Simplicity of the branching of representations of the groups GL$(n,q)$ the parabolic restriction: an elementary proof

It is well known that the branching of representations of the groups $\operatorname{GL}(n,q)$ under the parabolic restriction is simple. Apparently, an elementary proof of this important fact has not been found so far. We present such a proof, which uses a method not standard for the representation theory of finite groups.