Automorphic induction for \(\text{GL}(n,\mathbb C)\)
The goal of this article is to study automorphic induction over an archimedean local field, and apply this to complete the global theory for \(\text{GL}(n)\). Let \(F\) be an archimedean local field. \textit{R. P. Langlands} [Math. Surv. Monogr. 31, 101--170 (1989; Zbl 0741.22009)] classified the irreducible \((\mathfrak{g},K)\)-modules for \(\text{GL}(n,F)\) in terms of \(n\)-dimensional representations of the Weil group \(W_F\) of \(F\). Since \(W_{\mathbb C}\) is of index \(2\) in \(W_{\mathbb R}\), given an \(n\)-dimensional representation of \(W_{\mathbb C}\), one may induce it to obtain a \(2n\)-dimensional representation of \(W_{\mathbb R}\). By the Langlands classification, this gives a way of assigning to each irreducible \((\mathfrak{g},K)\)-module \(\tau\) of \(\text{GL}(n,{\mathbb C})\) an isomorphism class of irreducible \((\mathfrak{g},K)\)-modules \(\tau^{{\mathbb C}/{\mathbb R}}\) of \(\text{GL}(2n,{\mathbb R})\), that depends only on the isomorphism class of \(\tau\). This is the operation of automorphic induction. In this article, the author is concerned with irreducible \((\mathfrak{g},K)\) modules which may appear as the archimedean components of cuspidal automorphic representations of \(\text{GL}(n)\) over a number field. Such components are unitary, and by a result of \textit{J. A. Shalika} [Ann. Math. (2) 100, 171--193 (1974; Zbl 0316.12010)], they are necessarily generic, that is, they have a Whittaker model. The author shows that if \(\tau\) is unitary and generic, then the automorphic induction \(\tau^{{\mathbb C}/{\mathbb R}}\) is determined by \(\tau\) via a character identity. (In fact, the author works slightly more generally, treating a cyclic algebra over an archimedean local field.) This is the exact analogue of a nonarchimedean result of the author and \textit{R. Herb} [Duke Math. J. 78, No. 1, 131--192 (1995; Zbl 0849.11092)], valid for cyclic extensions of a \(p\)-adic base field. The last section gives an application to automorphic induction over a number field, by combining the result here with the author's work [``Induction automorphe pour \(\text{GL}(n)\) sur les corps des nombres'', preprint, per biblio]. As the author notes in this paper's abstract, ``This completes the theory of automorphic induction for local and global representations of \(\text{GL}(n)\) over number fields.''
scientific article; zbMATH DE number 5708598
scientific article