Explicit presentation of an Iwasawa algebra: The case of pro-<i>p</i> Iwahori subgroup of SL<sub> <i>n</i> </sub>(ℤ<sub> <i>p</i> </sub>)
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Abstract
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. We previously determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> </m:math> {\mathbb{Z}_{p}} which were uniform pro- p groups in the sense of Dixon, du Sautoy, Mann and Segal. In this paper, for prime <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>></m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:mrow> </m:math> {p>n+1} , we determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro- p Iwahori subgroup of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>GL</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>p</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\mathrm{GL}_{n}(\mathbb{Z}_{p})} which is not, in general, a uniform pro- p group.
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