Let $R$ be a root system, $R^+$ a choice of positive roots, $W$ the Weyl group, $\Lambda$ its weight lattice, $\Lambda^+$ the cone of dominant weights, and $\rho = \frac{1}{2} \sum_{\alpha \in R^+} \alpha$ the half sum of positive roots. Define the algebra of exponential polynomials $E[\Lambda] = \text{span}\{e^{\lambda}| \lambda \in \Lambda \}$ and $E[\Lambda]_W$ the $W$ invariant subspace of $E[\Lambda]$.
For $k \geq 0$ (not necessarily an integer), define the Weyl-Denominator with multiplicity $k$ as $$\Delta^k = \prod_{\alpha \in R^+} (e^{\alpha/2} - e^{-\alpha/2} )^{k} $$ and define the scalar-product on $E[\Lambda]$ as $\langle f, g \rangle_{k}$ = the constant term of $\Delta^k f g$. Finally, for each $\lambda \in \Lambda^+$, define the symmetric monomials $$M_{\lambda} = \sum_{w \in W} e^{w(\lambda)} $$ It can be shown that for each $\lambda \in \Lambda^+$, there exists a unique polynomial $S_k(\lambda)$ such that
$1.)$ $S_k(\lambda) = M_{\lambda} + \sum_{\mu < \lambda} c_{\mu}M_{\mu} $
$2.)$ $\langle S_{k}(\lambda), M_{\mu} \rangle_{k} = 0$ for all $\mu < \lambda$
Moreover, $\{S_k(\lambda)\}_{\lambda \in \Lambda^+}$ is an orthogonal basis for $E[\Lambda]_W$. Moreover, if we replace $M_{\lambda}$ in the conditions above with $e^{\lambda}$, then we get a collection $\{E_k{(\lambda)}\}_{\lambda \in \Lambda}$ which turns out to be an orthogonal basis for $E[\Lambda]$. I believe these exponential polynomials are referred to as symmetric Jacobi Polynomials of Heckman and Opdam (or non-symmetric Jacobi Polynomials for $\{E_{k}(\lambda)\}_{\lambda \in \Lambda}$.). Please correct me if I am wrong about the naming.
The Weyl Character Formula states that if $k=1$ and $S_1(\lambda) = S_{\lambda}$, then
$$\Delta \cdot S_{\lambda} = \sum_{w \in W} \text{sgn}(w) e^{w(\mu)+\rho} $$
There are a lot of interpretations of what this means, but one thing being said here is that regardless how many exponential terms make up $S_{\lambda}$, the seemingly larger exponential sum $\Delta S_{\lambda}$ is, surprisingly, a small sum of $|W|$ exponentials with pretty simple coefficients too.
Now I am very sure I've seen an extension of this result to $S_{k}(\lambda)$ for any positive integer $k$ but I cannot for the life of me find it again. So my questions are:
1.) Is there a result stating something like $\Delta^k S_k(\lambda)$ is some $W$-orbit sum of a small number exponentials? What is the statement formally or where can I find it?
2.) Are there extensions of the Weyl Character Formula to the non-symmetric case $E_k(\lambda)$?
