Dimension of zero weight space: An algebro-geometric approach

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 highest weight of the irreducible representation varies over the set of dominant integral weights of 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 quasi-polynomial functions on the polyhedral cone of dominant integral weights.The main tool we use are the Geometric Invariant Theory and the Riemann–Roch theorem.