Existence of lattices in Kac-Moody groups over finite fields

TY - JOUR

T1 - Existence of lattices in Kac-Moody groups over finite fields

AU - Carbone, Lisa

AU - Garland, Howard

N1 - Funding Information: The first author was supported in part by NSF g rant #DMS-9800604. The authors would like to thank B. Remy and T. Januszkiewicz for their correspondence, and for informing us of their results. Thanks to A. Lubotzky for encourag ing us to undertake this work and for explaining his constructions to us, and to H. Bass and I. Grojnowski for helpful discussions.

PY - 2003/10

Y1 - 2003/10

N2 - 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 non-uniform 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.

AB - 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 non-uniform 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.

KW - Kac-Moody Lie algebra

KW - Kac-Moody group

KW - Lattices

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U2 - https://doi.org/10.1142/S0219199703001117

DO - https://doi.org/10.1142/S0219199703001117

M3 - Article

VL - 5

SP - 813

EP - 867

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 5

ER -