Transformation Groups ( IF 0.7 ) Pub Date : 2021-02-12 , DOI: 10.1007/s00031-021-09642-3 ROMAN AVDEEV
Given a connected reductive complex algebraic group G and a spherical subgroup H ⊂ G, the extended weight monoid \( {\hat{\Lambda}}_G^{+}\left(G/H\right) \) encodes the G-module structures on spaces of global sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P ⊂ G, in this paper we obtain a description of \( {\hat{\Lambda}}_G^{+}\left(G/H\right) \) via the set of simple spherical roots of G/H together with certain combinatorial data explicitly computed from the pair (P;H). As an application, we deduce a new proof of a result of Avdeev and Gorfinkel describing \( {\hat{\Lambda}}_G^{+}\left(G/H\right) \) in the case where H is strongly solvable.
中文翻译:
球均质空间的扩展权重单调
给定连接的还原复杂代数群G ^和球形亚组ħ ⊂ ģ,扩展重量幺\({\帽子{\ LAMBDA}} _ G ^ {+} \左(G / H \右)\)编码G ^ - G / H上所有G线性化线束的全局截面空间上的模块结构。假设ģ是半单和简单地连接和ħ由以抛物线子组的规则嵌入指定P ⊂ ģ,在本文中,我们获得的描述\({\帽子{\ LAMBDA}} _ G ^ {+} \左( G / H \ right)\)通过一组简单的球根G / H以及从该对(P ; H)中明确计算出的某些组合数据。作为应用程序,我们推导了一个新的证明,即在H强烈的情况下,Avdeev和Gorfinkel描述\({\ hat {\ Lambda}} _ G ^ {+} \ left(G / H \ right)\)的结果可解决的。