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--- abstract: | Let $F$ be a $p$-adic field, and $G_n$ one of the groups $GL(n,F)$, $GSO(2n-1,F)$, $GSp(2n,F)$, or $GSO(2(n-1),F)$. Using the mirabolic subgroup or analogues of it, and related “derivative” functors, we give an asymptotic expansion of functions in the Whittaker model of generic representations of $G_n$, with respect to a minimal set of characters of subgroups of the maximal torus. Denoting by $Z_n$ the center of $G_n$, and by $N_n$ the unipotent radical of its standard Borel subgroup, we characterize generic representations occurring in $L^2(Z_nN_n\backslash G_n)$ in terms of these characters.\ This is related to a conjecture of Lapid and Mao for general split groups, asserting that the generic representations occurring in $L^2(Z_nN_n\backslash G_n)$ are the generic discrete series; we prove it for the group $G_n$. author: - 'Nadir MATRINGE[^1]' title: '**Derivatives and asymptotics of Whittaker functions**' --- Introduction {#introduction .unnumbered} ============ Let $G_n$ be the points of one of the groups $GL(n)$, $GSO(2n-1)$, $GSp(2n)$, or $GSO(2(n-1))$ over a $p$-adic field $K$. The main result (Theorem \[restorus\]) of this work describes the asymptotic behaviour of the restriction of Whittaker functions to the standard maximal torus, in terms of a family of characters which is minimal in some sense. From results of [@LM], this restriction can be described for split reductive groups in terms of cuspidal exponents.\ Here, after having defined analogues of the mirabolic subgroup for the groups $G_n$, and the corresponding derivative functors, following [@CP] (where the case of completely reducible derivatives is treated for $GL(n)$), we choose to describe the restriction of Whittaker functions to the torus in terms of central exponents of the derivatives.\ This description, inspired by [@B], is better adapted to understanding when the Whittaker model of a unitary generic representation is a subspace of $L^2(Z_n N_n\backslash G_n)$ (these representations are conjectured to be generic discrete series by Lapid and Mao).\ In the first section, we review the groups in question and define their mirabolic subgroups. We also give a decomposition of the unipotent radical of the standard Borel subgroup, and a description of how nondegenerate characters of this radical behave with respect to this decomposition.\ In Section 2, we give properties of the derivative functors, and use them to prove our asymptotic expansion of Whittaker functions, which is Theorem \[restorus\].\ In Section 3, we characterize generic representations with Whittaker model included in $L^2(Z_n N_n\backslash G_n)$ in terms of central exponents of the derivatives, in Corollary \[l2g\]. We then prove in Theorem \[l2’\] the conjecture 3.5 of [@LM]. Mirabolic subgroup and nondegenerate characters =============================================== Let $F$ be nonarchimedean local field, we denote by ${\mathfrak{O}_F}$ its ring of integer, and by ${\mathfrak{P}_F}={\varpi_F}{\mathfrak{O}_F}$ the maximal ideal of this ring, where ${\varpi_F}$ is a uniformiser of $F$.\ We give a list of groups, and describe some of their properties which will be used in the sequel:\ - **Case A:**\ The group $G_0$ is trivial, and for $n\geq1$, the group $G_n$ is $GL(n,F)$.\ We consider the maximal torus of $G_{n}$ consisting of diagonal matrices, it is isomorphic