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

Abstract

The coefficient forms ℓka and the para-Eisenstein series αk are simplicial Drinfeld modular forms. We study the attached simplicial complexes BTr(ℓka) and BTr(αk), which are full subcomplexes of the Bruhat-Tits building BTr of PGL(r,K∞). They are connected (if the rank r is larger than 2), strongly equidimensional of codimension 1 in BTr, boundaryless, and satisfy a symmetry property under the non-trivial involution of the Dynkin diagram Ar−1.