  We study the ring of differential operators D(X) on the basic affine space
X=G/U of a complex semisimple group G with maximal unipotent subgroup U. One of
the main results shows that the cohomology group H^*(X,O_X) decomposes as a
finite direct sum of non-isomorphic simple X-modules, each of which is
isomorphic to a twist of O(X) by an automorphism of D(X).
  We also use D(X) to study the properties of D(Y) for highest weight varieties
Y. For example we prove under mild hypotheses that Y is D-simple in the sense
that O(Y) is a simple D(Y)-module and produce an irreducible G-module of
differential operators on Y of degree -1 and specified order.
