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

Abstract

We consider the Belkale-Kumar cup product ${\odot }_t$ on $H^{\ast }(G/P)$ for a generalized flag variety $G/P$ with parameter $t \in \C^m$, where $m=\dim (H^2(G/P))$. For each $t\in \C^m$, we define an associated parabolic subgroup $P_K\supset P$. We show that the ring $(H^{\ast }(G/P),{\odot }_t)$ contains a graded subalgebra $A$ isomorphic to $H^{\ast }(P_K/P)$ with the usual cup product, where $P_K$ is a parabolic subgroup associated to the parameter $t$. Further, we prove that $(H^{\ast }(G/P_K),{\odot }_0)$ is the quotient of the ring $(H^{\ast }(G/P),{\odot }_t)$ 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 in corpus (1)

• The Belkale-Kumar cup product and relative Lie algebra cohomology