Automorphic Integral Transforms for Classical Groups II: Twisted Descents

The paper (Jiang, Automorphic forms: L-functions and related geometry: assessing the legacy of I.I. Piatetski-Shapiro. Contemporary mathematics, vol 614. American Mathematical Society, Providence, RI, 2014, pp 179–242) forms Part I of the theory of Automorphic Integral Transforms for Classical Groups, where the first named author made a conjecture on how the global Arthur parameters may govern the structure of the Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. This leads to a better understanding of the automorphic kernel functions with which the integral transforms yield conjecturally the endoscopic correspondences for classical groups. In this paper, we discuss the Twisted Automorphic Descent method and its variants that construct concrete modules for irreducible cuspidal automorphic representations of general classical groups. When the global Arthur parameters are generic, the details of the theory are referred to Jiang et al. (2015, accepted by IMRN), Jiang and Zhang (2015, submitted; 2015, in preparation), which extend the automorphic descent method of Ginzburg-Rallis-Soudry (The descent map from automorphic representations of GL(n) to classical groups. World Scientific, Singapore, 2011) to great generality.