# Small G-varieties

Abstract : An affine varieties with an action of a semisimple group $G$ is called "small" if every non-trivial $G$-orbit in $X$ is isomorphic to the orbit of a highest weight vector. Such a variety $X$ carries a canonical action of the multiplicative group $\mathbb{K}^*$ commuting with the $G$-action. We show that $X$ is determined by the $\mathbb{K}^*$-variety $X^U$ of fixed points under a maximal unipotent subgroups $U$ of $G$. Moreover, if $X$ is smooth, then $X$ is a $G$-vector bundle over the quotient $X// G$. If $G$ is of type $A_n$ ($n>1$), $C_n$, $E_6$, $E_7$ or $E_8$, we show that all affine $G$-varieties up to a certain dimension are small. As a consequence we have the following result. If $n>4$, every smooth affine $SL_n$-variety of dimension $<2n$ is an $\mathrm{SL}_n$-vector bundle over the smooth quotient $X//\mathrm{SL}_n$, with fiber isomorphic to the natural representation or its dual.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-02939731
Contributor : Susanna Zimmermann Connect in order to contact the contributor
Submitted on : Tuesday, September 15, 2020 - 5:58:39 PM
Last modification on : Wednesday, October 14, 2020 - 4:01:38 AM

### Identifiers

• HAL Id : hal-02939731, version 1
• ARXIV : 2009.05559

### Citation

Hanspeter Kraft, Andriy Regeta, Susanna Zimmermann. Small G-varieties. 2020. ⟨hal-02939731⟩

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