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This is a refinement of this old question of mine. In order to find an answer, I've been working my way through q-alg/9511026, which contains all the information I need.

In this paper, the authors associate to an affine Lie algebra $\mathfrak g$ the so-called orbit algebra $\check g$, whose highest weights seem to be in one-to-one correspondence with the orbits of $\mathfrak g$-weights under the action of some element of $\mathcal O(\mathfrak g)$, the outer automorphism group of $\mathfrak g$. I am having troubles to understand the correspondence precisely, though, and it would be great if someone could spell out the details for me.

Let me quote the relevant definitions. Given $\omega\in\mathcal O(\mathfrak g)$ the generator under consideration, and $H^i$ the generators of a Cartan sub-algebra of $\mathfrak g$, the authors define $$ P_\omega\colon\ \sum_i v_i H^i\mapsto \sum_{[i]} N_i v_i \check H^{[i]}\tag{4.3} $$ where $\check H^{[i]}$ generate a Cartan sub-algebra of $\check{\mathfrak g}$, and where $N_i$ is the length of the orbit of $i$ under $\omega$.

I'm not entirely sure I understand the expression above (I have a vague idea of what it's meant to represent, but the notation isn't helping). What I really need to understand, though, is the inverse operation, $(P^\star_\omega)^{-1}$, which maps an orbit of $\mathfrak g$-weights into a weight of $\check{\mathfrak g}$. Given a representative of such an orbit, say, $$ \lambda=\sum_{i=1}^{\operatorname{rank} \mathfrak g} \lambda_i\omega^i \in \Lambda_w(\mathfrak g) $$ we map it into a weight of $\check{\mathfrak g}$: $$ (P_\omega^\star)^{-1}(\lambda)=\sum_{i=1}^{\operatorname{rank} \check{\mathfrak g}} \check \lambda_i\check \omega^i\in \Lambda_w(\check{\mathfrak g}) $$ where $\omega^i,\check \omega^i$ are the fundamental weights of $\mathfrak g$ and $\check{\mathfrak g}$, respectively. What is this map, in explicit terms? What is $\{\check \lambda_i\}$ for a given $\{\lambda_i\}$?

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According to arXiv:hep-th/9601078 (a more recent paper by the same authors), the orbit algebra of $G=A_n$ is $\check G=A_n$, both at the same level (assuming we consider the full outer automorphism group $\mathcal O(\mathfrak g)=\mathbb Z_{n+1}$; were we to consider a sub-group $\mathbb Z_\ell$, the orbit algebra would be $\check G=A_{\ell-1}$, cf. §6.3, p.39). Surely I must be missing something, because the orbits of $A_n$ cannot be in one-to-one correspondence with the highest weights of $A_n$ (inasmuch as, in general, these orbits have more that one element, and therefore there are less orbits than weights themselves). It would be really helpful if someone could clear things up for me. Thanks!

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