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Existence of lattices in Kac-Moody groups over finite fields. (English) Zbl 1070.17010

V. Kac and R. Moody independently defined an infinite dimensional generalization of complex semisimple Lie algebras. Kac-Moody groups corresponding to these Kac-Moody algebras (in the algebraic definition given by J. Tits [J. Algebra 105, 542–573 (1987; Zbl 0626.22013)]) can be considered as infinite dimensional generalizations of semisimple algebraic groups over fields. It is then natural to ask whether some of the properties of semisimple algebraic groups and their subgroups are still relevant in the Kac-Moody setting. The theory of lattices (discrete subgroups of finite covolume) in semisimple Lie groups is well established. The paper under review addresses the existence and construction of lattices in Kac-Moody groups.
Given a Kac-Moody algebra \({\mathcal G}_\mathbb{Q}\) over \(\mathbb{Q}\) with a Cartan subalgebra \({\mathcal H}_\mathbb{Q}\) and a regular dominant integral weight \(\lambda\) in the dual \({\mathcal H}_\mathbb{Q}^*\) with respect to a choice of positive roots, the authors consider the corresponding irreducible highest weight module \(V^\lambda\) and show it contains a \(\mathbb{Z}\)-form \(V_\mathbb{Z}^\lambda\). For a field \(K\), they construct a group \(G_0\) in \(\operatorname{Aut}(V_\mathbb{Z}^\lambda \otimes K)\), which is a homomorphic image of the (minimal) Kac-Moody group associated to \({\mathcal G}\) by Tits. They complete \(G_0\) in the topology with base neighborhoods of the identity being fixators in \(G_0\) of finite subsets of \(V_K^\lambda= V_\mathbb{Z}^\lambda \otimes K\). Denote this completion \(G\). Let \(B_0\) \((B_0^-)\) be the Borel subgroups in \(G_0\) corresponding to the positive (negative) roots, and \(B\) \((B^-)\) be their completions and subgroups of \(G\). Let \(K\) be a finite field of cardinality \(q\). Then \(G\) is a locally compact group and \(B\) is an open compact subgroup. The authors prove that when the number \(l\) of simple roots of \(\mathcal G\) is 2 or \(q>l\), then the subgroup \(B^-\) of \(G\) is a non-uniform lattice. Similar results in a more general setting of quasi-split Kac-Moody groups were independently obtained by B. Remy [Groups: topological, combinatorial and arithmetic aspects, Cambridge Univ. Press, Cambridge, Lond. Math. Soc. Lect. Notes Ser. 311, 487–541 (2004; Zbl 1115.20025)] and [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 6, 475–478 (1999; Zbl 0933.22029)].
When the rank \(l=2\), the Kac-Moody group \(G\) acts on a locally finite tree \(X\). Generalizing the results of A. Lubotzky [Geom. Funct. Anal. 1, No. 4, 405–431 (1991; Zbl 0786.22011)], who constructed uncountable families of uniform and non-uniform lattices in algebraic groups acting on their Bruhat-Tits trees, the authors construct uncountable families of uniform and non-uniform lattices in \(G\). The uniform lattices are obtained via “Schottky” subgroups construction. The construction of nonuniform lattices employs a spherical Tits system obtained from the doubly transitive action of \(G\) on the ends of \(X\). The authors conjecture that there are uncountably many distinct conjugacy classes of non-uniform lattices and uncountably many distinct conjugacy classes of uniform lattices within \(G\) in this case.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20E42 Groups with a \(BN\)-pair; buildings
22E40 Discrete subgroups of Lie groups
22F50 Groups as automorphisms of other structures
20F65 Geometric group theory
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