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

We consider the Belkale-Kumar cup product $\odot_t$ on $H^*(G/P)$ 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 $(H^*(G/P), \odot_t)$ 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.

No reviews yet.

No comments yet.