Complexity of homogeneous spaces and growth of multiplicities

Details Complexity of homogeneous spaces and growth of multiplicities Complexity of homogeneous spaces and growth of multiplicities

2003-05-29

arxiv.org/abs/math/0305416v2

Mathematics

Algebraic Geometry

AmSLaTeX, 9 pages, 15 references

Scientific paper

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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