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

TL;DR

This paper provides an explicit presentation of the Iwasawa algebra for the pro-$p$ Iwahori subgroup of $GL_n(Z_p)$, extending previous work to a broader class of $p$-adic Lie groups for primes greater than $n+1$.

Contribution

It offers a new explicit presentation of the Iwasawa algebra for the pro-$p$ Iwahori subgroup of $GL_n(Z_p)$, generalizing earlier results to larger primes.

Findings

Explicit generators and relations for the Iwasawa algebra are determined.

The result applies for primes $p > n+1$, broadening previous cases.

The presentation aids in understanding the algebraic structure of these groups.

Abstract

Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.