The simplicity of branching of the principal series representations of the groups $GL(n,q)$ under the parabolic restrictions

We consider the parabolic restriction of representations of the group $GL(n+1,q)$ to the group $GL(n,q)$. The branching of representations under this restriction is simple. We present a direct proof of this fact in the case of the so-called principal series representations. This statement is reduced to the commutativity of the centralizer of the Hecke algebras $Z(H(n,q),H(n+1,q))$; we prove it using an auxiliary combinatorial theory.