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Transactions of the American Mathematical Society

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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
MathSciNet review: 3091267
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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 [Enhancements On Off] (What's this?)

  • [BK] Prakash Belkale and Shrawan Kumar, Eigenvalue problem and a new product in cohomology of flag varieties, Invent. Math. 166 (2006), no. 1, 185-228. MR 2242637 (2007k:14097)
  • [BW] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, second ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403 (2000j:22015)
  • [CE] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85-124. MR 0024908 (9:567a)
  • [EG] S. Evens and W. Graham, with an appendix jointly written with E. Richmond, The Belkale-Kumar cup product and relative Lie algebra cohomology, arXiv:1104.1415.
  • [HS] G. Hochschild and J.P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57, (1953), 591-603. MR 0054581 (14:943c)
  • [Kos1] Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329-387. MR 0142696 (26:265)
  • [Kos2] -, Lie algebra cohomology and generalized Schubert cells, Ann. of Math. (2) 77 (1963), 72-144. MR 0142697 (26:266)
  • [Lan] Serge Lang, Algebra, Revised third edition, Springer, 2002. MR 1878556 (2003e:00003)
  • [Wei] Charles Weibel, An introduction to homological algebra, Cambridge University Press, Cambridge, 1994. MR 1269324 (95f:18001)

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Additional Information

Sam Evens
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

William Graham
Affiliation: Department of Mathematics, University of Georgia, Boyd Graduate Studies Research Center, Athens, Georgia 30602

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.

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