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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?

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?

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, each applied to the $\lambda$ Borel-Weil line bundle on the flag manifold.

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?

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?

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!

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?

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, each applied to the $\lambda$ Borel-Weil line bundle on the flag manifold.

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?