### Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 813-867 |

Number of pages | 55 |

Journal | Communications in Contemporary Mathematics |

Volume | 5 |

Issue number | 5 |

DOIs | |

State | Published - Oct 1 2003 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Keywords

- Kac-Moody Lie algebra
- Kac-Moody group
- Lattices

### Cite this

*Communications in Contemporary Mathematics*,

*5*(5), 813-867. https://doi.org/10.1142/S0219199703001117

}

*Communications in Contemporary Mathematics*, vol. 5, no. 5, pp. 813-867. https://doi.org/10.1142/S0219199703001117

**Existence of lattices in Kac-Moody groups over finite fields.** / Carbone, Lisa; Garland, Howard.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Carbone, Lisa

AU - Garland, Howard

PY - 2003/10/1

Y1 - 2003/10/1

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

UR - http://www.scopus.com/inward/record.url?scp=0347529000&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0347529000&partnerID=8YFLogxK

U2 - 10.1142/S0219199703001117

DO - 10.1142/S0219199703001117

M3 - Article

AN - SCOPUS:0347529000

VL - 5

SP - 813

EP - 867

JO - Communications in Contemporary Mathematics

JF - Communications in Contemporary Mathematics

SN - 0219-1997

IS - 5

ER -