Exponential Hilbert series and the Stirling numbers of the second kind (Q1709518)

Given a graded algebra \(A\), with homogeneous components \(A_n,\;n\in\mathbb{N}\), its exponential Hilbert series is defined to be the formal power series \[ E_A(q)=\sum_{n\geq 0}\dim (A_n)\frac{q^n}{n!}. \] Let \(G\) be a semisimple, simply connected linear algebraic group over \(\mathbb{C}\), and fix a choice of Borel subgroup \(B\) inside a parabolic subgroup \(P\). The main purpose of this manuscript is to present results on exponential Hilbert series of \(G\)-equivariant embeddings of the flag varieties \(X=G/P\). The author computes a nice closed form for the exponential generating function in terms of finitely many differential operators and the Stirling polynomials. Also, the author proves that the above series converges to a product of a rational polynomial and an exponential. Finally, he is able to show that by summing the constant term and the linear coefficient of the above rational polynomial, the dimension of the representation is resulted.