Summary: We consider the dimensions of irreducible representations whose highest weights lie on a given lattice cone. We present a simple closed form for the multivariate formal power series which generates these dimensions. This closed form is a direct generalization of a formula for the Hilbert series of an equivariant embedding of a homogeneous variety, obtained by Gross and Wallach. We use this generalization to study multivariate and single variable Hilbert series for many varieties of interest in representation theory, including the Kostant cone and various determinantal varieties. We show how the classical Hilbert series of determinantal varieties may be obtained from the multivariate series by a simple recursive relationship. We also prove some combinatorial properties of the multivariate series.