On standard \(L\)-functions for unitary groups
Let \(K/k\) be a quadratic extension of number fields, and \(G_n\) be the associated quasi-split unitary group of \(2n\times 2n\) matrices. Then the \(L\)-group of \(G_n\) is a semidirect product \(^L G=GL_{2n}({\mathbb{C}})\rtimes \text{Gal}(K/k)\), and admits a \(4n\)-dimensional representation \(\rho\), the so-called standard representation. Let \({\mathcal A}\) denote the adeles of \(k\) and let \(\pi=\otimes \pi_v\) be a cuspidal automorphic representation of \(G_n({\mathcal A})\). Let \(S\) denote the set of places which are either archimedean or else finite but with the data ramified. In this paper the author shows that the partial \(L\)-function \(L_S(s,\pi,\rho)=\prod_{v\not\in S}L(s,\pi_v,\rho),\) when multiplied by a suitable normalizing factor (a product of Dedekind zeta functions), has meromorphic continuation to the entire complex plane with at most a finite number of poles. A similar result for the standard \(L\)-function of the split orthogonal group \(G=O_{n.n}\) was proved by \textit{S. Gelbart, I. I. Piatetski-Shapiro} and \textit{S. Rallis} [Explicit constructions of automorphic \(L\)-functions, Lect. Notes Math. 1254. Springer-Verlag (1987; Zbl 0612.10022), Chapter 1], using their doubling method to write a suitable Rankin-Selberg integral. A similar Rankin-Selberg integral is given here, and the author notes that the unfolding into local integrals and the unramified computation are similar to that work. The main new ingredient is the study of the local integrals at the remaining `bad' places. The author rewrites these integrals in terms of Godement-Jacquet integrals, and obtains their continuations. At a finite ramified place, he moreover shows that the local integral may be made constant by means of a suitable choice of test function.
scientific article; zbMATH DE number 1022282