Strong multiplicity theorems for 𝐺𝐿(𝑛)

Gilbert, George T.

doi:10.1090/S0002-9947-1987-0891635-9

Strong multiplicity theorems for $\textrm {GL}(n)$
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Trans. Amer. Math. Soc. 302 (1987), 561-576 Request permission

Abstract:

Let $\pi = \otimes {\pi _\upsilon }$ be a cuspidal automorphic representation of $GL(n,{F_A})$, where ${F_A}$ denotes the adeles of a number field $F$. Let $E$ be a Galois extension of $F$ and let $\{ g\}$ denote a conjugacy class of the Galois group. The author considers those cuspidal automorphic representations which have local components ${\pi _\upsilon }$ whenever the Frobenius of the prime $\upsilon$ is $\{ g\}$, 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].

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An analytic proof of the multiplicity one theorem

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