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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].

Strong multiplicity theorems for 𝐺𝐿(𝑛)