Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants

Teleman, Constantin; Woodward, Christopher

Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 713-748.

L'ensemble des classes de conjugaison apparaissant dans un produit de classes de conjugaison d'un groupe de Lie 1-connexe compact peut être identifié avec un polytope convexe dans une chambre pour le groupe affine de Weyl. Nous démontrons que les inégalités linéaires définissant ce polytope correspondent aux invariants de Gromov- Witten pour les variétés de drapeaux généralisées. Ceci généralise les résultats de Agnihotri, du deuxième auteur et de Belkale sur les valeurs propres d'un produit de matrices unitaires et la cohomologie quantique des grassmanniennes.

The set of conjugacy classes appearing in a product of conjugacy classes in a compact, $1$-connected Lie group $K$ can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety $G/P$, where $G$ is the complexification of $K$ and $P$ is a maximal parabolic subgroup. This generalizes the results for $SU\left(n\right)$ of Agnihotri and the second author and Belkale on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians.

Classification : 14L30, 14N35, 05E99
Keywords: conjugacy classes, parabolic bundles, quantum cohomology
Mot clés : classes de conjugaison, fibrés paraboliques, cohomologie quantique

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