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

TL;DR

This paper explores the Belkale-Kumar cup product on cohomology of flag varieties, revealing a subalgebra structure and quotient relations via Hochschild-Serre spectral sequence techniques.

Contribution

It establishes a connection between the Belkale-Kumar product and classical cohomology through spectral sequence methods, introducing a new perspective on the algebraic structure.

Findings

Identifies a graded subalgebra isomorphic to classical cohomology

Shows the quotient structure relating different Belkale-Kumar products

Uses Hochschild-Serre spectral sequence to prove structural results

Abstract

We consider the Belkale-Kumar cup product $\odot_t$ on $H^*(G/P)$ for a generalized flag variety $G/P$ with parameter $t \in \C^m$, 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.