These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local...These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.展开更多
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 ...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.展开更多
These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varie...These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.展开更多
We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects ...We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety.展开更多
Braverman and Kazhdan(2000)introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula.Their conjectures should imply that quite general Langlands L-functions have ...Braverman and Kazhdan(2000)introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula.Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands'functoriality conjecture.As an evidence for their conjectures,Braverman and Kazhdan(2002)considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved.The connection between the two papers is made explicit in the work of Li(2018).In this paper,we consider a special case of the setting in Braverman and Kazhdan's later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper.Along the way we provide analytic control on the Schwartz space we construct;this analytic control was conjectured to hold(in a slightly different setting)in the work of Braverman and Kazhdan(2002).展开更多
文摘These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.
基金partially supported by the DFG Schwerpunktprogramm 1388–Darstellungstheoriesupport from the National Science Foundation(USA)through grant DMS 1407394the PSC-CUNY Research Award Program
文摘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.
文摘These notes are an introduction to wonderful varieties. We discuss some general results on their geometry, their role in the theory of spherical varieties, several aspects of the combinatorics arising from these varieties, and some examples.
文摘We review in these notes the theory of equivariant embeddings of spherical homogeneous spaces. Given a spherical homogeneous space G/H, the normal equivariant embeddings of G/H are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties and which encode several geometric properties of the corresponding variety.
基金partially supported by a National Science Foundation Grant(Grant No.ID:DMS-1601303)partially supported by a National Science Foundation Grant(Grant No.ID:DMS-1500966)+1 种基金Simons Foundation Collaboration Grant for MathematiciansSimons Fellowship
文摘This is a survey of some recent results on spherical tropical geometry.
基金supported by the National Science Foundation of the USA(Grant Nos.DMS-1405708 and DMS-1901883)supported by the National Science Foundation of the USA(Grant Nos.DMS-1702218 and DMS-1848058)a start-up fund from the Department of Mathematics at Purdue University。
文摘Braverman and Kazhdan(2000)introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula.Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands'functoriality conjecture.As an evidence for their conjectures,Braverman and Kazhdan(2002)considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved.The connection between the two papers is made explicit in the work of Li(2018).In this paper,we consider a special case of the setting in Braverman and Kazhdan's later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper.Along the way we provide analytic control on the Schwartz space we construct;this analytic control was conjectured to hold(in a slightly different setting)in the work of Braverman and Kazhdan(2002).
