On Drinfeld modular forms of higher rank VI: The simplicial complex associated with a coefficient form

The coefficient forms ${}_{a} \ell_{k}$ and the para-Eisenstein series $\alpha_{k}$ are simplicial Drinfeld modular forms. We study the attached simplicial complexes $\mathcal{BT}^{r}( {}_{a} \ell_{k})$ and $\mathcal{BT}^{r}(\alpha_{k})$, which are full subcomplexes of the Bruhat-Tits building $\mathcal{BT}^{r}$ of $\mathrm{PGL}(r, K_{\infty})$. They are connected (if the rank $r$ is larger than 2), strongly equidimensional of codimension 1 in $\mathcal{BT}^{r}$, boundaryless, and satisfy a symmetry property under the non-trivial involution of the Dynkin diagram $A_{r-1}$.