Asymptotic Weyl Character Formula

Question

Let $G$ be a complex semi-simple group along with a chosen pair of opposite Borel subgroups (so we get all the root-theoretic data we need). Let $\lambda$ be a dominant weight, and let $V(\lambda)$ be the corresponding highest weight representation. I'm interested in a formula for the character of $V(n \lambda)$ as $n \rightarrow \infty$. Of course to have this limit converge, I need to normalize things. So I consider the character of $V(n\lambda)$ as a measure on the Lie algebra of the torus, which I rescale to live on precisely the convex hull of the weights appearing on $V(\lambda)$, and which I scale by a fixed power of $n$. Then I consider the weak limit of this measure. This measure is also the Duistermaat-Heckman measure of the flag variety corresponding to the line bundle $\mathcal{O}(\lambda)$ (perhaps I should restrict to $\lambda$ regular to be correct).

The Fourier transforms of these measure appearing in the limit are precisely the character in the usual sense (given by the Weyl character formula), so the Fourier transform of the limiting measure should be calculable. So I am asking for the Fourier transform of this measure.

In the case of $SL_2$, for $\lambda$ equal to the fundamental weight $\omega$, this is just the Duistermaat-Heckman measure of $\mathbb{P}^1$. This is just the Lebesgue measure supported on the interval $[-\omega,\omega]$, which clearly agrees with what we know about characters of $SL_2$ representations. And it's Fourier transform is (up to some constants) the sinc function $\frac{\sin(x)}{x}$, which can also be seen from the Weyl character formula since $\frac{\sin(x)}{x} = \frac{e^{ix} - e^{-ix}}{2ix}$ However, I do not know how to write down such a formula in other types. I expect that there should be some formula with roughly the form of the Weyl character formula. Does anyone know any such formula?

Answer 1

I personally consider a "character" to be function on the group, whereas you speak about it living on the convex hull of the weights, which is in the dual of the Lie algebra of the torus.

Anyway, the point is that you already understand Heckman's asymptotics of the weight multiplicity function on the weight lattice, as a measure on the vector space containing it, and you're asking for the corresponding limit of the Fourier transforms.

This is easy to get from the WCF directly: the relevant limit is $1 - \exp(-x) \sim x$. The limits of the Weyl character (thought of as a function on $T$) in its two forms $$ \sum_{w\in W} w \cdot \frac{t^{\lambda}}{\prod_{\beta \in \Delta_+}(1 - t^{-\beta})} = \frac{\sum_{w\in W} (-1)^w t^{w\cdot(\lambda+\rho)-\rho}}{\prod_{\beta\in\Delta_+}(1-t^{-\beta})}$$ are the functions on $\mathfrak t$ $$ \sum_{w\in W} w \cdot \frac{\exp(-\lambda)}{\prod_{\beta\in \Delta_+} \beta} = \frac{\sum_{w\in W} (-1)^w \exp(-w\cdot\lambda)}{\prod_{\beta\in \Delta_+} \beta}$$ The $\rho$-shift disappears entirely!

Instead of thinking of this as a limiting process, you could compare [Atiyah-Bott '65] localization in K-theory to [Atiyah-Bott '84] localization in cohomology, applied to the $\lambda$ Borel-Weil line bundle on the flag manifold or the Chern character thereof.

As for the $\sin(x)$ formula you want, that's special to the case $\lambda$ a multiple of $\rho$. Check out Why do flag manifolds, in the P(V_rho) embedding, look like products of P^1s?