Functorial relationships between quantum cohomology of complete and partial flag varieties

TL;DR

This paper introduces a natural filtration on the quantum cohomology of G/B that respects its structure and relates it to the cohomology of partial flag varieties, connecting quantum and classical cases.

Contribution

It constructs a filtration on quantum cohomology of G/B with an associated graded algebra isomorphic to a tensor product involving quantum cohomology of G/P and a graded algebra of QH(P/B).

Findings

The filtration respects the quantum product structure.

The associated graded algebra relates quantum cohomologies of G/B, G/P, and P/B.

Specializes to classical cohomology via the Leray spectral sequence.

Abstract

We give a natural filtration F on quantum cohomology QH(G/B) of G/B, which respects the quantum product structure. Its associated graded algebra is isomorphic to the tensor product of QH(G/P) and a corresponding graded algebra of QH(P/B) after localization. When the quantum parameter goes to zero, this specializes to the filtration on the classical cohomology H(G/B) from the Leray spectral sequence associated to the fibration P/B --> G/B --> G/P.