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On the local Bump-Friedberg $$L$$-function. II. (English) Zbl 1431.11066
Summary: Let $$F$$ be a $$p$$-adic field with residue field of cardinality $$q$$. To each irreducible representation of $$\mathrm{GL}(n, F)$$, 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 [D. Bump and S. Friedberg, Isr. Math. Conf. Proc. 3, 47–65 (1990; Zbl 0712.11030)], and studied in Part I [the author, J. Reine Angew. Math. 709, 119–170 (2015; Zbl 1398.11080)]. This work is in fact the continuation of [the author, loc. cit.].
The result is a consequence of the fact that if $$\delta$$ is a discrete series representation of $$\mathrm{GL}(2m, F)$$, and $$\chi$$ is a character of Levi subgroup $$L=\mathrm{GL}(m,F)\times \mathrm{GL}(m,F)$$ which is trivial on $$\mathrm{GL}(m, F)$$ 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.

##### MSC:
 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields
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