Embeddings of semisimple complex Lie groups and cohomological components of modules

TL;DR

This paper investigates the conditions under which cohomological pullbacks between flag manifolds, arising from embeddings of semisimple complex Lie groups, are nonzero, revealing new structure theorems and applications to invariant algebra generators.

Contribution

It establishes a necessary and sufficient condition for the nonvanishing of cohomological pullbacks and analyzes specific embeddings, including regular, diagonal, and non-regular cases, using Lie algebra cohomology.

Findings

Derived a criterion for nonvanishing of cohomological pullbacks

Proved a structure theorem for cohomological pairs of highest weights

Connected invariant algebra generators to cohomological components

Abstract

Let G --> G' be an embedding of semisimple complex Lie groups, let B and B' be a pair of nested Borel subgroups, and let f:G/B --> G'/B' be the associated equivariant embedding of flag manifolds. We study the pullbacks of cohomologies of invertible sheaves on G'/B' along the embedding f. Let O' be a G'-equivariant invertible sheaf on G'/B', and let O be its restriction to G/B. Consider the G-equivariant pullback on cohomology p : H(G'/B',O') --> H(G/B,O). The Borel-Weil-Bott theorem implies that the two cohomology spaces above are irreducible modules of G' and G respectively. By Schur's lemma, p is either surjective or zero. In this paper we establish a necessary and sufficient condition for nonvanishing of p, and apply it to the study of regular and diagonal embeddings. We also prove a structure theorem about the set of cohomological pairs of highest weights. We also study in detail two cases of embeddings which are neither regular nor diagonal. The first is the case of homogeneous rational curves in complete flag manifolds, and the second is the embedding of the complete flag manifold of G into the complete flag manifold of G'=SL(Lie(G)), via the adjoint representation of G. We show that the generators of the algebra of invariants in the polynomial algebra on Lie(G) can be realized as cohomological components. Our methods rely on Kostant's theory of Lie algebra cohomology.