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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. 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Tsanov","submitted_at":"2010-05-23T19:37:54Z","abstract_excerpt":"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. 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