Embeddings of semisimple complex Lie groups and cohomological components of modules

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 pullbackπλ˜:H(G˜/B˜,O˜(λ˜))→H(G/B,O(λ)). The Borel–Weil–Bott theorem and Schurʼs lemma imply that πλ˜ is either surjective or zero. If πλ˜ is nonzero, the image of the dual map (πλ˜)⁎ is a G-irreducible component in a G˜-irreducible module, called a cohomological component.We establish a necessary and sufficient condition for nonvanishing of πλ˜. 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.