کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6414895 | 1630541 | 2013 | 29 صفحه PDF | دانلود رایگان |
Let GâªGË be an embedding of semisimple complex Lie groups, BâBË a pair of nested Borel subgroups and G/BâªGË/BË the associated embedding of flag manifolds. Let OË(λË) be an equivariant invertible sheaf on GË/BË and O(λ) be its restriction to G/B. Consider the G-equivariant pullbackÏλË:H(GË/BË,OË(λË))âH(G/B,O(λ)). The Borel-Weil-Bott theorem and Schurʼs lemma imply that ÏÎ»Ë is either surjective or zero. If ÏÎ»Ë is nonzero, the image of the dual map (ÏλË)â is a G-irreducible component in a GË-irreducible module, called a cohomological component.We establish a necessary and sufficient condition for nonvanishing of ÏλË. Also, we prove a theorem on the structure of the set of pairs of dominant weights (μ,μË) with V(μ)âVË(μË) cohomological. Here V(μ) and VË(μË) 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.
Journal: Journal of Algebra - Volume 373, 1 January 2013, Pages 1-29