1
vote
1answer
33 views

Torus orbit closures and rank-1 subtori

Suppose I have a connected complex torus $K$ acting on a quasi-affine complex variety $X$. Suppose also that I have $p,q\in X$ such that the orbit $Kq$ is closed in $X$ and $q\in ...
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0answers
80 views

“Invariant integral” for linearly reductive groups, and the Reynolds operator

Let $G$ be an affine algebraic group. Let $$\lambda\colon G\to GL(k[G]),\quad \lambda(g)(f)=(h\mapsto f(g^{-1}h)),$$ be the left-translation, which is a rational representation of $G$ on its ...
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2answers
151 views

Diagonalizable linear algebraic group is isomorphic to $(\mathbb{C}^*)^r\times A$, for some finite abelian group $A$

I have three questions about algebraic groups. Let $D$ be a linear algebraic group. Then the following are equivalent: $D$ is diagonalizable. $\mathop{Hom}(D,\mathbb{C}^*)$ is finitely generated ...