The arithmetic theory of loop groups. II: The Hilbert-modular case. (English) Zbl 0974.22020

This paper is a sequel to the author’s Part I [Publ. Math., Inst. Hautes Etud. Sci. 52, 5-136 (1980; Zbl 0475.17004)]. The author generalizes the construction of fundamental domains in that paper to quotients of arithmetic loop groups \(\widehat G_{J}^\lambda \) (defined in Section 3 of the paper under review), when \(J\) is the ring of integers in a real Galois extension \(k\) of \({\mathbb Q}\) (so in the aforementioned paper the case \(J={\mathbb Z}\) was treated). In the present case, the fundamental domain appears to be a union of a finite number of translates of a Siegel set (Theorem 11.3). This number is bounded by a power of the class number of \(k\), and equal to the number of elements in a certain, finite double coset space (Theorem 7.1 and Appendix B). The key tool is the Hilbert-modular basis constructed in Section 4.


22E67 Loop groups and related constructions, group-theoretic treatment
20G25 Linear algebraic groups over local fields and their integers
17B65 Infinite-dimensional Lie (super)algebras
22E40 Discrete subgroups of Lie groups
14L17 Affine algebraic groups, hyperalgebra constructions
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)


Zbl 0475.17004
Full Text: DOI


[1] Artin, E., Galois theory, Notre Dame Math. Lectures, 2 (1955)
[2] Borel, A., Introduction aux groupes arithmétiques (1969), Hermann: Hermann Paris · Zbl 0186.33202
[3] Bourbaki, N., Groupes et algebres de Lie (1968), Hermann: Hermann Paris · Zbl 0186.33001
[4] Garland, H., The arithmetic theory of loop algebras, J. Algebra, 53, 480-551 (1978) · Zbl 0383.17012
[5] Garland, H., The arithmetic theory of loop groups, Publ. Math., 52, 181-312 (1980)
[6] Garland, H., A Cartan decomposition for p-adic loop groups, Math. Ann., 302, 151-175 (1995) · Zbl 0837.22013
[7] Godement, R., Domaines fondamentaux des groupes arithmetiques, Semin. Bourbaki, 257 (1963) · Zbl 0136.30101
[8] Iwahori, N.; Matsumoto, H., On some Bruhat decompositions and the structure of the Hecke rings of p-adic Chevalley groups, Publ. Math. I.H.E.S., 25, 237-280 (1965) · Zbl 0228.20015
[9] Raghunathan, M. S., A note on generators for arithmetic subgroups of algebraic groups, Pacific J. Math., 152, 365-373 (1991) · Zbl 0793.20045
[11] Vaserstein, On the group \(SL_2\), Math. USSR Sbornik, 18 (1972) · Zbl 0359.20027
[12] Venkataramana, T. N., On systems of generators of arithmetic subgroups of higher rank groups, Pacific J. Math., 166, 193-212 (1994) · Zbl 0822.22005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.