Automorphic Integral Transforms for Classical Groups II: Twisted Descents

Local root numbers and spectrum of the local descents for orthogonal groups : p-adic case

“…For any π ∈ Π(G n ) with a generic L-parameter, and for any ∈ {0, 1, · · · , r} with r being the F -rank of G n , if the twisted Jacquet module J O (π) is nonzero, then J O (π) has a tempered irreducible quotient as a representation of G O n (F ), and so does the -th local descent D O (π). In other words, if [25]). In such a generality, if the representation π in Proposition 1.6 is unramified, then for any index , the -th local descent D O (π) has a tempered, unramified irreducible quotient.…”

“…In such a generality, if the representation π in Proposition 1.6 is unramified, then for any index , the -th local descent D O (π) has a tempered, unramified irreducible quotient. While the Generic Summand Conjecture ( [24] and [25]) asserts that for any irreducible cuspidal automorphic representation of G n with a generic global Arthur parameter, its global descent at the first occurrence index is cuspidal and contains at least one irreducible summand that has a generic global Arthur parameter. This assertion serves a base for the construction of explicit modules for irreducible cuspidal automorphic representations of general classical groups with generic global Arthur parameters as developed in [24] and [25].…”

“…While the Generic Summand Conjecture ( [24] and [25]) asserts that for any irreducible cuspidal automorphic representation of G n with a generic global Arthur parameter, its global descent at the first occurrence index is cuspidal and contains at least one irreducible summand that has a generic global Arthur parameter. This assertion serves a base for the construction of explicit modules for irreducible cuspidal automorphic representations of general classical groups with generic global Arthur parameters as developed in [24] and [25]. We will discuss the global impact of the results obtained in this paper to the Generic Summand Conjecture in [24] and [25] in our future work.…”

The Fourier-Jacobi Periods : The case of $Mp(2n+2r) \times Sp(2n)$

“…We follow the similar approach here. As announced in [22] and [23], we mention that D. Jiang and L. Zhang are preparing a paper [24] which deals with the theory of twisted automorphic descents for Fourier-Jacobi case as an analogue of [22], which deals with that of Bessel case. Like [22,Theorem 5.3], once it is completed, it will contain our main theorem under the hypothesis on the unramified computation of the local zeta integral related to the Fourier-Jacobi functionals, which is an ongoing work of D. Jiang, D. Soudry and L. Zhang ( [21].)…”

On the non-vanishing of the central value of certain $L$-functions: unitary groups

“…Fourier coefficients of automorphic forms associated to unipotent orbits as developed in [10] and [11], 2. the global zeta integrals of tensor product type as developed in [12] and [13], 3. the theory of twisted automorphic descents as developed in [13] and [14], and 4. the endoscopic classification of the discrete spectrum as in [1], [17], and [15].…”