Invertible analytic functions on Drinfeld symmetric spaces and universal extensions of Steinberg representations

TL;DR

This paper establishes a new duality between invertible analytic functions on Drinfeld spaces and universal extensions of Steinberg representations, with applications to theta cocycles and Galois invariants.

Contribution

It demonstrates that invertible functions correspond to universal extensions of Steinberg representations, extending Gekeler's duality result.

Findings

Invertible functions are dual to universal extensions of Steinberg representations.

Lifting obstructions of theta cocycles relate to $\\mathcal{L}$-invariants of Galois representations.

Method applies to both Hilbert modular forms and definite unitary groups.

Abstract

Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld's upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon--Vonk can be computed in terms of $\mathcal{L}$-invariants of the associated Galois representation. The same argument applies to theta cocycles for definite unitary groups.