Coherence of Augmented Iwasawa Algebras

TL;DR

This paper characterizes when augmented Iwasawa algebras of p-adic Lie groups are coherent, showing coherence occurs only for specific low-rank or solvable cases, with implications for classical groups.

Contribution

It provides a complete classification of coherence for augmented Iwasawa algebras of p-adic Lie groups based on root system rank and solvability.

Findings

Coherence occurs only for rank one root systems.

GL(n) has coherent augmented Iwasawa algebra iff n ≤ 2.

Certain solvable p-adic Lie groups have coherent augmented Iwasawa algebras.

Abstract

The augmented Iwasawa algebra of a p-adic Lie group is a generalisation of the Iwasawa algebra of a compact p-adic Lie group. We prove that a split-semisimple group over a p-adic field has a coherent augmented Iwasawa algebra if and only if its root system is of rank one. We deduce that the general linear group of degree n has a coherent augmented Iwasawa algebra precisely when n is at most two. We also characterise when certain solvable p-adic Lie groups have a coherent augmented Iwasawa algebra.