On representations of the maximal unramified Galois extension of a field of positive characteristic (Q1091431)

Assume the following data are given: k, an algebraically closed field of positive characteristic; \(K\supset k\), an extension field; \(O=\prod O_ v\), the product being taken over the valuation rings \(O_ v\) in the completions \(K_ v\) of K with respect to the discrete valuations v which are trivial on k and whose residue fields are isomorphic to k; and, finally, \(K_ T\), the maximal Galois extension of K that is unramified at every v. The main result of the paper provides, in terms of group theory, a parametrization of the set of all \(GL_ n(k)\)-equivalence classes of representations of \(Gal(K_ T/K)\) into \(GL_ n(k)\) whose images are isomorphic to some subgroups of \(GL_ n({\mathbb{F}}_ q)\). Here \(O\cap K=k\) is assumed. The result applies to the case of the function field K of an algebraic variety over k that is complete and normal, and so retells an observation that was already made by \textit{H. Lange} and \textit{U. Stuhler} [Math. Z. 156, 73-83 (1977; Zbl 0343.14011)].