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- W4378673076 abstract "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 $X$." @default.
- W4378673076 created "2023-05-30" @default.
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- W4378673076 date "2014-06-22" @default.
- W4378673076 modified "2023-09-27" @default.
- W4378673076 title "Demazure roots and spherical varieties: the example of horizontal SL(2)-actions" @default.
- W4378673076 doi "https://doi.org/10.48550/arxiv.1406.5744" @default.
- W4378673076 hasPublicationYear "2014" @default.
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