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

Abstract

A family of global zeta integrals representing a product of tensor product (partial) L-functions: $$L^S(s, \pi \times \tau_1)L^S(s,\pi \times \tau_2)\cdots L^S(s, \pi \times \tau_r)$$ is established in this paper, where π is an irreducible cuspidal automorphic representation of a quasi-split classical group of Hermitian type and $${\tau_1,\ldots,\tau_r}$$ are irreducible unitary cuspidal automorphic representations of $${{\rm GL}_{a_1},\ldots,{\rm GL}_{a_r}}$$ , 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 $${\tau_1,\ldots,\tau_r}$$ 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.