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Hengfei Lu

Publications and source records attributed to Hengfei Lu.

14 recordsLinked to original sources
arXiv2025

On completeness of local intertwining periods

In this paper we study the problem of explicitly describing the space of invariant linear forms on induced distinguished representations in terms of invariant linear forms on the inducing representation. More precisely, for certain tempered reductive symmetric pairs (G,H) over a local field of characteristic zero, which we call unimodular in this paper, we study under which condition on the inducing representation, the space of H-invariant linear forms on a parabolically induced representation of G is generated by regularized intertwining periods attached to admissible parabolic orbits in G{H, as defined in the work of Matringe--Offen--Yang. We conjecture that it is the case when the inducing representation is square-integrable. Under this assumption we actually conjecture that one can replace regularized by normalized intertwining periods. We then verify the conjecture on known examples, and prove it for various pairs where G has semi-simple split rank one.

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arXiv2024

The sign of linear periods

Let $G$ be a group with subgroup $H$, and let $(\pi,V)$ be a complex representation of $G$. The natural action of the normalizer $N$ of $H$ in $G$ on the space $\mathrm{Hom}_H(\pi,\mathbb{C})$ of $H$-invariant linear forms on $V$, provides a representation $\chi_{\pi}$ of $N$ trivial on $H$, which is a character when $\mathrm{Hom}_H(\pi,\mathbb{C})$ is one dimensional. If moreover $G$ is a reductive group over a local field, and $\pi$ is smooth irreducible, it is an interesting problem to express $\chi_{\pi}$ in terms of the possibly conjectural Langlands parameter $\phi_\pi$ of $\pi$. In this paper we consider the following situation: $G=\mathrm{GL}_m(D)$ for $D$ a central division algebra of dimension $d^2$ over a local field $F$ of characteristic zero, $H$ is the centralizer of a non central element $\delta\in G$ such that $\delta^2$ is in the center of $G$, and $\pi$ has generic Jacquet-Langlands transfer to $\mathrm{GL}_{md}(F)$. In this setting the space $\mathrm{Hom}_H(\pi,\mathbb{C})$ is at most one dimensional. When $\mathrm{Hom}_H(\pi,\mathbb{C})\simeq \mathbb{C}$ and $H\neq N$, we prove that the value of the $\chi_{\pi}$ on the non trivial class of $\frac{N}{H}$ is $(-1)^m\epsilon(\phi_\pi)$ where $\epsilon(\phi_\pi)$ is the root number of $\phi_{\pi}$. Along the way we extend many useful multiplicity one results for linear and Shalika models to the case of non split $G$. When $F$ is $p$-adic we also classify standard modules with linear periods and Shalika models, which are new results even when $D=F$.

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arXiv2022

Modulo $\ell$ distinction problems

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution, with $\ell$ an odd prime different from $p$. We start by proving a general theorem allowing to lift supercuspidal $\overline{\mathbb{F}}_{\ell}$-representations of $\mathrm{GL}_n(F)$ distinguished by an arbitrary closed subgroup $H$ to a distinguished supercuspidal $\overline{\mathbb{Q}}_{\ell}$-representation. Given a quadratic field extension $E/F$ and an irreducible $\overline{\mathbb{F}}_{\ell}$-representation $\pi$ of $\mathrm{GL}_n(E)$, we verify the Jacquet conjecture in the modular setting that if the Langlands parameter $\phi_\pi$ is irreducible and conjugate-self-dual, then $\pi$ is either $\mathrm{GL}_n(F)$-distinguished or $(\mathrm{GL}_n(F),\omega_{E/F})$-distinguished (where $\omega_{E/F}$ is the quadratic character of $F^\times$ associated to the quadratic field extension $E/F$ by the local class field theory), but not both, which extends one result of S\'echerre to the case $p=2$. We give another application of our lifting theorem for supercuspidal representations distinguished by a unitary involution, extending one result of Zou to $p=2$. After that, we give a complete classification of the $\mathrm{GL}_2(F)$-distinguished representations of $\mathrm{GL}_2(E)$. Using this classification we discuss a modular version of the Prasad conjecture for $\mathrm{PGL}_2$. We show that the "classical" Prasad conjecture fails in the modular setting. We propose a solution using non-nilpotent Weil-Deligne representations. Finally, we apply the restriction method of Anandavardhanan and Prasad to classify the $\mathrm{SL}_2(F)$-distinguished modular representations of $\mathrm{SL}_2(E)$.

