Double descent in classical groups

Let A be the ring of adeles of a number field F. Given a self-dual irreducible, automorphic, cuspidal representation τ of GL_n(A), with a trivial central character, we construct its full inverse image under the weak Langlands functorial lift from the appropriate split classical group G. We do this by a new automorphic descent method, namely the double descent. This method is derived from the recent generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan [CFGK17], which represent the standard L-functions for G×GL_n. Our results are valid also for double covers of symplectic groups.