Existence of Lattices in Kac Moody Groups over Finite Fields
نویسنده
چکیده
Let g be a Kac–Moody Lie algebra. We give an interpretation of Tits’ associated group functor using representation theory of g and we construct a locally compact “Kac–Moody group” G over a finite field k. Using (twin) BN-pairs (G,B,N) and (G,B−, N) for G we show that if k is “sufficiently large”, then the subgroup B− is a non-uniform lattice in G. We have also constructed an uncountably infinite family of both uniform and nonuniform lattices in rank 2. We conjecture that these form uncountably many distinct conjugacy classes in G. The basic tool for the construction of non-uniform lattices in rank 2 is a spherical Tits system for G which we also construct.
منابع مشابه
Lattices in Kac - Moody Groups
Initially, we set out to construct non-uniform ‘arithmetic’ lattices in Kac-Moody groups of rank 2 over finite fields, as constructed by Tits ([Ti1], [Ti2]) using the BruhatTits tree of a Tits system for such groups. This attempt succeeded, and in fact, the construction we used can be applied to higher rank Kac-Moody groups over sufficiently large finite fields, and their buildings (Theorem 1.7...
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