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Register a new userIrreducibility of Theta Lifting for Unitary Groups
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Authors
Chenyan Wu
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This article shows that for unitary dual reductive pairs the first occurrence of theta lift of an irreducible cuspidal automorphic representation is irreducible. It also proves a refined tower property for theta lifts and the involutive property for twisted theta lifts.
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We give an explicit construct of a harmonic weak Maass form $F_{Theta}$ that is a lift of $Theta^3$, where $Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of $Theta^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the holomorphic part of $F_{Theta}$ are associated to class numbers of real quadratic fields.
We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r-th central derivative of non-singular Fourier coefficients of a normalized Siegel--Eisenstein series, and (2) the degree of special cycles of virtual dimension 0 on the moduli stack of unitary shtukas with r legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.
We evaluate regularized theta lifts for Lorentzian lattices in three different ways. In particular, we obtain formulas for their values at special points involving coefficients of mock theta functions. By comparing the different evaluations, we derive recurrences for the coefficients of mock theta functions, such as Hurwitz class numbers, Andrews spt-function, and Ramanujans mock theta functions.
Following the idea of [GJS09] for orthogonal groups, we introduce a new family of period integrals for cuspidal automorphic representations $sigma$ of unitary groups and investigate their relation with the occurrence of a simple global Arthur parameter $(chi,b)$ in the global Arthur parameter $psi_sigma$ associated to $sigma$, by the endoscopic classification of Arthur ([Art13], [Mok13], [KMSW14]). The argument uses the theory of theta correspondence. This can be viewed as a part of the $(chi,b)$-theory outlined in [Jia14] and can be regarded as a refinement of the theory of theta correspondences and poles of certain $L$-functions, which was outlined in [Ral91].
We derive a precise relation of poles of Eisenstein series associated to the cuspidal datum $chiotimessigma$ and lowest occurrence of theta lifts of a cuspidal automorphic representation $sigma$ of a unitary group, where $chi$ is conjugate self-dual character. We also give a refined result on non-vanishing of periods of Eisenstein series and first occurrence of theta lifts. This gives constraints on existence of $(chi,b)$-factors in the global $A$-parameter of $sigma$.
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