Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces

Pierre-Emmanuel Chaput (1) , Laurent Manivel (2) , Nicolas Perrin (3)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 IF - Institut Fourier
3 IMJ - Institut de Mathématiques de Jussieu

Let $X$ be a rational homogeneous space and let $QH^*(X)_{loc}^\times$ be the group of invertible elements in the small quantum cohomology ring of $X$ localised in the quantum parameters. We generalise results of arXiv:math/0609796 and realise explicitly the map $\pi_1({\rm Aut}(X))\to QH^*(X)_{loc}^\times$ described in arXiv:dg-ga/9511011. We even prove that this map is an embedding and realise it in the equivariant quantum cohomology ring $QH^*_T(X)_{loc}^\times$. We give explicit formulas for the product by these elements. The proof relies on a generalisation, to a quotient of the equivariant homology ring of the affine Grassmannian, of a formula proved by Peter Magyar arXiv:0705.3826. It also uses Peterson's unpublished result -- recently proved by Lam and Shimozono in arXiv:0705.1386 -- on the comparison between the equivariant homology ring of the affine Grassmannian and the equivariant quantum cohomology ring.

Mots clés: quantum cohomology; homogeneous space; affine Weyl group

Domaines: Géométrie algébrique [math.AG]

Pierre-Emmanuel Chaput, Laurent Manivel, Nicolas Perrin. Affine symmetries of the equivariant quantum cohomology ring of rational homogeneous spaces. Mathematical Research Letters, 2009, 16, pp.7-21. ⟨hal-00330621⟩