arXiv subjects

Jaeho Haan

Publications and source records attributed to Jaeho Haan.

10 recordsLinked to original sources

arXiv2026

Special periods and some non-tempered cases of the Gan-Gross-Prasad conjecture

In this paper, we establish a relationship between special periods and special L-values of automorphic representations of classical groups, and prove the non-tempered global Gan--Gross--Prasad conjecture in several cases. Our approach consists of two main steps. First, inspired by Rallis' tower property, we study the interaction between special periods and the tower property for the genericity of global theta lifts. Second, we investigate the relationship between the analytic properties of L-functions and special periods via the Rankin--Selberg integral method. Combining these results with non-vanishing criteria for global theta lifts in terms of various L-values, we prove three explicit higher-corank families of non-tempered cases of the global Gan--Gross--Prasad conjecture.

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arXiv2024

The tower property on the genericity of global theta lifts

In this paper, we examine the tower property concerning the genericity of global theta lifts between various classical groups, drawing inspiration from Rallis' tower property. By exploring the relationship between the analytic properties of $L$-functions and special Bessel and Fourier-Jacobi periods, we demonstrate that the first occurrence of global theta lifts between dual reductive groups preserves genericity. As an application, we establish the global Gan-Gross-Prasad conjecture for $\SO_{2n+1} \times \SO_{2}$ under the assumption that $\SO_{2}$ is split and its representation is trivial.

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arXiv2023

Regularized Periods and the global Gan-Gross-Prasad conjecture : The case of $U(n+2r) \times U(n)$

In this paper, we introduce regularized trilinear periods on certain non-reductive groups. It has two direct applications. Firstly, it enables us to define the regularized Bessel periods and the regularized Fourier-Jacobi periods for all classical and metaplectic groups. Secondly, by using the properties of the regularized Fourier-Jacobi periods, we can prove one direction of the full Gan-Gross-Prasad conjecture on skew-hermitian unitary groups.

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arXiv2023

The local converse theorem for quasi-split $O_{2n}$ and $SO_{2n}$

Let $F$ be a non-archimedean local field of characteristic not equal to 2. In this paper, we prove the local converse theorem for quasi-split $\O_{2n}(F)$ and $\SO_{2n}(F)$, via the description of the local theta correspondence between $\O_{2n}(F)$ and $\Sp_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $\gamma$-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of $\O_{2n}(\A)$ and $\SO_{2n}(\mathbb{A})$, respectively, where $\A$ is a ring of adele of a global number field $L$.

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arXiv2022

The local converse theorem for $Mp_{2n}$ : the generic case

In this paper, we establish the local converse theorem and the stability of local gamma factors for $\Mp_{2n}$ via the precise local theta correspondence between $\Mp_{2n}$ and $\SO_{2n+1}$ over local fields of characteristic not equal to 2. We also prove the rigidity theorem for irreducible generic cuspidal automorphic representations of $\Mp_{2n}$ over number fields.

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arXiv2022

Fourier-Jacobi periods and the non-tempered Gan--Gross--Prasad conjecture for $\Mp_{2n} \times \Sp_{2m}$

In this paper, we establish one direction of the non-tempered global Gan--Gross--Prasad (GGP) conjecture for the symplectic-metaplectic pairs $\Sp_{2n} \times \Mp_{2m}$ and $\Mp_{2n} \times \Sp_{2m}$. Our study focuses on two distinct families of non-tempered global $A$-parameters spanning all coranks, where standard relative trace formula methods remain unavailable. Our approach proceeds along two distinct lines of attack. First, we treat the case where both members of the pair are non-tempered, utilizing regularized Fourier-Jacobi periods of residual Eisenstein series and establishing a reciprocal non-vanishing theorem. Second, we address the setting where only the metaplectic member is non-tempered and its central $L$-value vanishes, relying heavily on the global theta correspondence and explicit seesaw identities. Along the way, we establish one direction of the tempered GGP conjecture for these pairs in arbitrary coranks, and prove a dichotomy for the global associated $L$-packet of the metaplectic parameter $[1] \boxplus M'$. We show that this packet lies entirely in the residual spectrum or entirely in the cuspidal spectrum, a behavior determined uniformly by the non-vanishing of the central $L$-value $L(1/2, M')$.

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arXiv2017

The local Gan-Gross-Prasad conjecture for $U(n+1) \times U(n)$ : a non-generic case

The local Gan-Gross-Prasad conjecture of unitary groups, which is now settled by the works of Plessis, Gan and Ichino, says that for a pair of generic $L$-parameters of $(U(n+1),U(n))$, there is a unique pair of representations in their associated Vogan $L$-packets which produces the Bessel model. In this paper, we examined the conjecture for a pair of $L$-parameters of $\big(U(n+1), U(n)\big)$ as fixing a special non-generic parameter of $U(n+1)$ and varing tempered $L$-parameters of $U(n)$. We observed that there still exist a Gan-Gross-Prasad type formulae depending on the choice of $L$-parameter of $U(n)$.

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arXiv2016

On the Fourier-Jacobi model for some endoscopic Arthur packet of $U(3) \times U(3)$ : the non-generic case

For a generic $L$-parameter of $U(n)\times U(n)$, it is conjectured that there is a unique representation in their associated relevant Vogan $L$-packet which produces the unique Fourier-Jacobi model. We investigated this conjecture for some non-generic $L$-parameters of $U(3)\times U(3)$ and discovered that this conjecture is still true for some non-generic $L$-parameter and false for some non-generic $L$-parameter. In the case when it holds, we specified such representation under the local Langlands correspondence for unitary group.

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arXiv2016

The local Gan-Gross-Prasad conjecture for $U(3) \times U(2)$ : the non-generic case

In this paper, we investigate the local Gan-Gross-Prasad conjecture for some pair of representations of $U(3)\times U(2)$ involving a non-generic representation. For a pair of generic $L$-parameters of $(U(n),U(n-1))$, it is known that there is a unique pair of representations in their associateed Vogan $L$-packets which produces the unique Bessel model of these $L$-parameters. We showed that this is not ture for some pair of $L$-parameters involving a non-generic one. On the other hand, we give the precise local theta correspondence for $(U(1),U(3))$ not at the level of $L$-parameters but of individual representations in the framework of the local Langlands correspondence for unitary group. As an applicaiton of these results, we prove an analog of Ichino-Ikeda local conjecture for some non-tempered case.

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arXiv2015

The Bessel Period of U(3) and U(2) involving a non-tempered representation

In \cite{Ha}, Neal Harris has given a refined Gross-Prasad conjecture for unitary group as an analogue of Ichino and Ikeda's paper \cite{Ich} concerning special orthogonal groups. In his paper, he stated a conjecture under the assumption that the pair of given representations should be tempered. In this paper, we consider a specific pair involving a non-tempered one. In this case, an analogous formula still exists but the central critical $L$-value is slightly different with the one in the conjecture. As a corollary, this verifies that the tempered condition is indispensable in formulating the conjecture.

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