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

Garland, Howard

J. Algebra 209, No. 2, 446-532 (1998).

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.

Reviewer: Vladimir L. Popov (Moskva)

MSC:

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)

Keywords:

loop group; fundamental domain; Galois extension

Citations:

Zbl 0475.17004

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