Abstract:
Let $H$ be an algebraic subgroup of a connected algebraic group $G$ defined over an algebraically closed field $k$ of characteristic zero. For a dominant weight $\lambda$ of $G$, let $V_\lambda$ be a simple $G$-module with highest weight $\lambda$, $d_\lambda=\dim V_\lambda$, and denote by $k[G/H]_{(\lambda)}$ the isotypic $\lambda$-component in $k[G/H]$. For $G/H$ quasi-affine, we show that the ratio $k[G/H]_{(\lambda)}/d_\lambda$ grows no faster than a polynomial in $\|\lambda\|$ whose degree is the complexity of the homogeneous space $G/H$. If $H$ is reductive and connected, this yields an estimate of branching coefficients for the pair $(G,H)$ in terms of the complexity of $G/B_H$, where $B_H$ is a Borel subgroup of $H$. We classify all affine homogeneous spaces $G/H$ such that $G$ is simple and the comlexity of $G/B_H$ is at most 1. Some explicit descriptions of branching rules are also given.
Key words and phrases:Complexity of a homogeneous space, branching rule, Grosshans subgroup, algebra of covariants.
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