Arithmetic quotients of the Bruhat-Tits building for projective general linear group in positive characteristic

Abstract

Let $d \ge 1$. We study a subspace of the space of automorphic forms of $\mathrm{GL}_d$ over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place $\infty$ of $F$, and we consider the subspace $\mathcal{A}_{\mathrm{St}}$ consisting of automorphic forms such that the local component at $\infty$ of the associated automorphic representation is the Steinberg representation (to be made precise in the text). We have two results. One theorem (Theorem 16) describes the constituents of $\mathcal{A}_{\mathrm{St}}$ as automorphic representation and gives a multiplicity one type statement. For the other theorem (Theorem 12), we construct, using the geometry of the Bruhat-Tits building, an analogue of modular symbols in $\mathcal{A}_{\mathrm{St}}$ integrally (that is, in the space of $\mathbb{Z}$-valued automorphic forms). We show that the quotient is finite and give a bound on the exponent of this quotient.