Root subgroups on affine spherical varieties

Abstract

Given a connected reductive algebraic group $G$ and a Borel subgroup $B\subseteq G$, we study $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical $G$-variety $X$ we obtain a complete description of all $G$-normalized one-parameter additive group actions on $X$ and show that the open $G$-orbit in $X$ can be connected with every $G$-stable prime divisor via a suitable choice of a $B$-normalized one-parameter additive group action. Finally, when $G$ is of semisimple rank $1$, we obtain a complete description of all $B$-normalized one-parameter additive group actions on affine spherical $G$-varieties having an open orbit of a maximal torus $T\subseteq B$.

arXiv commentv3: 31 pages, revised according to the referee's suggestions, numbering of sections changed

References in corpus (19)

• Flexible varieties and automorphism groups
• Affine T-varieties of complexity one and locally nilpotent derivations
• Complete toric varieties with reductive automorphism group
• Ga-actions of fiber type on affine T-varieties
• Additive actions on toric varieties
• Polyhedral divisors and SL_2-actions on affine T-varieties
• Infinite transitivity, finite generation, and Demazure roots
• Commutative algebraic monoid structures on affine spaces
• New and old results on spherical varieties via moduli theory
• Orbits of strongly solvable spherical subgroups on the flag variety
• The moduli scheme of affine spherical varieties with a free weight monoid
• On the irreducible components of moduli schemes for affine spherical varieties
• Commutative algebraic monoid structures on affine surfaces
• Demazure roots and spherical varieties: the example of horizontal SL(2)-actions
• Wonderful varieties of type B and C
• On extended weight monoids of spherical homogeneous spaces
• On the existence of $B$-root subgroups on affine spherical varieties
• Commuting homogeneous locally nilpotent derivations
• Characterizing quasi-affine spherical varieties via the automorphism group

Cited by in corpus (1)

• On the existence of $B$-root subgroups on affine spherical varieties