On Some Families of Smooth Affine Spherical Varieties of Full Rank

Abstract

Let G be a complex connected reductive group. I. Losev has shown that a smooth affine spherical G-variety X is uniquely determined by its weight monoid, which is the set of irreducible representations of G that occur in the coordinate ring of X. In this paper we use a combinatorial characterization of the weight monoids of smooth affine spherical varieties to classify: (a) all such varieties for G = SL(2) × C × and (b) all such varieties for G simple which have a G-saturated weight monoid of full rank. We also use the characterization and F. Knop's classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of C. Woodward's result that every reflective Delzant polytope is the moment polytope of such a manifold.Correspondingly, the varieties X a can be considered as "building blocks" for the manifold M. Let us recall briefly in which sense, referring to Section 2 of [Kno11] for details: the manifold M admits an open cover M = a U a , where the union is over all vertices a of P M and U a is K-stable for all a. Moreover, the subset U a is a K-equivariant fibre bundle of the form K × Ka Y a , where Y a is a K a -stable open subset of X a intersecting the closed G(a)-orbit.Example 1.1. Let K = SU(2). Then one can identify su(2) * with R 3 and the coadjoint action of SU(2) with the standard action through SUHamiltonian manifold under the diagonal action of SU(2) with moment mapOne can check that M is multiplicity free. For T R we can take the subgroup of K made up of diagonal matrices. Then T R ∼ = U(1) and Λ ∼ = Z. We can choose the identification su(2) * ≡ R 3 in such a way that Lie(T R ) * gets identified with R(0, 0, 1) and the weight lattice Λ with Z(0, 0, 1). As a dominant Weyl chamber t + we can then choose R ≥0 (0, 0, 1). To lighten notation, we will identify Λ with Z and t + with R ≥0 . Elementary computations show thatwhere b ∈ R >0 depends on the choice of v ∈ su(2) * , that L R = T R and that the generic isotropy group K M is equal to the center of SU(2). Consequently,We confirm that the conditions (1.4) are satisfied at the two vertices of P M . We begin with the vertex 0. Since K 0 = SU(2), we have that G(0) = SL(2). Observe that C 0 P = t + . Consider the smooth affine spherical SL(2)-varietywhere T is the subset of SL(2) made up of diagonal matrices. As is well known, the weight monoid of X 0 is 2N ⊂ N = Λ ∩ t + , whence (1.4) holds at the vertex 0. Next we consider the vertex b of P M . Then K b = T R and consequently G(b) = T . Note that C b P M = −t + . Straightforward computations show that for the smooth affine spherical T -varietythe conditions in (1.4) hold again. Remark 1.2. (a) Generalizing Example 1.1, Corollary 11.4 in [Kno11] shows how one can recover P. Iglésias's classification of multiplicity free Hamiltonian SU(2)-manifolds in [Igl91] from Knop's theorems above. (b) Similarly, Delzant's classification of symplectic toric manifolds in [Del88] by Delzant polytopes is an immediate consequence of Knop's results (see [Kno11, Corollary 11.3].) (c) In [Kno16, Section ...

Quasi-Hamiltonian model spaces

“…Part 4. G-saturated smooth affine spherical varieties of full rank This section will recall the combinatorial smoothness criterion from [PVS15] and give a brief overview of the classification of smooth affine spherical G-saturated varieties of full rank in [PPVS18]. These are precisely the local models in X 0 for quasi-Hamiltonian model spaces.…”

“…As an instructive example of how to apply the smoothness criterion, we investigate the family of monoids for G of type C n and Σ N (Γ) = S + . We want to show that Γ is smooth if and only if Γ = Λ + (This is a subcase of [PPVS18], Lemma 3.20. We show a more detailed proof here).…”

“…Next we recall the classification of G-saturated smooth affine spherical varieties of full rank with G simple and simply connected from [PPVS18]. As said, this is the complete list of possible local models in X 0 for quasi-Hamiltonian model Kτ -spaces.…”

“…The lattice Λ intersected with the tangent cone C X 0 (P) gives a well known smooth monoid with spherical roots S + 0 . We know from [PPVS18] that all smooth G-saturated weight monoids for A n have either all 2α or all sums of consecutive simple roots as spherical roots. For n ≥ 3, {α 1 , α 2 } or {α n−1 , α n } is contained in every local root system.…”

“…This gives candidates for the lattices Λ M , these are a subclass of smooth affine spherical varieties of full rank. This was done in [PPVS18] and is summarized in part 4. (2) We decide which of these lattices fulfill theorem 1.2.…”