L-functions for \(SO_ n\times GL_ k\). (English) Zbl 0684.22009

Ginzburg, David

J. Reine Angew. Math. 405, 156-180 (1990).

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

MSC:

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

Keywords:

connected reductive algebraic group; automorphic cuspidal representation; Eisenstein series; Langlands’ conjecture; standard representation; Rankin-Selberg integrals; unramified local integrals