Complexity of homogeneous spaces and growth of multiplicities

The complexity of a homogeneous space $G\mathrm{/}H$ under a reductive group $G$ is by definition the codimension of generic orbits in $G\mathrm{/}H$ 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 $G\mathrm{/}H$. 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 $G\mathrm{/}H$, but only on its complexity.