User:TakuyaMurata/Linear algebraic group action

Throughout the article, G is a linear algebraic group and X a smooth scheme (or a stack) on which G acts.

If X is a quasi-affine variety and if G is a unipotent group, then its orbits on X are closed.^[1]

Let $f:G\to X,g\mapsto gx$ be the orbit map at x. The differential at the identity element $df_{1}$ is surjective if and only if $G_{x}$ and $\operatorname {ker} df_{1}$ have the same dimension.^[2]

The action of G on X is called principal if. A principal action is free, but the converse does not hold in general.

Quotient

Adjoint representation

Notes

^ Borel 1991, Proposition 4.10
^ Richardson 1967, Lemma 2.1

References

A. Borel, Linear algebraic groups
Richardson, Conjugacy Classes in Lie Algebras and Algebraic Groups, The Annals of Mathematics, ISSN 0003-486X, 07/1967, Volume 86, Issue 1, pp. 1 - 15
A. Bialłynicki-Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2), 98:480–497, 1973.

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[1] Borel 1991, Proposition 4.10

[2] Richardson 1967, Lemma 2.1

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