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 
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/
@article{ZNSL_2009_373_a7,
     author = {E. E. Goryachko},
     title = {The simplicity of the branching of representations of the groups $\mathrm{GL}(n,q)$ under the parabolic restrictions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {124--133},
     year = {2009},
     volume = {373},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a7/}
}
TY  - JOUR
AU  - E. E. Goryachko
TI  - The simplicity of the branching of representations of the groups $\mathrm{GL}(n,q)$ under the parabolic restrictions
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2009
SP  - 124
EP  - 133
VL  - 373
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a7/
LA  - ru
ID  - ZNSL_2009_373_a7
ER  - 
%0 Journal Article
%A E. E. Goryachko
%T The simplicity of the branching of representations of the groups $\mathrm{GL}(n,q)$ under the parabolic restrictions
%J Zapiski Nauchnykh Seminarov POMI
%D 2009
%P 124-133
%V 373
%U http://geodesic.mathdoc.fr/item/ZNSL_2009_373_a7/
%G ru
%F ZNSL_2009_373_a7

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

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.

[1] A. M. Vershik, S. V. Kerov, “Ob odnoi beskonechnomernoi gruppe nad konechnym polem”, Funkts. anal. i ego pril., 32:3 (1998), 3–10 | DOI | MR | Zbl

[2] A. M. Vershik, A. Yu. Okunkov, “Novyi podkhod k teorii predstavlenii simmetricheskikh grupp. II”, Zap. nauchn. semin. POMI, 307, 2004, 57–98 | MR | Zbl

[3] E. E. Goryachko, “Prostota vetvleniya osnovnoi serii predstavlenii grupp $\mathrm{GL}(n,q)$ pri parabolicheskikh ogranicheniyakh”, Zap. nauchn. semin. POMI, 331, 2006, 43–59 | MR | Zbl

[4] A. Aizenbud, D. Gourevitch, Multiplicity free Jacquet modules, arxiv: 0910.3659v1[math.RT]

[5] I. M. Gelfand, D. A. Kajdan, “Representations of the group $\mathrm{GL}(n,K)$ where $K$ is a local field”, Lie Groups and Their Representations, Akad. Kiadó, Budapest, 1975, 95–118 | MR

[6] A. M. Vershik, S. V. Kerov, “Four drafts on the representation theory of the group of infinite matrices over a finite field”, Zap. nauchn. semin. POMI, 344, 2007, 5–36 | MR

[7] A. V. Zelevinsky, Representations of Finite Classical Groups. A Hopf Algebra Approach, Lect. Notes Math., 869, 1981 | MR | Zbl