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Complexity of homogeneous spaces and growth of multiplicities

Dmitri A. Timashev
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Abstract

The complexity of a homogeneous space $G/H$ under a reductive group $G$ is by definition the codimension of generic orbits in $G/H$ of a Borel subgroup $B\subseteq G$. We give a representation-theoretic interpretation of this number as the exponent of growth for multiplicities of simple $G$-modules in the spaces of sections of line bundles on $G/H$. For this, we show that these multiplicities are bounded from above by the dimensions of certain Demazure modules. This estimate for multiplicities is uniform, i.e., it depends not on $G/H$, but only on its complexity.

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Dmitri A. Timashev. 2003-09-09. Complexity of homogeneous spaces and growth of multiplicities. https://arxiv.org/abs/math/0305416

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