Geometric Invariant Theory and Generalized Eigenvalue Problem

Details Geometric Invariant Theory and Generalized Eigenvalue Problem Geometric Invariant Theory and Generalized Eigenvalue Problem

2007-04-17

arxiv.org/abs/0704.2127v3

Inventiones mathematicae 180, 2 (2010) 389-441

Mathematics

Algebraic Geometry

comments welcome

Scientific paper

10.1007/s00222-010-0233-3

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.

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