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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product


Authors: Sam Evens and William Graham
Journal: Trans. Amer. Math. Soc. 365 (2013), 5833-5857
MSC (2010): Primary 17B56, 14M15, 20G05
Published electronically: August 2, 2013
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Abstract: We consider the Belkale-Kumar cup product $ \odot _t$ on $ H^*(G/P)$ for a generalized flag variety $ G/P$ with parameter $ t \in \mathbb{C}^m$, where $ m=\dim (H^2(G/P))$. For each $ t\in \mathbb{C}^m$, we define an associated parabolic subgroup $ P_K \supset P$. We show that the ring $ (H^*(G/P), \odot _t)$ contains a graded subalgebra $ A$ isomorphic to $ H^*(P_K/P)$ with the usual cup product, where $ P_K$ is a parabolic subgroup associated to the parameter $ t$. Further, we prove that $ (H^*(G/P_K), \odot _0)$ is the quotient of the ring $ (H^*(G/P), \odot _t)$ with respect to the ideal generated by elements of positive degree of $ A$. We prove the above results by using basic facts about the Hochschild-Serre spectral sequence for relative Lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of Hochschild and Serre.


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Additional Information

Sam Evens
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: sevens@nd.edu

William Graham
Affiliation: Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, Georgia 30602
Email: wag@math.uga.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05792-3
PII: S 0002-9947(2013)05792-3
Received by editor(s): January 1, 2012
Published electronically: August 2, 2013
Additional Notes: The first author was supported by the National Security Agency
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.