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Multiplicities in the Branching Rules and the Complexity of Homogeneous Spaces

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 λ of G, let Vλ be a simple G-module with highest weight λ, dλ = dimVλ, and denote by k[G/H](λ) the isotypic λ-component in k[G/H]. For G/H quasi-affine, we show that the ratio dim k[G/H](λ)/dλ grows no faster than a polynomial in ‖λ‖ 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/BH , where BH is a Borel subgroup of H. We classify all affine homogeneous spaces G/H such that G is simple and the complexity of G/BH is at most 1. Some explicit descriptions of branching rules are also given. 2000 Math. Subj. Class. 14L30, 20G05.