✦ LIBER ✦

Compact Subgroups of Linear Algebraic Groups

✍ Scribed by Richard Pink

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 497 KB
Volume: 206
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1998.7439

No coin nor oath required. For personal study only.

✦ Synopsis

The general problem underlying this article is to give a qualitative classification Ž . of all compact subgroups ⌫ ; GL F , where F is a local field and n is arbitrary.

It is natural to ask whether ⌫ is an open compact subgroup of H E , where H is a linear algebraic group over a closed subfield E ; F. We show that ⌫ indeed has this form, up to finite index and a finite number of abelian subquotients. When ⌫ is Zariski dense in a connected semisimple group, we give a precise openness result Ž . for the closure of the commutator group of ⌫. In the case char F s 0 the answers have long been known by results of Chevalley and Weyl. The motivation for this work comes from the positive characteristic case, where such results are needed to study Galois representations associated to function fields. We also derive openness results over a finite number of local fields.

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