  Let G be a connected, adjoint, simple algebraic group over the complex
numbers with a maximal torus T and a Borel subgroup B containing T. The study
of zero weight spaces in irreducible representations of G has been a topic of
considerable interest; there are many works which study the zero weight space
as a representation space for the Weyl group. In this paper, we study the
variation on the dimension of the zero weight space as the irreducible
representation varies over the set of dominant integral weights for T which are
lattice points in a certain polyhedral cone. The theorem proved here asserts
that the zero weight spaces have dimensions which are piecewise polynomial
functions on the polyhedral cone of dominant integral weights.
