The Rankin-Selberg method for automorphic distributions. (English) Zbl 1302.11033

Miller, Stephen D.; Schmid, Wilfried

Kobayashi, Toshiyuki (ed.) et al., Representation theory and automorphic forms. Based on the symposium, Seoul, Korea, February 14–17, 2005. Basel: Birkhäuser (ISBN 978-0-8176-4505-2/hbk). Progress in Mathematics 255, 111-150 (2008).

Summary: This paper describes our method of pairing automorphic distributions.We present a third technique for obtaining the analytic properties of automorphic \(L\)-functions, in addition to the existing methods of integral representations (Rankin-Selberg) and Fourier coefficients of Eisenstein series (Langlands-Shahidi). We recently used this technique to establish new cases of the full analytic continuation of the exterior square \(L\)-functions. The paper here gives an exposition of our method in two special, yet representative cases: the Rankin-Selberg tensor product \(L\)-functions for \(\mathrm{PGL}\setminus \mathrm{PGL}(2,\mathbb R)\), as well as for the exterior square \(L\)-functions for \(\mathrm{GL}(4,\mathbb Z)\setminus \mathrm{GL}(4,\mathbb R)\).
For the entire collection see [Zbl 1124.11004].

MSC:

11F70	Representation-theoretic methods; automorphic representations over local and global fields
11F66	Langlands \(L\)-functions; one variable Dirichlet series and functional equations
22E30	Analysis on real and complex Lie groups

Keywords:

analytic \(L\)-functions; integral representations; Rankin-Selberg; exterior square; automorphic distributions