Bounds for (k,c) model functions over p-adic fields

Let $F$ be a $p$-adic field, let $k,c\ge 1$, and let $\rho$ be a $(k,c)$ representation of ${\mathrm{G}\mathrm{L}}_{kc}(F)$: a complex representation of finite length whose wave front set has $(k^c)$ as its unique maximal orbit, and which admits a $(k,c)$ functional that is unique up to scaling. We prove that the functions in the $(k,c)$ model of $\rho$ are of moderate growth, by relating this model to the Zelevinsky model of $\rho$, and obtain as a consequence a bound for the restriction of these functions to the standard Levi subgroup associated with the composition $(c^k)$ of $kc$. As applications, we study convergence problems of several integrals involving $(k,c)$ models: integrals from the generalized doubling method, and a new family of integrals which extends the Rankin–Selberg integrals for Speh representations of Lapid and Mao.