The simplicity of the branching of representations of the groups $\mathrm{GL}(n,q)$ under the parabolic restrictions

We present a direct proof of the simplicity of the branching of representations of the groups $\mathrm{GL}(n,q)$ under the parabolic restrictions. The proof consists of three steps: first, we reduce the problem to the statement that a certain pair of finite groups is a Gelfand pair, then we obtain a criterion for establishing this fact, which generalizes the classical Gelfand criterion, and, finally, we check the obtained criterion with the help of some matrix computations. Bibl. – 7 titles.