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

TL;DR

This paper investigates the geometric and combinatorial properties of simplicial complexes associated with certain Drinfeld modular forms, revealing their connectivity, symmetry, and structural features within the Bruhat-Tits building.

Contribution

It introduces and analyzes the simplicial complexes linked to coefficient forms and para-Eisenstein series, highlighting their topological and symmetry properties in higher rank Drinfeld modular forms.

Findings

Connected for rank > 2

Strongly equidimensional of codimension 1

Boundaryless and symmetric under Dynkin involution

Abstract

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 \(A_{r-1}\).