Bernstein-Zelevinsky derivatives, branching rules and Hecke algebras

Let $G$ be a split reductive group over a $p$-adic field $F$. Let $B$ be a Borel subgroup and $U$ the maximal unipotent subgroup of $B$. Let $\psi$ be a Whittaker character of $U$. Let $I$ be an Iwahori subgroup of $G$. We describe the Iwahori-Hecke algebra action on the Gelfand-Graev representation $(\mathrm{ind}_{U}^{G}\psi)^I$ by an explicit projective module. As a consequence, for $G=GL(n,F)$, we define and describe Bernstein-Zelevinsky derivatives of representations generated by $I$-fixed vectors in terms of the corresponding Iwahori-Hecke algebra modules. Furthermore, using Lusztig's reductions, we show that the Bernstein-Zelevinsky derivatives can be determined using graded Hecke algebras. We give two applications of our study. Firstly, we compute the Bernstein-Zelevinsky derivatives of generalized Speh modules, which recovers a result of Lapid-M\'inguez and Tadi\'c. Secondly, we give a realization of the Iwahori-Hecke algebra action on some generic representations of $GL(n+1,F)$, restricted to $GL(n,F)$, which is further used to verify a conjecture on an Ext-branching problem of D. Prasad for a class of examples.