N ov 2 00 2 Globally maximal arithmetic groups

Let G be a linear algebraic group defined over Q, and assume that G(R) is compact and meets every connected component of G(C). Let Q̂ := Ẑ ⊗ Q be the ring of finite adèles. Every arithmetic subgroup Γ of G(Q) is finite, and is obtained by choosing an open, compact subgroup K of G(Q̂) and defining Γ = K ∩G(Q) in G(Q̂). We note that G(Q) is discrete and co-compact in G(Q̂). In this paper, we consider the cases where the arithmetic subgroup Γ is contained in a unique maximal compact subgroup Kp of G(Qp), for all primes p. We call such Γ globally maximal; examples are provided by finite groups Γ with globally irreducible representations V over Q where G is a classical group O(V ), SUK(V ), or SUD(V ), according to whether the commuting algebra of V is Q, an imaginary quadratic field K or a definite quaternion algebra D. Other, in general not globally irreducible, examples are provided by the finite absolutely irreducible rational matrix groups that are “lattice sparse” of even type (see [5]). These are finite subgroups Γ ≤ GLn(Q) for which the natural representation is absolutely irreducible such that all Γ-invariant lattices can be obtained from any Γ-invariant lattice L, by successively taking the dual lattice, scalar multiples, intersections and sums of lattices that are already constructed (there are many such groups, e.g. for n = 24 there are 34 such maximal finite groups). Here the algebraic group G is G = O(V ) and the maximal compact subgroup G(Qp) containing Γ is O(L⊗ Zp) for any Γ-invariant lattice L. Another simple example of a globally maximal Γ is the group Γ = S4 = 2 ⋊ SL2(2), which has a unique irreducible representation V of dimension 3 and