  We present a closed form for a multi-variate generating function for the
dimensions of the irreducible representations of a semisimple, simply connected
linear algebraic group over $\mathbb{C}$ whose highest weights lie in a
finitely generated lattice cone in the dominant chamber. This result
generalizes the formula for the Hilbert series of an equivariant embedding of a
homogeneous projective variety. As a special case, we show how the
multi-variate series can be used to compute the Hilbert series of the
determinantal varieties.
