Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over $\mathbb{Z}_p$

Abstract

It is a general principle that objects coming from semi-simple, simply connected (split) groups have explicit presentations like Serre's presentation of semi-simple algebras and Steinberg's presentation of Chevalley groups. In this paper we give an explicit presentation (by generators and relations) of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over Zp, extending the proof given by Clozel for the group Γ 1 (SL 2 (Zp)), the first congruence kernel of SL 2 (Zp) for primes p > 2 .

Normal elements of completed group algebras over ${\rm SL}_3(\mathbb{Z}_p) $

“…Then G(k) is called the k-th congruence kernel of G(Z p ) which satisfies a descending filtration G(1) ⊇ G(2) ⊇ G(3) ⊇ • • • . Ray [11] give an explicit presentation (by generators and relations) of the completed group algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over Z p , extending the proof given by Clozel for the group Γ 1 (SL 2 (Z p )), the first congruence kernel of SL 2 (Z p ) for primes p > 2. This immediately gives rise to the following question.…”

“…In this section, we will present several potential topics for future further research. Motivated by our current work, Clozel's systematic work [1,2,3] and Ray's papers [11,12] , it is natural to propose several questions in this line.…”

“…Indeed, for any odd prime p, Clozel in his paper [1] gives explicit presentations for the afore-mentioned two completed group algebra over the first congruence subgroup of SL 2 (Z p ), which is Γ 1 (SL 2 (Z p )) = ker(SL 2 (Z p ) −→ SL 2 (F p )). More recently, Ray [11,12] extended Clozel's work to the cases of semi-simple, simply connected Chevalley groups over Z p and pro-p Iwahori subgroups of GL n (Z p ).…”

Explicit presentation of an Iwasawa algebra: The case of pro-p Iwahori subgroup of SL n (ℤ p )

“…Schneider and Teitelbaum use the Iwasawa algebra to study the category of Q p -Banach representations of compact p-adic Lie groups [ST02]. In [Ray16], we found an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of a semi-simple, simply connected Chevalley group over Z p . In this section, our goal is to extend the method to give an explicit presentation of the Iwasawa algebra for the pro-p Iwahori subgroup of SL(n, Z p ) generalizing the work of Clozel for n = 2 [Clo17].…”