Isometry type of alcoves in affine Coxeter complexes

Question

Let $W$ be an irreducible affine Coxeter group (say of type $\widetilde{X}_n$), and let $\Sigma$ be the associated Coxeter complex. Thus, $\Sigma$ is an $n$-dimensional Euclidean space tesselated by isometric copies of a given simplex $A$ (namely, by the alcoves of $\Sigma$). I would like to know what the isometry type of $A$ is, depending on $\widetilde{X}_n$ (at least for the classical types $\widetilde{A}_n$, $\widetilde{B}_n$, $\widetilde{C}_n$ and $\widetilde{D}_n$).

More specifically, if $1$ is the length of the shortest edge of $A$, what are the other possible lengths for the edges of $A$?

(For instance, in type $\widetilde{A}_2$, all edges have length $1$).

Answer 1

Go to Bourbaki, Groupes et Algebres de Lie, Ch.4-6 where the fundamental weights $\varpi_1, \ldots , \varpi_n$ of $X_n$ are listed. The vertices of the simplex are $$0 \ \mbox{ and } \ x_k \varpi_k, \ k=1,\ldots,n$$ where $x_k>0$ is such that $x_k \varpi_k$ lies on the fixed hyperplane of the last Coxeter generator of the affine Weyl group. This hyperplane is given by the equation $$ \langle z , \alpha_0^{\vee}\rangle =1 $$ where $\alpha_0^\vee$ is highest coroot. The highest roots are also listed in the same book. Don't forget to swap $B_n \leftrightarrow C_n$, when looking for the highest coroot, because the coroot system is Langlands dual to the root system.

No, I don't know where this calculation has been carried out. Please, compute the length of all the edges of this simplex yourself, and put it here for the benefit of the whole humankind:-))