Abstract
Let F be a p-adic field with residue field of cardinality q. To each irreducible representation of GL(n, F), we attach a local Euler factor \(L^{BF}(q^{-s},q^{-t},\pi )\) via the Rankin–Selberg method, and show that it is equal to the expected factor \(L(s+t+1/2,\phi _\pi )L(2s,\Lambda ^2\circ \phi _\pi )\) of the Langlands’ parameter \(\phi _\pi \) of \(\pi \). The corresponding local integrals were introduced in Bump and Friedberg (The exterior square automorphic L-functions on \(\mathrm{GL}(n)\) 47–65, 1990), and studied in Matringe (J Reine Angew Math doi:10.1515/crelle-2013-0083). This work is in fact the continuation of Matringe (J Reine Angew Math doi:10.1515/crelle-2013-0083). The result is a consequence of the fact that if \(\delta \) is a discrete series representation of GL(2m, F), and \(\chi \) is a character of Levi subgroup \(L=GL(m,F)\times GL(m,F)\) which is trivial on GL(m, F) embedded diagonally, then \(\delta \) is \((L,\chi )\)-distinguished if an only if it admits a Shalika model. This result was only established for \(\chi =\mathbf {1}\) before.
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Acknowledgments
I thank the referee for his careful reading and comments. This work was supported by the research project ANR-13-BS01 -0012 FERPLAY.
