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

TL;DR

This paper provides an explicit presentation of the Iwasawa algebra for the first congruence kernel of a semi-simple, simply connected Chevalley group over _p, extending known results for specific groups.

Contribution

It offers a new explicit generators-and-relations presentation of the Iwasawa algebra for these groups, generalizing previous work on _2(_p).

Findings

Explicit presentation of Iwasawa algebra for the first congruence kernel.

Extension of Clozel's proof to broader class of Chevalley groups.

Generalization from _2(_p) to all semi-simple, simply connected Chevalley groups.

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 $\mathbb{Z}_p$, extending the proof given by Clozel for the group $\Gamma_1(SL_2(\mathbb{Z}_p))$, the first congruence kernel of $SL_2(\mathbb{Z}_p)$ for primes $p>2$.