Non-existence of non-trivial normal elements in the Iwasawa Algebra of Chevalley groups

Abstract

For a prime p > 2, let G be a semi-simple, simply connected, split Chevalley group over Zp, G(1) be the first congruence kernel of G and Ω G(1) be the mod-p Iwasawa algebra defined over the finite field Fp. Ardakov, Wei, Zhang [2] have shown that if p is a "nice prime " (p ≥ 5 and p ∤ n + 1 if the Lie algebra of G( 1) is of type An), then every non-zero normal element in Ω G(1) is a unit. Furthermore, they conjecture in their paper that their nice prime condition is superfluous. The main goal of this article is to provide an entirely new proof of Ardakov, Wei and Zhang's result using explicit presentation of Iwasawa algebra developed by the second author of this article and thus eliminating the nice prime condition, therefore proving their conjecture. Contents1. Introduction 1 2. Basic Setup on Lazard Ordered Basis 3 3. Calculation of the Lowest Degree Terms of Commutators 4 4. Partial Differential Equations 7 5. Main Result and Its Proof 9 6. Proof of Claim 5.3 13 6.1. Applications to center 18 6.2. Future questions 19 References 20