algebraic groups by Onishchik and Vinberg, Theorem 3 in Section 5.2.1 on page 240 says: Let $S$ be a real structure on a simply connected complex semisimple Lie group $G$. …

I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed. …

$ \newcommand{\g}{{\mathfrak g}} \newcommand{\h}{{\mathfrak h}} \newcommand{\t}{{\mathfrak t}} \newcommand{\C}{{\mathbb C}} \newcommand{\Ad}{{\rm Ad}} \newcommand{\ad}{{\rm ad}} $Theorem. Let $G$ be …

For the case of a connected semisimple compact Lie group $G$, see this preprint, Section 3, where, following ideas of Kac and Vinberg, we describe set of conjugacy classes of $n$-th roots of a given …

The question is essentially about ${\mathbf{R}}$-groups, so we shall assume that $G$ is defined over $\mathbf{R}$. It is not true that for a given connected reductive ${\mathbf{R}}$-group $G$, there …

Question 1. Does Élie Cartan's paper Les groupes réels simples, finis et continus, Ann. Sci. École Norm. Sup. (3) 31 (1914), 263–355 contain a classification of $\Bbb C$-linear involutions of simple …

Let $X=G/H$, where $G$ is a connected linear algebraic group over the field $\mathbf{C}$ of complex numbers and $H\subset G$ is an algebraic subgroup. In general, we can write the algebraic variety $X …

(0) The group ${\rm Aut}(\mathfrak{g}_0)$ does not have to be connected (even over $\mathbb C$), take $\mathfrak{g}_0=\mathfrak{su}(2,2)$ as a counter-example. So let $G={\rm Inn}(\mathfrak g_0)$, tha …

Let $G$ be a simply connected simple algebraic group over the field of complex numbers $\mathbb C$. Let $H$ be a symmetric subgroup of $G$. This means that there exists an automorphism of order 2 $\si …