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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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