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 \(A_{ r-1}\).
Categories: math.NT