The dualizing module and top-dimensional cohomology group of $\text{GL}_n(\mathcal{O})$

TL;DR

This paper investigates the dualizing module of the general linear group over a number ring, revealing it can differ from the Steinberg module and deriving cohomology results in the virtual cohomological dimension.

Contribution

It identifies conditions under which the dualizing module of isplaystyleGL_n(\u2115) varies from the Steinberg module, extending the understanding of its cohomological properties.

Findings

Dualizing module sometimes equals the Steinberg module.

Introduces a variant dualizing module considering orientation.

Provides vanishing and nonvanishing cohomology theorems.

Abstract

For a number ring $\mathcal{O}$, Borel and Serre proved that $\text{SL}_n(\mathcal{O})$ is a virtual duality group whose dualizing module is the Steinberg module. They also proved that $\text{GL}_n(\mathcal{O})$ is a virtual duality group. In contrast to $\text{SL}_n(\mathcal{O})$, we prove that the dualizing module of $\text{GL}_n(\mathcal{O})$ is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of $\text{GL}_n(\mathcal{O})$ in its virtual cohomological dimension.