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 semi-simple group $G$. We are interested in the set of pairs $(\mu,\nu)$ of dominant characters for $G$ and $H$ such that $V_\mu \otimes V_\nu$ contains nonzero $H$-invariant vectors. This set of pairs $(\mu,\nu)$ generates a convex cone $C$ in a finite dimensional vector space. Using methods of variation of quotient in Geometric Invariant Theory, we obtain a list of linear inequalities which characterize $C$. This list is a generalization of the list that Belkale and Kumar obtained in the case when $G=H^s$. Moreover, we prove that this list in no far to be minimal (and really minimal in the case when $G=H^s$). We also give a description of some lower faces of $C$; if $G=H^s$ these description gives an application of the Belkale-Kumar product on the cohomology group of all the projective $G$-homogeneous spaces. Some of the results are more general than in the abstract and are obtained in the general context of Geometric Invariant Theory.

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. 2007. ⟨hal-00142075v1⟩