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L-functions for \(SO_ n\times GL_ k\). (English) Zbl 0684.22009
We prove Langlands’ conjecture for the standard representation of the groups \(SO_{2n+1}\times GL_ k\) and \(SO_{2n}\times GL_ k\) where \(k<n\). We construct Rankin-Selberg integrals for these groups, prove their convergence, show that they are Euclidean and compute the unramified local integrals. These integrals are a natural generalization of the ones constructed for \(SO_{2n+1}\times GL_ n\) and \(SO_{2n}\times GL_ n\) be Gelbart and Piatetski-Shapiro.
Reviewer: D.Ginzburg

22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11R39 Langlands-Weil conjectures, nonabelian class field theory
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11R42 Zeta functions and \(L\)-functions of number fields
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