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Wee Teck Gan

Publications and source records attributed to Wee Teck Gan.

At least 19 recordsLinked to original sources
arXiv2026

Local Intertwining Relations and Co-tempered $A$-packets of Classical Groups

The local intertwining relation is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. We prove several instances of the local intertwining relation for quasi-split classical groups and the twisted general linear group, as they are required in the inductive proof of the endoscopic classification for quasi-split classical groups due to Arthur and Mok. In addition, we construct the co-tempered local $A$-packets by Aubert duality and verify their key properties by purely local means, which provide the seed cases needed as an input to the inductive proof. Together with further technical results that we establish, this makes the endoscopic classification conditional only on the validity of the twisted weighted fundamental lemma.

math.NT↗
arXiv2026

Generalised Whittaker models as instances of relative Langlands duality II: Plancherel density and global periods

In an earlier paper of the authors, a general family of instances of the relative Langlands duality of Ben-Zvi-Sakellaridis-Venkatesh [BZSV] were proposed and studied in the setting of branching problems for smooth representations. In this paper, we show the numerical conjectures of [BZSV] for the local Plancherel density, as well as an application to their conjectures on global periods, for this general family of instances.

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arXiv2026

Triality and adjoint lifting for GL(3)

Using the stable twisted trace formula for the triality automorphism, we show the adjoint lifting (to GL(8)) of cuspidal representations of GL(3) with a discrete series local component. We also describe the possible isobaric decompositions of the resulting automorphic representations on GL(8) and discuss applications towards Ramanujan bounds for GL(3) and the strong Artin conjecture for certain 3-dimensional Galois representations.

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arXiv2025

Triality and Functoriality

We use the triality automorphism of simple algebraic groups of type $D_4$ to prove some new instances of global Langlands functorial lifting. In particular, we prove the (weak) spin lifting from ${\rm GSp}_6$ to ${\rm GL}_8$ and the tensor product lifting from ${\rm GL}_2 \times {\rm GSp}_4$ to ${\rm GL}_8$. As an arithmetic application, we establish the expected properties of the spinor L-function attached to an arbitrary Siegel modular cusp form for ${\rm Sp}_6(\mathbb{Z})$ generating a holomorphic discrete series.

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arXiv2025

Fourier Transform and the minimal representation of $E_7$

We consider the minimal representation of the adjoint split group $E_7$ over a p-adic field. The representation has a model in a space of functions on a 17 dimensional cone $Ω$, and elements of the unique parabolic subgroup Q with abelian radical act by simple geometric formulas. We write a formula for the action of an involutive element $s$, conjugating $Q$ to the opposite parabolic $\bar Q$. The resulting integral operator, called a Fourier transform on $Ω$, is related to generalized Fourier transform, defined by Braverman and Kazhdan.

math.RT↗
arXiv2023

Generalised Whittaker models as instances of relative Langlands duality

The recent proposal by Ben-Zvi, Sakellaridis and Venkatesh of a duality in the relative Langlands program, leads, via the process of quantization of Hamiltonian varieties, to a duality theory of branching problems. This often unexpectedly relates two a priori unrelated branching problems. We examine how the generalised Whittaker (or Gelfand-Graev) models serve as the prototypical example for such branching problems. We give a characterization, for the orthogonal and symplectic groups, of the generalised Whittaker models possibly contained in this duality theory. We then exhibit an infinite family of examples of this duality, which, provably at the local level via the theta correspondence, satisfy the conjectural expectations of duality.

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arXiv2023

Similitude exceptional theta correspondences

We construct and develop a similitude version of exceptional theta correspondences and show that the Howe duality theorem follows from that for the "isometry" case. We also extend basic tools such as the seesaw identity associated to seesaw dual pairs to the similitude setting.

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arXiv2023

A theory of $γ$-factors for $G_2 \times GL_r$

We construct a theory of local gamma factors for $G_2 \times GL_r$ using a functorial lifting from $G_2$ to $GL_7$. This theory of gamma factors is uniquely characterized by a usual list of properties, showing that it is the only possible candidate. Moreover, this theory of gamma factors is compatible with the Galois theoretic one under the local Langlands correspondence for $G_2$.

math.NT↗
arXiv2023

Twisted GGP Problems and Conjectures

In an earlier work, we considered a family of restriction problems for classical groups (over local and global fields) and proposed precise answers to these problems using the local and global Langlands correspondence. These restriction problems were formulated in terms of a pair $W \subset V$ of orthogonal, Hermitian, symplectic, or skew-Hermitian spaces. In this paper, we consider a twisted variant of these conjectures in one particular case -- that of a pair of skew-Hermitian spaces $W = V$.

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arXiv2023

${\rm Spin}(7)$ is unacceptable

We classify the pairs of group morphisms $Γ\rightarrow {\rm Spin}(7)$ which are element conjugate but not globally conjugate. As an application, we study the case where $Γ$ is the Weil group of a $p$-adic local field, which is relevant to the recent approach to the local Langlands correspondence for ${\rm G}_2$ and ${\rm PGSp}_6$ by Gan and Savin. As a second application, we improve some result of Kret and Shin about ${\rm GSpin}_7$-valued Galois representations.

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arXiv2023

A family of Spin(8) dual pairs: the case of real groups

Exceptional groups of type $E_6$ contain dual pairs where one member is $\mathrm{Spin}(8)$, and the other is $T\rtimes \mathbb Z/2\mathbb Z$, where $T$ is a two-dimensional torus and the non-trivial element in $\mathbb Z/2\mathbb Z$ acts on $T$ by the inverse involution. We describe the correspondence of representations arising by restricting the minimal representation.

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arXiv2022

Unitary Friedberg-Jacquet periods

The main goal of this paper is to study the unitary Friedberg-Jacquet period through their connection with the unitary Shalika period. Locally we study the multiplicity of unitary Friedberg-Jacquet periods for discrete series. Globally we prove one direction of a conjecture of Xiao-Zhang, stating that the non-vanishing of a global unitary Friedberg-Jacquet period implies the non-vanishing of the central value of the (twisted) standard L-function.

math.RT↗
arXiv2022

Twisted Gan-Gross-Prasad conjecture for certain tempered L-packets

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

math.RT↗
arXiv2022

Local parameters of supercuspidal representations

For a connected reductive group $G$ over a non-archime\-dean local field $F$ of positive characteristic, Genestier and Lafforgue have attached a semisimple parameter $\CL^{ss}(π)$ to each irreducible representation $π$. Our first result shows that the Genestier-Lafforgue parameter of a tempered $π$ can be uniquely refined to a tempered L-parameter $\CL(π)$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of $\CL^{ss}(π)$ for unramfied $G$ and supercuspidal $π$ constructed by induction from an open compact (modulo center) subgroup. If $L^{ss}(π)$ is pure in an appropriate sense, we show that $\CL^{ss}(π)$ is ramified (unless $G$ is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal{L}^{ss}(π)$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is $\PP^1$ and a simple application of Deligne's Weil II.

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