Presentation of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$

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

“…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.…”

“…Are there any non-trivial normal elements in Ω G(1) ? For a prime p > n + 1, Ray [12] determine explicitly the presentation in the form of generators and relations of the completed group algebras Λ G and Ω G over the pro-p Iwahori subgroup G of GL n (Z p ). Let G be the pro-p Iwahori subgroup of GL n (Z p ), i.e.…”

“…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 ).…”

Globally analytic principal series representation and Langlands base change

“…(Cf. [12,III,3.3.2] for the rigid-analyticity and see theorem 2.2.1 and remark 2.2.2 of [14] for the order of the product i.e. an ordered Lazard basis of G, although in [14] we have taken G to be upper unipotent matrices modulo p but this does not matter).…”

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

“…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 ).…”