arXiv subjects
Publications and source records attributed to Jialiang Zou.
In this paper, we generalize the method of Gan-Ichino and Atobe in [GI16][A18] to the field of real numbers and prove the basic tempered case of the local Gan-Gross-Prasad conjecture for Fourier-Jacobi models of symplectic-metaplectic groups, based on the tempered case of the conjecture for Bessel models proved in [CL22] by Chen-Luo.
In this paper we investigate the theta lifting of type II dual pairs over a non-Archimedean local field, by combining the homological method of Adams--Prasad--Savin and the analytic method of Fang--Sun--Xue. We have three main results: 1. we determine completely the big theta lift of an irreducible representation when its Godement--Jacquet L-function is holomorphic at a critical point; 2. we compute the big theta lift of all characters, hence determine the space of eigendistributions on matrix spaces for all characters; 3. we show that the Weil representation is projective if and only if the dual pair is almost in the stable range.
Let G be an even orthogonal or unitary group over a number field. Based on the same observation used in arXiv:1705.10106, we prove the Arthur's multiplicity formula for the generic part of the automorphic discrete spectrum of G by using the theta lift. We also consider a class of non-generic A-parameters and obtain a multiplicity formula in this case. In particular, we obtain a description of the full automorphic discrete spectrum of even orthogonal or unitary groups with Witt index less or equal to one.
In this paper we consider the theta correspondence over a non-Archimedean local field. Using the homological method and the theory of derivatives, we show that under a mild condition the big theta lift is irreducible.
We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at $1$ can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we generalize the results of Aubert-Michel-Rouquier and Pan on theta correspondence between the Harish-Chandra series.
Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which was established in an early paper of Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this provides an alternative approach to the works of Kaletha-Minguez-Shin-White and Moeglin-Renard.
Using theta correspondence, we obtain a classification of irreducible representations of an arbitrary even orthogonal group (i.e. the local Langlands correspondence) by deducing it from the local Langlands correspondence for symplectic groups due to Arthur. Moreover,we show that our classifications coincide with the local Langlands correspondence established by Arthur and formulated precisely by Atobe-Gan for quasi-split even orthogonal groups.
In spirit of Gan-Ichino's work on the Arthur's multiplicity formula for metaplectic groups, we have established the Arthur's multiplicity formula for even orthogonal or unitary groups with Witt index less than or equal to one. In that multiplicity formula, some local packets defined using the stable range theta lifts are involved. In this paper, we prove that at non-Archimedean places, the definition of the local packets involved in that multiplicity formula is independent of the choice of the dual-pairs used in their construction. Moreover, at those places where the groups are quasi-split, we prove that the local packets involved are the same as the local A-packets defined by Arthur/ Mok.