The aim of this paper is to begin a study of the cohornology modules H-i(X (w) L-lambda) for non-dominant weights lambda on Schubert varieties X(w) in G/B. The aim is to set up a, combinatorial dictionary for describing the cohomology modules and give criteria for their vanishing. Here L-lambda denotes the line bundle on X(w) corresponding to the 1-dimensional representation of B given by the character lambda.
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