  We give upper bounds for the number of irreducible representations of
dimension at most n for a compact semisimple Lie group. In particular, we prove
that there are at most n irreducible representations of dimension at most n for
a simple compact Lie group. We use this prove that the number of conjugacy
classes of maximal subgroups of a compact simple Lie group is O(r) where r is
the Lie rank. We also give a short proof that the dimensions of the weight
spaces of a maximal torus are small relative to the dimension of an irreducible
module.
