Strong multiplicity theorems for 𝐺𝐿(𝑛)

Abstract

Let π = ⊗ π υ \pi = \otimes {\pi _\upsilon } be a cuspidal automorphic representation of G L ( n , F A ) GL(n,{F_A}) , where F A {F_A} denotes the adeles of a number field F F . Let E E be a Galois extension of F F and let { g } \{ g\} denote a conjugacy class of the Galois group. The author considers those cuspidal automorphic representations which have local components π υ {\pi _\upsilon } whenever the Frobenius of the prime υ \upsilon is { g } \{ g\} , showing that such representations are often easily described and finite in number. This generalizes a result of Moreno [Bull. Amer. Math. Soc. 11 (1984), pp. 180-182].