Symmetric square \(L\)-functions on \(GL(r)\). (English) Zbl 0753.11021

Bump, Daniel; Ginzburg, David

Ann. Math. (2) 136, No. 1, 137-205 (1992).

This paper is devoted to a study of the \(L\)-functions associated with automorphic representations of \(GL(r,k_ \mathbb{A})\) and the ‘symmetric square’ representation of the \(L\)-group \(GL(r,\mathbb{C})\) of \(GL(r)\). The analytic continuation and functional equation of this function have already been established through the theory of Eisenstein series by Shahidi but this method has some serious inherent restrictions, especially in regard to the analysis of the poles and zeros of the \(L\)- function. Here the authors give an ingenious generalization of a construction of Patterson and Piatetski-Shapiro which gives a Rankin- Selberg representation of these \(L\)-functions. It involves an integral of Shimura’s type. The interesting feature here is that neither the ‘theta function’ on \(GL(r)\), or rather its double cover, nor, for \(r>2\), the one on \(GL(r-1)\) used to construct the Eisenstein series has a Whittaker model in the usual sense. However both have certain unique degenerate Whittaker models and it is this that the authors exploit. This method leads to a new proof of the meromorphic continuation and of the functional equation. As is to be expected the method does not quite yield poles and zeros of the \(L\)-function, only when a certain number of factors are left out. However with this proviso the authors obtain very sharp results about the poles, which are rather exceptional.

Reviewer: S.J.Patterson (Göttingen)

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
22E55	Representations of Lie and linear algebraic groups over global fields and adèle rings

Keywords:

\(L\)-functions; automorphic representations; symmetric square; \(L\)-group; Rankin-Selberg representation; integral of Shimura type; degenerate Whittaker models; meromorphic continuation; functional equation; poles