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Globally maximal arithmetic groups

Benedict Gross·Gabriele Nebe
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Abstract

We begin with a review of the structure of simple, simply-connected complex Lie groups and their Lie algebras, describe the Chevalley lattice and the associated split group over the integers. This gives us a hyperspecial maximal compact subgroup of the p-adic Lie group and we describe the other maximal parahoric subgroups and their Lie algebras starting from the hyperspecial one. We then consider the Killing form on the Chevalley lattice and show that it is divisible by 2 times the dual Coxeter number. The same holds for the Lie algebras of the other maximal parahorics. We compute the discriminants of the resulting scaled forms. Finally we consider Jordan subgroups of the exceptional groups. We show that these Jordan subgroups are globally maximal and determine their maximal compact overgroups in the p-adic Lie group. The last section treats the Jordan subgroups of the classical groups.

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Benedict Gross, Gabriele Nebe. 2002-11-13. Globally maximal arithmetic groups. https://arxiv.org/abs/math/0211206

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