Normal elements in the Iwasawa algebras of Chevalley groups

For a prime p> 2 , let G be a semi-simple, simply connected, split Chevalley group over Z_p, G(1) be the first congruence kernel of G and Ω _G(1) be the mod-p Iwasawa algebra defined over the finite field F_p. Ardakov et al. (Adv Math 218: 865–901, 2008) 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 A_n), 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 et al. 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. We also propose some potential topics regarding to the normal elements and ideals in the Iwasawa algebras of the pro-p Iwahori subgroups of general linear group GL _n(Z_p) and discuss how to extend our current techniques and methods to the case of the pro-p Iwahori subgroups of GL _n(Z_p).