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$\newcommand{\diag}{{\rm diag}} \newcommand{\sH}{{\mathcal H}} \newcommand{\R}{{\mathbb R}} \newcommand{\HH}{\sf H} \newcommand{\V}{{\sf V}} \newcommand{\B}{{\sf B}} \newcommand{\C}{{\Bbb C}} $No, th …

Let $G={\rm SL}(n)\times {\rm SL}(n')$ and let $H\subset G$ denote the stabilizer of $J_k$ in $G$. We write ${\frak X}(G)$ for the character group of $G$. Then ${\frak X}(G)=0$. We have a canonical is …

$\DeclareMathOperator\Pic{Pic}\DeclareMathOperator\GL{GL}\DeclareMathOperator\GO{GO}$ Let $G$ be a connected linear algebraic group over an algebraically closed field $K$ of characteristic 0. Let $ …

Yes, assuming that $K$ is a connected compact Lie group. Indeed, fix $n$ such that $0\le n\le d={\rm dim}(M)$. The group $K$ acts on the integral cohomology group $H^n(M,\Bbb Z)$ trivially, because …

Consider the real diagonal $4\times 4$ - matrix $$I_{2,2}={\rm diag}(1,1,-1,-1)$$ and the corresponding special unitary group $$ G={\rm SU}(2,2)=\{g\in {\rm SL}(4,{\mathbb{C}})\ |\ g\cdot I_{2,2}\cd …

Although the question has been answered in comments (by Victor Petrov), I prefer to post an answer. I assume that $G={\rm U}(2,2)$ rather than $G={\rm SU}(2,2)$. My variety $G/U$ is the variety $X$ …

Let $G/H$ be a spherical homogeneous variety, where $G$ is a complex semisimple group. Assume that the subgroup $H$ is self-normalizing, i.e., $\mathcal{N}_G(H)=H$. Then by results of Brion and Pauer …

Let $k$ be an algebraically closed field of characteristic 0. Let $G$ be a connected linear algebraic group over $k$. Let $H\subset G$ be an algebraic $k$-subgroup. Let $Y$ be an algebraic variety ove …

I wish to "compute" ${{\rm Aut}}(G,X)$ for a spherical homogeneous space $X$ of $G$ in terms of the spherical datum of $X$. First let $G$ be any algebraic group over $\mathbb C$, and let $X$ be any le …

The result in my preprint mentioned in the question was erroneous (the mistake was noticed by a referee). It is possible to construct a quotient $X=G/H$ and and automorphism $\tau$ of $\mathbb{C}$ su …

Let $X$ be a smooth irreducible algebraic variety over the field of complex numbers ${\mathbb{C}}$. Let $x\in X({\mathbb{C}})$. Let $\tau$ be an automorphism of ${\mathbb{C}}$ (not necessarily continu …

This is a continuation of my question Quotient of a reductive group by a non-smooth central finite subgroup. Let $G$ be a smooth, connected, reductive $k$-group over a field $k$ of characteristic $p …

I answer the question in the comment of Tom Goodwillie: What is known when $H=1$? Theorem. Let $G$ be a connected linear algebraic group over ${\mathbb{C}}$. Let $\tau$ be an automorphism of ${\mathb …