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arXiv2021

The generalized linear period

Let $F$ be a non-archimedean local field of characteristic zero. We study the linear period problem for the pair $(G,H_{p,p+1})=(GL_{2p+1}(F), GL_{p}(F)\times GL_{p+1}(F))$ and we prove that any bi-$(H_{p,p+1},μ)$-invariant generalized function on $G$ is invariant under the matrix transpose when μis a good character. We also show that any $P\cap H_{p,p+1}$-invariant linear functional on an $H_{p,p+1}$-distinguished irreducible smooth representation of $G$ is also $H_{p,p+1}$-invariant when F is nonarchimedean, where $P$ is a standard mirabolic subgroup of $G$ with last row vector $(0,\cdots,0,1)$.

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arXiv2021

Multiplicity one for the pair (GL(n,D),GL(n,E))

Let F be a local field of characteristic zero. Let D be a quaternion algebra over F. Let E be a quadratic field extension of F. Let μ be a character of GL(1,E). We study the distinction problem for the pair (GL(n,D), GL(n,E)) and we prove that any bi-(GL(n,E), μ)-equivariant tempered generalized function on GL(n,D) is invariant with respect to an anti-involution. Then it implies that dimHom(π,μ) is at most 1 by the generalized Gelfand-Kazhdan criterion. Thus we give a new proof to the fact that (GL(2n,F),GL(n,E)) is a Gelfand pair when μ is trivial and D splits.

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arXiv2020

The action of a mirabolic subgroup on a symmetric variety

Let F be a local field of character zero. Let E be a quadratic field extension of F. We show that any P-invariant linear functional on a GL(n,E)-distinguished irreducible smooth admissible representation of GL(2n,F) is also GL(n,E)-invariant where P is a mirabolic subgroup of GL(n,E).

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arXiv2020

A New Proof to the Period Problems of GL(2)

We use the relations between the base change representations, theta lifts and Whittaker model, to give a new proof to the period problems of $GL(2)$ over a quadratic local field extension $E/F.$ And we classify both local and global $D^\times(F)-$distinguished representations $π^D$ of $D^\times(E),$ where $D^\times$ is an inner form of $GL_2$ defined over a nonarchimedean field or a number field $F.$

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arXiv2019

The distinction problem for metaplectic case

We use the theta lifts between Mp(2) and PD to study the distinction problems for the pair (Mp(2,E), SL(2,F )), where E is a quadratic field extension over a nonarchimedean local field F of characteristic zero and D is a quaternion algebra. With a similar strategy, we give a conjectural formula for the multiplicity of distinction problem related to the pair (Mp(2n,E),Sp(2n,F)).

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arXiv2019

The Prasad conjectures for $\mathrm{GSp_4}$ and $\mathrm{PGSp_4}$

In this paper, we use the theta correspondence between $\mathrm{GSp_4}$ and $\mathrm{GO(V)}$ to discuss the $\mathrm{GSp_4}$-distinction problems over a quadratic field extension $E/F.$ With a similar strategy, we study the period for the pair $(\mathrm{GSp_4(E)},\mathrm{GSp_{1,1}(F)}),$ where $\mathrm{GSp_{1,1}}$ is the unique inner form of $\mathrm{GSp_4}.$ Then we verify the Prasad conjecture for $\mathrm{PGSp_4(E)}$.

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arXiv2018

Theta correspondence and the Prasad conjecture for SL(2)

We use relations between the base change representations and theta lifts, to give a new proof to the local period problems of SL(2) over a nonarchimedean quadratic field extension E/F. Then we will verify the Prasad conjecture for SL(2). With a similar strategy, we obtain a certain result for the Prasad conjecture for Sp(4).

math.RT↗
arXiv2018

The SL(1,D)-distinction problem

We use the local theta correspondences between the quaternionic Hermitian groups and the quaternionic skew-Hermitian groups to understand the distinction problem for the symmetric pair SL(2,E)/SL(1,D), where E is a quadratic field extension of a nonarchimedean local field extension F and D is a 4-dimensional division quaternion algebra over F

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