The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product
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- by Sam Evens and William Graham PDF
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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.References
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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
- MR Author ID: 321363
- Email: wag@math.uga.edu
- Received by editor(s): January 1, 2012
- Published electronically: August 2, 2013
- Additional Notes: The first author was supported by the National Security Agency
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 5833-5857
- MSC (2010): Primary 17B56, 14M15, 20G05
- DOI: https://doi.org/10.1090/S0002-9947-2013-05792-3
- MathSciNet review: 3091267