Bessel functions for \(\text{GL}(3)\) over a \(p\)-adic field. (Q1880066)

Let \(k\) be a non-archimedean local field and \(G=GL(3,k)\). Let \((\pi,V)\) be an irreducible admissible representation of \(G\) that is generic. In this paper the author attaches a Bessel function to such \(\pi\) (actually one for each cell in the Bruhat decomposition) and studies the asymptotics of this function. In more detail: Let \(\psi\) be a non-degenerate character of the subgroup of upper triangular unipotents \(N\) of \(G\), let \(L\) be a non-zero \(\psi\)-Whittaker functional, and let \(v\to W_v=L(\pi(g)v)\) be a realization of \(\pi\) on a space of \(\psi\)-Whittaker functions (so \(W_v(ng)=\psi(n)W_v(g)\) for all \(n\in N\), \(g\in G\)). For \(g\) in the big cell \(Bw_0B\) of \(G\), let \(L_g\) be the functional defined by \(L_g(v)=\lim \int_{N_m} W_v(gn)\,\psi^{-1}(n)\,dn,\) where the integral is over an expanding sequence of compact open subgroups \(N_m\) that exhaust \(N\). Then \(L_g\) is a Whittaker functional, and from the uniqueness of the Whittaker functional it follows that there is a function \(j_{\pi,\psi}(g)\) such that \(L_g(v)=j_{\pi,\psi}(g)\,L(v).\) The function \(j_{\pi,\psi}(g)\) is the Bessel function of \(\pi\) for the big cell; it satisfies \(j_{\pi,\psi}(n_1gn_2)=\psi(n_1n_2)j_{\pi,\psi}(g)\) for all \(n_1,n_2\in N\) and \(g\in Bw_0B\). The author proves that the above definition (based on a limiting process) is valid, and that the Bessel function is locally constant on \(Bw_0B\). He also proves that the asymptotics of this function are the same as the asymptotics of certain orbital integrals that appear in the relative trace formula which were studied by \textit{H. Jacquet} and \textit{Y. Ye} [Trans. Am. Math. Soc. 348, No.3, 913-939 (1996; Zbl 0861.11033)]. Bessel functions for the double cover of \(GL(2,k)\) were constructed by Gelbart and Piatetski-Shapiro, and used to compute epsilon-factors. Additional work was carried out by Soudry, Cogdell and Piatetski-Shapiro and Averbuch. The author [Trans. Am. Math. Soc. 353, No.7, 2601-2614 (2001; Zbl 0976.22015)] has attached Bessel functions to every generic representation of a quasi-split reductive group over \(k\) using a different approach, based on distributions and modeled on Harish-Chandra's study of character functions. In a later paper [``Bessel distributions for \(GL(3)\) over the \(p\)-adics'', Pac. J. Math. 217, No.1, 11-27(2004)], the author compares these two approaches, and uses this to prove that \(j_{\pi,\psi}\) is locally integrable.