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

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