2014/06/22 by Kevin Langlois, Langlois, Kevin, Alexander Perepechko +1 · 1 citation
Mathematics · #14M25 14M27 14R20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1406.5744
openalex publication_date 2014/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a connected reductive group, and let X be an affine G-spherical variety. We show that the classification of \mathbbGa-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 \mathbbGa\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 \mathbbX(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 SL2 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 ℚ⊗ℤ\mathbbX(G) and a sublattice of \mathbbX(G). We conjecture that \rm Rt(X) can be described in a similar combinatorial way for an arbitrary affine spherical variety X.