Embeddings of semisimple complex Lie groups and cohomological components of modules

作者：Tsanov Valdemar V^*

来源：Journal of Algebra, 2013, 373: 1-29.

DOI：10.1016/j.jalgebra.2012.09.030

• Representation theory of Lie groups and Lie algebras
• Flag manifolds
• Homogeneous vector bundles
• Borel-Weil-Bott theorem
• Kostant theorem on Lie algebra cohomology

Let G hooked right arrow (G) over tilde be an embedding of semisimple complex Lie groups, B subset of (B) over tilde a pair of nested Borel subgroups and G/B hooked right arrow (G) over tilde/(B) over tilde the associated embedding of flag manifolds. Let (O) over tilde((lambda) over tilde) be an equivariant invertible sheaf on (G) over tilde/(B) over tilde and O(lambda) be its restriction to G/B. Consider the G-equivariant pullback %26lt;br%26gt;pi((lambda) over tilde) : H((G) over tilde/(B) over tilde, (O) over tilde((lambda) over tilde)) -%26gt; H(G/B, O(lambda)). %26lt;br%26gt;The Borel-Weil-Bott theorem and Schur%26apos;s lemma imply that pi((lambda) over tilde) is either surjective or zero. If pi((lambda) over tilde) is nonzero, the image of the dual map (pi((lambda) over tilde))* is a G-irreducible component in a (G) over tilde irreducible module, called a cohomological component. %26lt;br%26gt;We establish a necessary and sufficient condition for nonvanishing of pi((lambda) over tilde). Also, we prove a theorem on the structure of the set of pairs of dominant weights (mu, (mu) over tilde) with V(mu) subset of (V) over tilde((mu) over tilde) cohomological. Here V(mu) and (V) over tilde((mu) over tilde) 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 Kostant%26apos;s theory of Lie algebra cohomology.

• 出版日期2013-1-1