On local \(L\)-factors for Archimedean \(\mathrm{GL}(n)\) (Q6558482)

The authors consider the question on how the \(n\)-dimensional representations \(\phi\) of Weil groups of \({\mathbb C}\) and \({\mathbb R}\) are determined by their \(L\)-functions \(L(\phi\otimes \chi,s)\), where \(\chi\) is in a family of representations of the Weil group. Since the irreducible representations of the Weil groups are either one or two dimensional, and the \(L\)-functions can be explicitly described using Gamma functions, the paper can provide a precise answer to this question by a careful consideration of the poles of the \(L\)-functions.\N\NOver the complex field, \(\phi\) is uniquely determined by the family \(L(\phi\otimes \chi,s)\), where \(\chi\) is one-dimensional. Over the real field \(\phi\) is uniquely determined by the family \(L(\phi\otimes \chi,s)\), where \(\chi\) is one-dimensional or two dimensional. Over the real field, just using the family of \(L\)-functions, where \(\chi\) is one-dimensional, it is not enough to determine \(\phi\); however if we know \(\phi\) is of dimension 2 or 3, and with fixed central character, then \(\phi\) is determined by the family of \(L\)-functions over one-dimensional \(\chi\). Once \(n=4\), even with fixed central characters, there exists two different \(\phi_1,\phi_2\), where \(L(\phi_1\otimes \chi,s)=L(\phi_2\otimes \chi,s)\) for all one-dimensional \(\chi\).