Exponential Hilbert series and the Stirling numbers of the second kind

Abstract

We consider the exponential generating function whose coefficients encode the dimensions of irreducible highest weight representations which lie on a given ray in the dominant chamber of the weight lattice. This formal power series can be considered as an exponential version of the Hilbert series of a flag variety. In this context, we compute a simple closed form for the exponential generating function in terms of finitely many differential operators and the Stirling polynomials. We prove that this series converges to a product of a rational polynomial and an exponential, and that, by summing the constant term and linear coefficient of this polynomial, we recover the dimension of the representation.