Poisson Morphisms and Reduced Affine Poisson Group Actions

We,establish the concept of a quotient affine Poisson group,and study the reduced Poisson action of the quotient of an affine Poisson group G on the quotient of an affine Poisson G-variety V.The Poisson morphisms(including equivariant cases)between Poisson affine varieties are also discussed.

On Some Families of Smooth Affine Spherical Varieties of Full Rank

Let G be a complex connected reductive group. 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 Knop＇s classification theorem for multiplicity free Hamiltonian manifolds to give a new proof of Woodward＇s result that every reflective Delzant polytope is the moment polytope of such a manifold.

Fully Hodge-Newton Decomposable Shimura Varieties

The motivation for this paper is the study of arithmetic properties of Shimura varieties,in particular the Newton stratification of the special fiber of a suitable integral model at a prime with parahoric level structure.This is closely related to the structure of Rapoport–Zink spaces and of affine Deligne–Lusztig varieties.We prove a Hodge–Newton decomposition for affine Deligne–Lusztig varieties and for the special fibers of Rapoport–Zink spaces,relating these spaces to analogous ones defined in terms of Levi subgroups,under a certain condition(Hodge–Newton decomposability)which can be phrased in combinatorial terms.Second,we study the Shimura varieties in which every non-basic risogeny classis Hodge–Newton decomposable.We show that(assuming the axioms of He and Rapoport in Manuscr.Math.152(3–4):317–343,2017)this condition is equivalent to nice conditions on either the basic locus or on all the non-basic Newton strata of the Shimura varieties.We also give a complete classification of Shimura varieties satisfying these conditions.While previous results along these lines often have restrictions to hyperspecial(or at least maximal parahoric)level structure,and/or quasi-split underlying group,we handle the cases of arbitrary parahoric level structure and of possibly non-quasisplit underlying groups.This results in a large number of new cases of Shimura varieties where a simple description ofthe basiclocus can be expected.As a striking consequence of the results,we obtain that this property is independent of the parahoric subgroup chosen as level structure.We expect that our conditions are closely related to the question whether the weakly admissible and admissible loci coincide.