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On Linear Groups of Degree 2n Containing a Representation of the Special Linear Group of Degree n

Let n be an integer, n ≥ 2, and let a field P be a quadratic extension of an infinite field k. Regarding P as a k-vector space of dimension 2, we consider an n-dimensional P-vector space V as a 2n-dimensional k-vector space so the general linear group GL n (P) acting on V is embedded in the group GL 2n (k). Let a field K be an algebraic extension of k. In this article, we determine overgroups of the special linear group SL n (P) in the group GL 2n (K).