Presentation of the Iwasawa algebra of the first congruence kernel of a semi-simple, simply connected Chevalley group over Zp

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

“…Furthermore, we would like to point out that G(1) is a uniform pro-p group in the sense of [11]. A nice exposition can be found in Sections 2, 3 and 4 of the Arxiv version ( [20,Arxiv]) of [20]. Now, let g…”

“…In this section we are going to find out the lowest degree terms of all commutator relation between y α and y δ in Ω G (1) . The strategy is to use firstly Steinberg's Chevalley relation in G(1) following [27] (which gave us the explicit relations for the Iwasawa algebra in [20,Lemma 3.1]) and then determine the commutators of elements y α 's and y δ 's. Furthermore we determine the lowest degree terms of the commutators (cf.…”

“…For any odd prime p, Clozel in his paper [6] gave explicit presentations for the afore-mentioned two Iwasawa algebras 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 )). Ray [20][21][22][23] generalized Clozel's work to the following three cases: the first congruence kernel of a semi-simple, simply connected Chevalley group over Z p , general uniform pro-p groups, 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].…”