arXiv subjects

Geo Kam-Fai Tam

Publications and source records attributed to Geo Kam-Fai Tam.

10 recordsLinked to original sources

arXiv2026

Endoscopic liftings and parameters of epipelagic representations for classical groups

Let $G$ be a $p$-adic classical group (orthogonal, symplectic, or unitary) and $π$ be an epipelagic representation of $G$ in the sense of Reeder-Yu. Using Mœglin's theory of extended cuspidal supports and Bushnell-Kutzko's theory of covering types, we determine explicitly the endoscopic lift of $π$ to the general linear group, whose Langlands dual expresses the dual group of $G$ as a complex matrix group, in terms of the inducing type of $π$ that extends the character of the first Moy-Prasad filtration subgroup defined by a stable functional. We interpret the inducing type of $π$ via Stevens' construction of supercuspidal representations by skew semisimple strata and introduce what we will call epipelagic strata, requiring only that the residual characteristic $p$ be odd. As an application, we reprove M. Oi's results on the endoscopic lifts of simple supercuspidal representations, in the sense of Gross-Reeder, of quasi-split classical groups. Finally, on the Galois side, we show that the epipelagic Langlands parameters of $G$ constructed by Reeder-Yu can be recovered via the self-duality of Bushnell-Henniart's admissible triples and endoscopic embeddings of L-groups. We also establish some related results concerning epipelagic parameters that were previously proved under restrictions on $p$, including a parity result on rectifying characters and an adjoint Swan conductor result related to the Hiraga-Ichino-Ikeda conjecture.

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arXiv2026

Distinguishing Gauss sums

Let $\F_q$ be the field of order $q = p^f$ for prime $p$. If $G(χ)$ is the Gauss sum attached to a multiplicative character $χ$ on $\F_q^\times$, then $G(χ) = G(χ^p)$. We investigate the converse question: when does the equality of Gauss sums $G(χ_2) = G(χ_1)$ imply that $χ_2 = χ_1^{p^j}$ for some integer $j$. If $χ_2 = χ_1^{p^j}$, we say that $χ_1$ and $χ_2$ are Frobenius-conjugate. We use the Stickelberger factorization of ideals in cyclotomic fields to give an easily testable criterion for equality of Gauss sums, based on $p$-adic digit expansion. As an application, we develop several conditions under which Gauss sum equalities between characters on $\F_q^\times$ are explained by Frobenius-conjugacy. For example, if $χ_1$ has order $q-1$ or $\frac{1}{2}(q-1)$ and $G(χ_2) = G(χ_1)$, then $χ_1$ and $χ_2$ are Frobenius-conjugate. We also include several examples, based on the explicit evaluation of certain {\em pure} Gauss sums by R.J. Evans.

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arXiv2026

A conjectural construction of Arthur packets in Fargues-Scholze's categorical local Langlands correspondence

We present a conjectural construction of Arthur packets within Fargues-Scholze's framework for the categorical local Langlands correspondence (CLLC). This construction consists of three parts. We first provide an overview the main statement of the CLLC, the underlying moduli stacks -- $\mathrm{Par}_G$ of parameters and $\mathrm{Bun}_G$ of $G$-bundles -- on the two sides of the correspondence, and its relation to representations of reductive p-adic groups. We then review the geometric Satake correspondence in order to define Hecke operators and the spectral action of sheaves on $\mathrm{Par}_G$ on sheaves on $\mathrm{Bun}_G$, and to construct semisimple parameters using excursion data. Finally, we generalize the geometric construction of Arthur packets from pushing-forward skyscraper sheaves on the regular conormal bundle of $\mathrm{Par}_G$ over $\mathbb C$ to a conjectural analogous operation on the stack of singularities on $\mathrm{Par}_G$ over $\overline{\mathbb Q}_\ell$.

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arXiv2023

Depth Zero Supercuspidal Representations of Classical Groups into L-packets: the Typically Almost Symmetric Case

We classify what we call ``typically almost symmetric'' depth zero supercuspidal representations of classical groups into L-packets. Our main results resolve an ambiguity in the paper of Lust-Stevens \cite{Lust-Stevens} in this case, where they could only classify these representations in two or four, if not one, L-packets. By assuming the expected numbers of supercuspidal representations in the L-packets, we employ only simple properties of the representations to prove the main results. In particular, we do not require any deep calculations of character values.

