Small G-varieties

TL;DR

This paper classifies small affine G-varieties with semisimple group actions, showing they are determined by fixed point sets and, if smooth, are vector bundles over quotients, with classifications for certain group types and dimensions.

Contribution

It introduces the concept of small G-varieties, characterizes their structure via fixed points, and classifies them for specific groups and dimensions, extending understanding of affine G-varieties.

Findings

Small G-varieties are determined by fixed points under unipotent subgroups.

Smooth small G-varieties are G-vector bundles over their quotients.

All affine G-varieties of certain types and dimensions are small.

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.