The Borel-Moore homology of an arithmetic quotient of the Bruhat-Tits building of PGL of a non-archimedean local field in positive characteristic and modular symbols

TL;DR

This paper investigates the homology and Borel-Moore homology of arithmetic quotients of Bruhat-Tits buildings for PGL over positive characteristic fields, introducing modular symbols and linking homology limits to automorphic and cusp forms.

Contribution

It defines modular symbols in this context, relates homology to automorphic forms, and describes the structure of the Borel-Moore homology limit as an induced representation.

Findings

Image of homology to Borel-Moore homology map is generated by modular symbols.

Limit of homology corresponds to automorphic forms with specific invariance properties.

Limit of Borel-Moore homology contains an irreducible subquotient as an induced representation.

Abstract

We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, the limit of the Borel-Moore homology as the arithmetic group becomes small is isomorphic to the space of $\mathbb{Q}$-valued automorphic forms that satisfy certain conditions at a distinguished (fixed) place (namely that it is fixed by the Iwahori subgroup and the center at the place). We show that the limit of the homology with $\mathbb{C}$-coefficients is identified with the subspace consisting of cusp forms. We also describe an irreducible subquotient of the limit of Borel-Moore homology as an induced representation in a precise manner and give a multiplicity one type result.