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Pré-Publication, Document De Travail Année : 2014

Demazure roots and spherical varieties: the example of horizontal SL(2)-actions

Kevin Langlois
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Résumé

Let $G$ be a connected reductive group, and let $X$ be an affine $G$-spherical variety. We show that the classification of $\mathbb{G}_{a}$-actions on $X$ normalized by $G$ can be reduced to the description of quasi-affine homogeneous spaces under the action of a semi-direct product $\mathbb{G}_{a}\rtimes G$ with the following property. The induced $G$-action is spherical and the complement of the open orbit is either empty or a $G$-orbit of codimension one. These homogeneous spaces are parametrized by a subset ${\rm Rt}(X)$ of the character lattice $\mathbb{X}(G)$ of $G$, which we call the set of Demazure roots of $X$. We give a complete description of the set ${\rm Rt}(X)$ when $G$ is a semi-direct product of ${\rm SL}_{2}$ and an algebraic torus; we show particularly that ${\rm Rt}(X)$ can be obtained explicitly as the intersection of a finite union of polyhedra in $\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{X}(G)$ and a sublattice of $\mathbb{X}(G)$. We conjecture that ${\rm Rt}(X)$ can be described in a similar combinatorial way for an arbitrary affine spherical variety $X$.
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Dates et versions

hal-01011047 , version 1 (22-06-2014)

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  • HAL Id : hal-01011047 , version 1

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Kevin Langlois, Alexander Perepechko. Demazure roots and spherical varieties: the example of horizontal SL(2)-actions. 2014. ⟨hal-01011047⟩

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