Let $\Phi$ denote a root system and let $\mathfrak P$ denote the associated Kostant partition function. Thus $\mathfrak P(\lambda)$ is the number of ways of writing $\lambda$ as a sum of elements of $\Phi^+$. For example, if $\Phi=A_3$ then the longest root $\alpha_0$ can be written as $\alpha_0=(\alpha_1+\alpha_2+\alpha_3)$, $(\alpha_1)+(\alpha_2)+(\alpha_3)$, $(\alpha_1+\alpha_2)+\alpha_3$ or $\alpha_1 + (\alpha_2+\alpha_3)$. Thus $\mathfrak P(\alpha_0)=4$. There is a recursion $$\mathfrak P(\mu)=-\sum_{1\neq w\in W} (-1)^{l(w)}\mathfrak P(\mu+w(\rho)-\rho)$$ due to Kostant.
I can't find very many general results on $\mathfrak P$.
One knows that $\mathfrak P((p-1)\rho)$ is the dimension of the $0$ weight space of the Steinberg module (see Jantzen, RAGS, II.10.12). I conjecture that one also has $\mathfrak P(r\rho)$ is the dimension of the $0$ weight space in $L_{\mathbb C}(r\rho)$, where $L_{\mathbb C}(r\rho)$ is the irreducible representation of high weight $r\rho$ for a complex Lie algebra with root system $\Phi$ (which may be zero if $r$ is odd). A reference even for this basic statement would be good!
[RE-EDIT: This paragraph, including the conjecture, is completely wrong. See Chuck Hague's and Jim Humphreys' remarks below. Of course one must have $\dim H^0(\lambda)_0\leq \mathfrak P(\lambda)$ for all $\lambda$, so this gives a lower bound.]
I'd also like some data on asymptotics. I have used the Maple `Coxeter/Weyl' package to compute $\mathfrak P(2\rho)$ for $A_n$ up to $n=5$. The numbers are $1,3,15,219,7834\dots$ and correspond to sequence A007081 on the OEIS.
[EDIT: These calculated values are in fact the dimensions of the $0$ weight space of $H^0(2\rho)$. Since the conjecture that $\mathfrak P(2\rho)=\dim H^0(2\rho)_0$ is incorrect these do not coincide with the values of $\mathfrak P(2\rho)$ as claimed.]
I guess $\mathfrak P(2\rho) < n^n$. [EDIT: This looks very optimistic—suspect it's wrong for $n=4$.] Any ideas? What about $\mathfrak P (2r\rho)<(nr)^{nr}$ or something better? [EDIT: Similarly.] Lower bounds?
Best Answer
[EDIT] Concerning your specific question, my earlier answer was too offhand. After trying this with pencil and paper, I'm very doubtful that the multiplicity of the zero weight in the finite dimensional module with highest weight $r\rho$ for $r>0$ is always given by applying the partition function to $r\rho$. Sorry to have confused the issue here. Maybe I can identify a specific small counterexample, though a computer program makes such things easier.
In more detail, $\rho$ has two basic characterizations: as the sum of fundamental dominant weights (showing that it is dominant) and as the half-sum of all positive roots (showing at least that $2\rho$ lies in the root lattice, though $\rho$ itself might or might not). From representation theory it follows that the value of the partition function at $r\rho$ is the dimension of the zero weight space of the corresponding infinite dimensional Verma module.
Kostant's basic insight was that the finite dimensional simple quotient of any Verma module with dominant highest weight has formal character given by an alternating sum of naturally related Verma module characters (which later got encoded in the BGG resolution). So your underlying representation theory problem is to show that the alternating sum in this special case will not reduce the size of the zero weight space. But I don't see this following from Kostant's weight multiplicity formula.
In general, Kostant explained in the introduction to his 1959 paper here how his intuition developed toward the rigorous formula. Later work by Bernstein-Gelfand-Gelfand streamlined the derivation of this formula (and Cartier pointed out earlier the logical equivalence of the formula with Weyl's), but it may be useful to take a look back at the original paper. [Note that the symbol $g$ is used there for what later became standardized as $\rho$.]
Concerning asymptotics of the partition function, I'm not sure how much has been written down. But for example an appendix in Verma's 1966 Yale thesis makes some observations along this line; as I recall he developed a polynomial bound. Much later Lusztig (and then R.K. Brylinski) studied a $q$-version of the partition function as well. Such work is done in a purely theoretical mode, of course, but there may be other computational approaches of interest for your question.
[ADDED] The discussion has been getting more convoluted, starting with David's original attempt to quote Jantzen's II.10.12 (which turns out to be only marginally relevant to the question about Kostant's partition function). Now that I've gone back over that material in RAGS more carefully, I can see that it's one of those places where Jantzen's notation gets devious. In 10.11 he defines $\mathrm{St'}_r$ to be the module for $G_r T$ obtained from the usual Steinberg module (for $G$) of highest weight $(p^r-1)\rho$ by tensoring with that weight. So for example the zero weight of the Steinberg module becomes the $(p^r-1)\rho$ weight of the new module. This weight of course varies with increasing $r>0$, whereas Jantzen asks you to fix a weight $\lambda$ of the new module and then find large enough $r$, etc. As I say, it's devious.
Some of the comments indicate that the conjecture involving $2\rho$ and such will have counterexamples. But the basic question about asymptotic behavior of Kostant's function may be worth further study. As I mentioned above, this did come up in Verma's thesis where he proved existence of a polynomial bound when the partition function is expressed in terms of the $\ell$ coefficients of an element in the root lattice ($\ell$=rank). But his reason for doing all this got bypassed in later literature such as Dixmier's book.