Nonexistence of reflexive ideals in Iwasawa algebras of Chevalley type

TL;DR

This paper proves that for certain Chevalley type pro-p groups, their Iwasawa algebras lack nontrivial reflexive two-sided ideals, extending previous results from the A1 case to general root systems.

Contribution

The authors generalize the nonexistence of reflexive ideals in Iwasawa algebras from the A1 case to all Chevalley type root systems for p ≥ 5.

Findings

No nontrivial reflexive two-sided ideals in the specified Iwasawa algebras.

Extension of previous A1 case results to general Chevalley root systems.

Applicable for uniform pro-p groups associated with powerful Lie algebras.

Abstract

Let $\Phi$ be a root system and let $\Phi(\Zp)$ be the standard Chevalley $\Zp$-Lie algebra associated to $\Phi$. For any integer $t\geq 1$, let $G$ be the uniform pro-$p$ group corresponding to the powerful Lie algebra $p^t \Phi(\Zp)$ and suppose that $p\geq 5$. Then the Iwasawa algebra $\Omega_G$ has no nontrivial reflexive two-sided ideals. This was previously proved by the authors for the root system $A_1$.