  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.
