Let F be a number field with discriminant ∆F . Denote its (normalized) absolute values by SF , and write SF = Sfin · ∪ S∞, where S∞ denotes the collection of all archimedean valuations. For simplicity, we use v (resp. σ) to denote elements in Sfin (resp. S∞). Denote by A = AF the ring of adeles of F , by Glr(A) the rank r general linear group over A, and write A := Afin ⊕A∞ and GLr(A) := GLr(A)...

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