Strongly solvable spherical subgroups and their combinatorial invariants

A subgroup H of an algebraic group G is said to be strongly solvable if H is contained in a Borel subgroup of G. This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna's general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna's 1993 classification of strongly solvable wonderful subgroups, and the author's 2011 classification of strongly solvable spherical subgroups. We give a detailed presentation of all the three classifications and exhibit interrelations between the corresponding combinatorial invariants, which enables one to pass from one of these classifications to any other.Comment: v3: 58 pages, revised according to the referee's suggestions; v4: numbering of sections changed to agree with the published versio

Degenerations of spherical subalgebras and spherical roots

We obtain several structure results for a class of spherical subgroups of connected reductive complex algebraic groups that extends the class of strongly solvable spherical subgroups. Based on these results, we construct certain one-parameter degenerations of the Lie algebras corresponding to such subgroups. As an application, we exhibit explicit algorithms for computing the set of spherical roots of such a spherical subgroup.Comment: v2: 45 pages, revised extended version with new Section 6 containing an optimization of the initial algorith

On the irreducible components of moduli schemes for affine spherical varieties

We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $\Gamma$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_\Gamma$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_\Gamma$ to be irreducible and exhibit several examples where $\mathrm M_\Gamma$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_\Gamma$.Comment: v4: 26 pages, final versio

New and old results on spherical varieties via moduli theory

Given a connected reductive algebraic group $G$ and a finitely generated monoid $\Gamma$ of dominant weights of $G$, in 2005 Alexeev and Brion constructed a moduli scheme $\mathrm M_\Gamma$ for multiplicity-free affine $G$-varieties with weight monoid $\Gamma$. This scheme is equipped with an action of an `adjoint torus' $T_{\mathrm{ad}}$ and has a distinguished $T_{\mathrm{ad}}$-fixed point $X_0$. In this paper, we obtain a complete description of the $T_{\mathrm{ad}}$-module structure in the tangent space of $\mathrm M_\Gamma$ at $X_0$ for the case where $\Gamma$ is saturated. Using this description, we prove that the root monoid of any affine spherical $G$-variety is free. As another application, we obtain new proofs of uniqueness results for affine spherical varieties and spherical homogeneous spaces first proved by Losev in 2009. Furthermore, we obtain a new proof of Alexeev and Brion's finiteness result for multiplicity-free affine $G$-varieties with a prescribed weight monoid. At last, we prove that for saturated $\Gamma$ all the irreducible components of $\mathrm M_\Gamma$, equipped with their reduced subscheme structure, are affine spaces.Comment: v3: 45 pages, minor improvements, final versio

An epimorphic subgroup arising from Roberts' counterexample

In 1994, based on Roberts' counterexample to Hilbert's fourteenth problem, A'Campo-Neuen constructed an example of a linear action of a 12-dimensional commutative unipotent group H_0 on a 19-dimensional vector space V such that the algebra of invariants k[V]^{H_0} is not finitely generated. We consider a certain extension H of H_0 by a one-dimensional torus and prove that H is epimorphic in SL(V). In particular, the homogeneous space SL(V)/H provides a new example of a homogeneous space with epimorphic stabilizer that admits no projective embeddings with small boundary.Comment: v2: 9 pages, small correction