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 SLn(P) in the group GL(2n)(K).