Derivatives and exceptional poles of the local exterior square \(L\)-function for \(\mathrm{GL}_m\) (Q2304338)

This paper concerns the equality of local \(L\)-functions arising in two different ways for the exterior square \(L\)-function. Suppose that \(F\) is a non-archimedean local field, \(W_F\) its Weil-Deligne group, and let \(\phi\) be a complex representation of \(W_F\) of dimension \(m\) that is Frobenius semisimple. Then the local Langlands correspondence for GL\(_m(F)\), established by Harris-Taylor and Henniart, attaches to this data an irreducible admissible representation \(\pi(\phi)\) of GL\(_m(F)\). On the one hand, composing \(\phi\) with the exterior square representation \(\wedge^2\), one has the local Artin \(L\)-function \(L(s,\wedge^2\circ \phi)\). Alternatively, one may define a local \(L\)-function \(L(s,\pi(\phi),\wedge^2)\) as the greatest common divisor of the local Rankin-Selberg integrals for \(\pi(\phi)\) that appear in the Jacquet-Shalika integral representation [\textit{H. Jacquet} and \textit{Joseph. Shalika}, Perspect. Math. 11, 143--226 (1990; Zbl 0695.10025)] for the exterior square \(L\)-function. In this paper it is proved that these two local \(L\)-functions are equal. To do so, the author establishes multiplicativity results for these \(L\)-functions. This involves the study of Shalika functionals for Bernstein-Zelevinsky derivatives. The study of the local \(L\)-functions attached to the Jacquet-Shalika integrals was initiated by Cogdell and Piatetski-Shapiro in the 1990s. This paper is a continuation of the approach of these authors described in [\textit{J. W. Cogdell} and \textit{I. I. Piatetski-Shapiro}, Prog. Math. 323, 115--173 (2017; Zbl 1433.11054)]. In addition, a result of \textit{G. Henniart} [Int. Math. Res. Not. 2010, No. 4, 633--673 (2010; Zbl 1184.22009)] states that the Artin \(L\)-function \(L(s,\wedge^2\circ \phi)\) matches the Langlands-Shahidi exterior square \(L\)-function, and \textit{P. K. Kewat} and \textit{R. Raghunathan} [Math. Res. Lett. 19, No. 4, 785--804 (2012; Zbl 1333.11047)] show the matching of the Jacquet-Shalika exterior square \(L\)-function and the Langlands-Shahidi exterior square \(L\)-function for irreducible smooth square-integrable representations. One may also define a local exterior square \(L\)-factor by the integrals of \textit{D. Bump} and the reviewer [Isr. Math. Conf. Proc. 3, 47--65 (1990; Zbl 0712.11030)]. \textit{N. Matringe} [J. Reine Angew. Math. 709, 119--170 (2015; Zbl 1398.11080)] shows that the Bump-Friedberg local \(L\)-function matches the corresponding Artin \(L\)-function.