Hartley´s Theorem on Representations of the General Linear Groups and Classical Groups

Authors: A. E. ZALESSKI

Abstract: We suggest a new proof of Hartley's theorem on representations of the general linear groups GL_n(K) where K is a field. Let H be a subgroup of GL_n(K) and E the natural GL_n(K)-module. Suppose that the restriction E|_H of E to H contains a regular KH-module. The theorem asserts that this is then true for an arbitrary GL_n(K)-module M provided dim M>1 and H is not of exponent 2. Our proof is based on the general facts of representation theory of algebraic groups. In addition, we provide partial generalizations of Hartley's theorem to other classical groups.

Keywords: subgroups of classical groups, representation theory of algebraic groups

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