Abstract
AbstractAn affine variety with an action of a semisimple group G is called “small” if every nontrivial 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 subgroup
$U \subset G$
. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient
$X /\!\!/ G$
.If G is of type
${\mathsf {A}_n}$
(
$n\geq 2$
),
${\mathsf {C}_{n}}$
,
${\mathsf {E}_{6}}$
,
${\mathsf {E}_{7}}$
, or
${\mathsf {E}_{8}}$
, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If
$n \geq 5$
, every smooth affine
$\operatorname {\mathrm {SL}}_n$
-variety of dimension
$< 2n-2$
is an
$\operatorname {\mathrm {SL}}_n$
-vector bundle over the smooth quotient
$X /\!\!/ \operatorname {\mathrm {SL}}_n$
, with fiber isomorphic to the natural representation or its dual.
Publisher
Canadian Mathematical Society
Reference31 articles.
1. [18] Kraft, H. , Letter to Michel Brion, 1980.
2. Linearizing certain reductive group actions
3. Sur les groupes linéaires algébriques dont l’algèbre des invariants est libre;Kac;C. R. Math.,1976
4. CONTRACTION OF THE ACTIONS OF REDUCTIVE ALGEBRAIC GROUPS