TY - JOUR

T1 - Normal elements in the Iwasawa algebras of Chevalley groups

AU - Han, Dong

AU - Ray, Jishnu

AU - Wei, Feng

N1 - Publisher Copyright:
© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2021/7

Y1 - 2021/7

N2 - 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 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 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 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(Zp) and discuss how to extend our current techniques and methods to the case of the pro-p Iwahori subgroups of GL n(Zp).

AB - 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 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 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 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(Zp) and discuss how to extend our current techniques and methods to the case of the pro-p Iwahori subgroups of GL n(Zp).

UR - http://www.scopus.com/inward/record.url?scp=85086567068&partnerID=8YFLogxK

U2 - 10.1007/s00229-020-01214-1

DO - 10.1007/s00229-020-01214-1

M3 - Article

AN - SCOPUS:85086567068

SN - 0025-2611

VL - 165

SP - 415

EP - 451

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

IS - 3-4

ER -