Geometric Invariant Theory and Generalized Eigenvalue Problem

Nicolas Ressayre (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Let $H$ be a connected reductive subgroup of a complex connected reductive group $G$. Fix maximal tori and Borel subgroups of $H$ and $G$. Consider the pairs $(V,V')$ of irreducible representations of $H$ and $G$ such that $V$ is a submodule of $V'$. We are interested in the cone $LR(G,H)$ generated by the pairs of dominant weights of such a pair of representations. Our main result gives a minimal set of inequalities describing $LR(G,H)$ as a part of the dominant chamber. In way, we obtain results about the faces of the Dolgachev-Hu's $G$-ample cone and variations of this cone.

Mots clés: Geometric Invariant Theory; Eigenvalue Problem; Product of Belkale-Kumar; Moment polytope

Domaines: Géométrie algébrique [math.AG]; Théorie des représentations [math.RT]

Nicolas Ressayre. Geometric Invariant Theory and Generalized Eigenvalue Problem. Inventiones Mathematicae, 2010, 180 (2), pp.389-441. ⟨10.1007/s00222-010-0233-3⟩. ⟨hal-00142075v3⟩