Refined Dynkin Data for Nilpotent Elements

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Let B be a Borel subgroup in a complex semisimple Lie group G and let g be the Lie algebra corresponding to G. For each nilpotent element x ∈ g we define an element cl B ( x) of the Cartan algebra of g as a homology class of an appropriate curve on the flag variety G/B. For x "generic" in its G-orbit O the class cl B ( x) coincides with the element defining the weighted Dynkin diagram corresponding to O . We obtain some information about cl B( x) and, in particular, calculate it completely in the case of regular x.