A Product of Tensor Product L-functions of Quasi-split Classical Groups of Hermitian Type

• Published in: Geometric and functional analysis Vol. 24; no. 2; pp. 552 - 609
• Main Authors: Jiang, Dihua, Zhang, Lei
• Format: Journal Article
• Language: English
• Published: Basel Springer Basel 01.04.2014
• Subjects: Analysis; Mathematics; Mathematics and Statistics; Secondary 11F85; functions; Bessel periods of Eisenstein series; 22E50; global zeta integrals; Primary 11F70; 22E55; classical groups of Hermitian type; tensor product
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-014-0266-7

A family of global zeta integrals representing a product of tensor product (partial) L -functions: L S ( s , π × τ 1 ) L S ( s , π × τ 2 ) ⋯ L S ( s , π × τ r ) is established in this paper, where π is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and τ 1 , … , τ r are irreducible unitary cuspidal automorphic representations of GL a 1 , … , GL a r , respectively. When r  = 1 and the classical group is an orthogonal group, this family was studied by Ginzburg et al. (Mem Am Math Soc 128: viii+218, 1997 ). When π is generic and τ 1 , … , τ r are not isomorphic to each other, such a product of tensor product (partial) L -functions is considered by Ginzburg et al. (The descent map from automorphic representations of GL( n ) to classical groups, World Scientific, Singapore, 2011 ) in with different kind of global zeta integrals. In this paper, we prove that the global integrals are eulerian and finish the explicit calculation of unramified local zeta integrals in a certain case (see Section 4 for detail), which is enough to represent the product of unramified tensor product local L -functions. The remaining local and global theory for this family of global integrals will be considered in our future work.