Geodesic
The simplicity of the branching of representations of the groups $\mathrm{GL}(n,q)$ under the parabolic restrictions
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 124-133 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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E. E. Goryachko. The simplicity of the branching of representations of the groups $\mathrm{GL}(n,q)$ under the parabolic restrictions. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVII, Tome 373 (2009), pp. 124-133. http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a7/

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