On the local Bump–Friedberg L-function II

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.