Globally maximal arithmetic groups.
(submitted) 

Benedict Gross  
Dept. of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 01238, USA
and 
Gabriele Nebe
 Abteilung Reine Mathematik, Universit\"at Ulm, 89069 Ulm, Germany


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.