Abstract
Let G be a connected reductive algebraic group. We prove that for a quasi-affine G-spherical variety the weight monoid is determined by the weights of its non-trivial 𝔾 a -actions that are homogeneous with respect to a Borel subgroup of G. As an application we get that a smooth affine spherical variety that is non-isomorphic to a torus is determined by its automorphism group (considered as an ind-group) inside the category of smooth affine irreducible varieties.
Highlights
In this article, we work over an algebraically closed field k of characteristic zero if it is not specified otherwise.In [Kra[17], Th. 1.1], Kraft proved that An is determined by its automorphism group Aut(An) seen as an ind-group inside the category of connected affine varieties and in [KRvS19, Main Th.], this result was partially generalized in case Aut(An) is seen only as an abstract group
A], the last results are widely generalized in the following sense: An is completely characterized through the abstract group Aut(An) inside the category of connected affine varieties
This characterization of the sphericity is stable under group isomorphisms of automorphism groups that preserve algebraic groups and we get Main Theorem A(1)
Summary
We work over an algebraically closed field k of characteristic zero if it is not specified otherwise. We show that if G is not a torus, an irreducible normal quasi-affine variety with a faithful G-action is G-spherical if and only if the dimension of all generalized root subgroups of Aut(X) with respect to B is bounded (see Definition 7.1, Proposition 7.3 and Lemma 7.6) This characterization of the sphericity is stable under group isomorphisms of automorphism groups that preserve algebraic groups and we get Main Theorem A(1). Note that for a quasi-affine G-spherical variety X the following fact holds: the algebraic quotient Xaff //U is an affine toric variety, where U denotes the unipotent radical of a Borel subgroup of G Using this fact and our study of the homogeneous Ga-actions presented, we prove Theorem 8.2. The authors thank the anonymous referees for very helpful suggestions and comments
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