Derivatives and asymptotics of Whittaker functions (Q2889339)

The author proves a conjecture of \textit{E. Lapid} and \textit{Z. Mao} [Represent. Theory 13, 63--81 (2009; Zbl 1193.22016)] for several groups; recently, this has also been proved independently (for all split groups) by Patrick Delorme.NEWLINENEWLINELet \(F\) be a nonarchimedean local field. Let \(G_n\) denote one of the groups \(GL(n,F), GSO(2n-1,F), GSp(2n-2,F)\) or \(GSO(2n-2,F)\). The author defines analogues of mirabolic subgroups and the corresponding derivative functors for these groups \(G_n\). Then, he describes the restriction of the Whittaker functions to the standard maximal torus in terms of central exponents of the derivatives -- this follows ideas of Bernstein. Finally, he proves for the groups \(G_n\), the conjecture of Lapid \& Mao which amounts to the assertion that the generic representations occurring in \(L^2(Z_nN_n \backslash G_n)\) are the generic discrete series. Here \(Z_n\) is the center of \(G_n\) and \(N_n\) is the unipotent radical of the standard Borel subgroup of \(G_n\).