# Non-Abelian Duistermaat-Heckman Measure (not just a reference request)

Let $V$ be a representation of some torus $T$. It is then well-known that the Duistermaat-Heckman measure for $P(V)$ is the weak limit of the properly rescaled distribution of multiplicities of weights in $\mathrm{Sym}^n(V)$.

I've seen many allusions to the fact that the analogous statement is true for general compact Lie groups $K$ (i.e., one pushes further forward to a positive Weyl chamber and compares with the irrep distribution), e.g., in Allen Knutson's reply to my last question and in the appendix of Guillemin-Prato's 1990 paper, but could not find an explicit statement of this in the literature.

Do you know whether this statement is true at all, and do you maybe even have a reference?

The more general statement you want is, let $X$ be projective with a $K$-equivariant ample line bundle ${\mathcal O}(1)$. For each $n$, let $\mu_n$ be a measure on ${\mathfrak t}^*_+$, $$\mu_n := \sum_{\lambda \in {\mathfrak t}^*_+} \frac{\dim Hom_K(V_{n\lambda}, \Gamma(X;{\mathcal O}(n))}{n^{\dim X}} \delta_{\lambda}.$$ (Note that $V_{n\lambda}$ only means anything if $n\lambda$ is integral.) Then $\lim_{n\to \infty} \mu_n$ is the nonabelian DH measure for $K$ acting on $X$.
One approach to proving this to degenerate $X$ to $X' := (X//N \times G//N) // T$, where $G = K^{\mathbb C}$ and $N$ is a maximal unipotent group. Then the nonabelian DH measure of $X$ is the nonabelian DH measure of $X'$ is the {\em abelian} DH measure of $X//N$. (Here $G//N$ goes by the name "Gel$'$fand variety", $X//N$ by "imploded cross-section", and $X'$ by "Vinberg asymptotic cone".) At that point the theorem has to be checked once and for all for $G//N$.