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Globally maximal arithmetic groups. (English) Zbl 1113.20040

Let \(G\) be a linear algebraic group defined over \(\mathbb Q\), and assume that \(G(\mathbb R)\) is compact. Let \(\widehat{\mathbb Q}:=\widehat{\mathbb Z}\otimes\mathbb Q\) be the ring of finite adèles. Every arithmetic subgroup \(\Gamma\) of \(G(\mathbb Q)\) is finite, and is obtained by choosing an open, compact subgroup \(K\) of \(G(\widehat{\mathbb Q})\) and defining \(\Gamma=K\cap G(\mathbb Q)\) in \(G(\widehat{\mathbb Q})\).
The authors consider the cases where the arithmetic subgroup \(\Gamma\) is contained in a unique maximal compact subgroup \(K_p\) of \(G(\mathbb Q_p)\), for all primes \(p\). Such \(\Gamma\) is called globally maximal; examples are provided by finite groups \(\Gamma\) with globally irreducible representations \(V\) over \(\mathbb Q\), and by the finite absolutely irreducible rational matrix groups that are “lattice sparse” of even type.
As further examples are considered the cases of (normalizers of) Jordan subgroups \(\Gamma\) in Lie algebras. The following Jordan subgroups are considered: \(2^3.\text{SL}_3(2)\leq G_2\), \(3^3.\text{SL}_3(3)\leq F_2\), and \(2^5.\text{SL}_5(2)\leq 2^5.2^{10}.\text{SL}_5(2)\leq E_8\). The \(\Gamma\)-invariant lattices are also determined. Moreover, using an obvious generalization of the notion of globally maximal groups to arbitrary number fields, it is shown that the Jordan subgroups of the classical groups are also globally maximal.

MSC:

20G30 Linear algebraic groups over global fields and their integers
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
11E57 Classical groups
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References:

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