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arXiv2023

Endoscopic liftings of epipelagic representations for classical groups

Let $G$ be a p-adic classical group (orthogonal, symplectic, unitary) and $π$ be an epipelagic representation of $G$ defined by Reeder-Yu. Using Mœglin's theory of extended cuspidal supports and Bushnell-Kutzko's theory of covering types, we determine the endoscopic lift of $π$ into the general linear group whose dual expresses the dual group of $G$ as a complex matrix group, explicitly in terms of the inducing type of $π$ that is extended from the character of the first Moy-Prasad filtration subgroup defined by a stable functional. We interpret the inducing type of $π$ via Stevens' construction of supercuspidal representations by skew semi-simple strata, and introduce the so-called epipelagic strata, which only require the residual characteristic p to be odd. As an application, we reprove the results of M. Oi on the endoscopic lifts of Gross-Reeder's simple supercuspidal representations of quasi-split classical groups.

math.RT↗

arXiv2020

Base change for ramified unitary groups: the strongly ramified case

We describe a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. Roughly speaking, we require the underlying stratum of a given supercuspidal representation to be skew maximal simple, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

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arXiv2013

On Certain Admissible Embeddings of L-groups

Let $F$ be a local field and $E/F$ be a separable extension of degree $n$. Regard $T=\text{Res}_{E/F} \mathbb{G}_m$ as an elliptic maximal torus of $G=\mathrm{GL}_n$. We can construct an admissible embedding of L-groups ${}^LT\hookrightarrow {}^LG$ using Langlands-Shelstad $χ$-data. Such embedding gives rise to an induced representation of the Weil group $W_F$ of $F$ from a character of $W_E$. The relation between induced representations and admissible embeddings provides a different interpretation of the work of Bushnell-Henniart on the essentially tame local Langlands correspondence.

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arXiv2013

On finite symplectic modules arising from supercuspidal representations

Let $F$ be a non-Archimedean local field with finite residue field. Let $\mathcal{A}^{et}_n(F)$ be the collection of isomorphism classes of essentially tame irreducible supercuspidal representations of $\mathrm{GL}_n(F)$ studied by Bushnell-Henniart. It is known that we can parameterize $\mathcal{A}^{et}_n(F)$ by the collection $P_n(F)$ of equivalence classes of admissible pairs $(E, ξ)$ consisting of a tamely ramified extension $E/F$ of degree $n$ and an $F$-admissible character $ξ$ of $E^\times$. We are interested in a finite symplectic module $V = V(ξ)$ arising from the construction of the supercuspidal representation from the character $ξ$. This module $V$ is known to admit an orthogonal decomposition with respect to a symplectic form depending on $ξ$. We work with a fixed ambient module $U$ containing $V$ and show that $U$ decomposes in a way analogous to the root space decomposition of the Lie algebra $\mathfrak{gl}_n(F)$. We then obtain a complete orthogonal decomposition of the submodule $V$ by restriction. Such decomposition relates the finite symplectic module of a supercuspidal and certain admissible embedding of L-groups. This relation provides a different interpretation on the essentially tame local Langlands correspondence.

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arXiv2013

Admissible embedding of L-groups and essentially tame local Langlands correspondence

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Based on the previous results of the author, we can describe the Langlands parameter of an essentially tame supercuspidal representation of $G(F)$ by those admissible embeddings of L-groups constructed by Langlands and Shelstad. We therefore provide a different interpretation on Bushnell and Henniart's essentially tame local Langlands correspondence.

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arXiv2012

Transfer relations in essentially tame local Langlands correspondence

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Bushnell and Henniart described the essentially tame local Langlands correspondence of $G(F)$ using rectifiers, which are certain characters defined on tamely ramified elliptic maximal tori of $G(F)$. They obtained such result by studying the automorphic induction character identity. We relate this identity to the spectral transfer character identity, based on the theory of twisted endoscopy of Kottwitz, Langlands, and Shelstad. In this article, we establish the following two main results. (i) To show that the automorphic induction character identity is equal to the spectral transfer character identity, when both are normalized by the same Whittaker data. (ii) To express the essentially tame local Langlands correspondence using admissible embeddings constructed by Langlands-Shelstad $χ$-data, and to relate Bushnell-Henniart's rectifiers to certain transfer factors.

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