On simply-connected groups over \(\mathbb{Z}\), with \(G(\mathbb{R})\) compact. (English) Zbl 0969.20023

Gross, Benedict H.

Kim, Myung-Hwan (ed.) et al., Integral quadratic forms and lattices. Proceedings of the international conference on integral quadratic forms and lattices, Seoul National University, Seoul, Korea, June 15-19, 1998. Dedicated to the memory of Dennis Ray Estes. Providence, RI: American Mathematical Society. Contemp. Math. 249, 113-118 (1999).

The main object of the paper under review is a semisimple simply connected group \(G\) defined over the field \(\mathbb{Q}\) of rational numbers. The author studies integral models \(\mathbf G\) of \(G\). By definition, this is a group scheme over \(\mathbb{Z}\) with generic fibre isomorphic to \(G\). In an earlier paper [Invent. Math. 124, No. 1-3, 263-279 (1996; Zbl 0846.20049)] the author established the following important fact: if \(G\) is split over \(\mathbb{Q}_p\) for all \(p\) and anisotropic over \(\mathbb{R}\), one can choose \(\mathbf G\) to be a smooth affine group \(\mathbb{Z}\)-scheme, with good reduction at all primes \(p\) (this means that all special fibres are reductive groups over \(\mathbb{F}_p\)). Throughout below by integral model we mean a scheme with these properties.
The main observation of the author in this paper is that all models of \(G\) lie, in a sense, in the same genus. Let us make this more precise. Fix a model \(\mathbf G\) of \(G\). If \(\mathbf G'\) is a smooth group scheme with everywhere good reduction and generic fibre \(G'\) admitting a \(\mathbb{Q}\)-isomorphism \(f'\colon G'\to G\), the pair \(({\mathbf G}',f')\) is said to be in the genus of \(\mathbf G\) if the hyperspecial maximal compact subgroups \(f'({\mathbf G}'(\widehat\mathbb{Z}))\) and \({\mathbf G}(\widehat\mathbb{Z})\) are conjugate in the locally compact group \(G(\widehat\mathbb{Q})\) (here \(\widehat\mathbb{Z}=\prod_p\mathbb{Z}_p\), \(\widehat\mathbb{Q}=\widehat\mathbb{Z}\otimes\mathbb{Q}\)). Two models in the genus are said to be equivalent if there is an isomorphism compatible with the isomorphism of their generic fibres. One then defines a natural action of \(G(\mathbb{Q})\) on the equivalence classes of models in the genus and shows that the set of orbits with respect to this action can be identified with the finite double coset space \(G(\mathbb{Q})\backslash G(\widehat\mathbb{Q})/{\mathbf G}(\widehat\mathbb{Z})\). This description allows the author to prove that given any model \(\mathbf G\) of \(G\) and any smooth group scheme \(\mathbf G'\) with everywhere good reduction and generic fibre isomorphic to \(G\), there is an isomorphism \(f'\colon G'\to G\) such that \(({\mathbf G}',f')\) is a model in the genus of \(\mathbf G\).
This result, together with the mass formula, is applied to the description of \(G(\mathbb{Q})\)-orbits of integral models. Yet another application is the proof of existence and uniqueness of \(\mathbf G\)-stable lattices in certain irreducible \(\mathbb{Q}\)-representations of \(G\).
For the entire collection see [Zbl 0931.00029].

Reviewer: B.Kunyavskii (Ramat Gan)

MSC:

20G30	Linear algebraic groups over global fields and their integers
14L15	Group schemes
11E57	Classical groups
22E40	Discrete subgroups of Lie groups
20G05	Representation theory for linear algebraic groups

Keywords:

simply connected groups; integral models; mass formula; group schemes; generic fibres; stable lattices

Citations:

Zbl 0846.20049