On the local Bump-Friedberg L function II

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 $L^{BF}(q^{-s},q^{-t},π)$ via the Rankin-Selberg method, and show that it is equal to the expected factor $L(s+t+1/2,ϕ_π)L(2s,Λ^2\circ ϕ_π)$ of the Langlands' parameter $ϕ_π$ of $π$. The corresponding local integrals were introduced in [BF], and studied in [M15]. This work is in fact the continuation of [M15]. The result is a consequence of the fact that if $δ$ is a discrete series representation of $GL(2m,F)$, and $χ$ is a character of Levi subgoup $L=GL(m,F)\times GL(m,F)$, trivial on $GL(m,F)$ embedded diagonally, then $δ$ is $(L,χ)$-distinguished if an only if it admits a Shalika model, a result which was only established for $χ=1$ before.

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