Embeddings of semisimple complex Lie groups and cohomological components of modules
Embeddings of semisimple complex Lie groups and cohomological components of modules
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半单复李群的嵌入和模的上同调分量
DOI：
10.1016/j.jalgebra.2012.09.030
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发表时间：
2010
影响因子：
0.9
通讯作者：
V. Tsanov
中科院分区：
文献类型：
--
作者：
V. Tsanov
设G´G≈一个半简单复李群的嵌入，B´B≈一对嵌套的Borel子群，G/B´G≈/B≈标志流形的关联嵌入。设O ~ （λ ~）是G ~ /B ~上的等变可逆束，O （λ）是它对G/B的约束。考虑到G-equivariant回落πλ˜:H (G˜/ B˜,O˜(λ˜))→H (G / B、O(λ))。Borel-Weil-Bott定理和Schur引理暗示π λ≈要么满射要么为零。如果π λ≈非零，对偶映射（π λ≈）的像是G ~不可约模中的G不可约分量，称为上同调分量。建立了π λ￣不灭的充分必要条件。此外，我们证明了V (μ)∧~ （μ ~）上同调的优势权（μ, μ ~）对集合结构的一个定理。这里V （μ）和V ~ （μ ~）表示各自的最高权重模块。简化的专门化是为规则和对角嵌入制定的。特别地，我们给出了Dimitrov和Roth最近定理的另一种证明。除了正则和对角情况外，我们还研究了等嵌有理曲线，并证明了半单李代数上不变多项式代数的生成子可以用上同调分量的形式得到。我们的方法依赖于柯士坦的李代数上同调理论。
Abstract Let G↪ G˜ be an embedding of semisimple complex Lie groups, B⊂ B˜ a pair of nested Borel subgroups and G/B↪ G˜/B˜ the associated embedding of flag manifolds. Let O˜(λ˜) be an equivariant invertible sheaf on G˜/B˜ and O (λ) be its restriction to G/B. Consider the G-equivariant pullback π λ˜: H (G˜/B˜, O˜(λ˜))→ H (G/B, O (λ)). The Borel–Weil–Bott theorem and Schurʼs lemma imply that π λ˜ is either surjective or zero. If π λ˜ is nonzero, the image of the dual map (π λ˜)⁎ is a G-irreducible component in a G˜-irreducible module, called a cohomological component. We establish a necessary and sufficient condition for nonvanishing of π λ˜. Also, we prove a theorem on the structure of the set of pairs of dominant weights (μ, μ˜) with V (μ)⊂ V˜(μ˜) cohomological. Here V (μ) and V˜(μ˜) denote the respective highest weight modules. Simplified specializations are formulated for regular and diagonal embeddings. In particular, we give an alternative proof of a recent theorem of Dimitrov and Roth. Beyond the regular and diagonal cases, we study equivariantly embedded rational curves and we also show that the generators of the algebra of ad-invariant polynomials on a semisimple Lie algebra can be obtained as cohomological components. Our methods rely on Kostantʼs theory of Lie algebra cohomology.