Some Results on the Cohomology of Line Bundles on the Three Dimensional Flag Variety

Muhammad Fazeel Anwar
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Muhammad Fazeel Anwar: Department of Mathematics, Sukkur IBA University, Sukkur 65200, Pakistan

Mathematics, 2019, vol. 7, issue 3, 1-5

Abstract: Let k be an algebraically closed field of prime characteristic and let G be a semisimple, simply connected, linear algebraic group. It is an open problem to find the cohomology of line bundles on the flag variety G / B , where B is a Borel subgroup of G . In this paper we consider this problem in the case of G = S L 3 ( k ) and compute the cohomology for the case when 〈 λ , α ∨ 〉 = − p n a − 1 , ( 1 ≤ a ≤ p , n > 0 ) or 〈 λ , α ∨ 〉 = − p n − r , ( r ≥ 2 , n ≥ 0 ) . We also give the corresponding results for the two dimensional modules N α ( λ ) . These results will help us understand the representations of S L 3 ( k ) in the given cases.

Keywords: representation theory; algebraic groups; cohomology; line bundles (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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