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@WillSawin: Dear Will, thanks for your comments! Maybe I will first try to calculate the U-equivariant morphisms between the line bundles as you suggested. Is it true that the $U$-invariant space in $Ext^i(\mathcal{L}(\lambda), \mathcal{L}(\mu))$ (using Borel-Weil-Bott) calculates the $U$-equivariant morphisms? Or the latter is more complicated than that?
