  We describe a new formula for weight multiplicities and characters of
semisimple Lie algebras. Our formula expresses these weight multiplicities as
sums of positive rational numbers. In fact, the formula works more generally
for the Jacobi polynomials of Heckman and Opdam, which are a one parameter
deformation of these characters. As a consequence we prove that after suitable
normalization, the coefficients of these polynomials are themselves polynomials
in the parameter with positive integral coefficients. This generalizes our
earlier joint work with F. Knop on Jack polynomials.
