Searcharxiv⌕ Search

arXiv subjects

Kei Yuen Chan

Publications and source records attributed to Kei Yuen Chan.

At least 19 recordsLinked to original sources
arXiv2026

Stability of the smooth Casselman-Jacquet functor

We establish and prove several results for the smooth Casselman-Jacquet submodule and quotient functors for real reductive groups. In addition to exactness, surjectivity, transitivity and globalization results, we establish a stability property for the intersection of Jacquet subspaces. Our approach is based on a family of seminorms on the Casselman-Jacquet quotient module. As an application, we establish a full version of the real Bernstein-Zelevinsky filtrations for smooth Fr\'echet representations of moderate growth.

math.RT↗
arXiv2026

An inductive Ext non-vanishing theorem for the $p$-adic general linear group

We study some homological properties of the parabolic induction functor for the $p$-adic general linear group. We obtain an embedding theorem of Ext-groups in the context of parabolic induction. As an application, we establish and prove a variation of the non-tempered Gan-Gross-Prasad conjecture in homological branching laws for $p$-adic general linear groups.

math.RT↗
arXiv2026

Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments II: Minimal sequences

Let $F$ be a non-Archimedean local field. For any irreducible smooth representation $\pi$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, we have an operation $D_{\mathfrak m}(\pi)$ to construct a simple quotient $\tau$ of a Bernstein-Zelevinsky derivative of $\pi$. This article continues the previous one to study the following poset \[ \mathcal S(\pi, \tau) :=\left\{ \mathfrak n : D_{\mathfrak n}(\pi)\cong \tau \right\} , \] where $\mathfrak n$ runs for all the multisegments. Here the partial ordering on $\mathcal S(\pi, \tau)$ comes from the Zelevinsky ordering. We show that the poset has a unique minimal multisegment. Along the way, we introduce two new ingredients: fine chain orderings and local minimizability.

math.RT↗
arXiv2026

Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments III: properties of minimal sequences

Let $F$ be a non-Archimedean local field. For an irreducible smooth representation $\pi$ of $\mathrm{GL}_n(F)$ and a multisegment $\mathfrak m$, one associates a simple quotient $D_{\mathfrak m}(\pi)$ of a Bernstein-Zelevinsky derivative of $\pi$. In the preceding article, we showed that \[ \mathcal S(\pi, \tau) :=\left\{ \mathfrak m : D_{\mathfrak m}(\pi)\cong \tau \right\} , \] has a unique minimal element under the Zelevinsky ordering, where $\mathfrak m$ runs for all multisegments. The main result of this article includes commutativity and subsequent property of the minimal sequence. At the end of this article, we conjecture some module structure arising from the minimality.

math.RT↗
arXiv2025

Discrete series for the graded Hecke algebra of type $H_{4}$

This article confirms the prediction that the set of discrete series central character for the graded (affine) Hecke algebra of type $H_4$ coincides with the set of the Heckman-Opdam central characters. Combining with previous cases of Kazhdan-Lusztig, Kriloff, Kriloff-Ram, Opdam-Solleveld, Ciubotaru-Opdam, this completes the classification of discrete series for all the graded Hecke algebras of positive parameters. Main tools include construction of calibrated modules and construction of certain minimally induced modules for discrete series. We also study the anti-sphericiity and Ext-branching laws for some discrete series.

math.RT↗
arXiv2025

Algorithms for parabolic inductions and Jacquet modules in $\mathrm{GL}_n$

In this article, we present algorithms for computing parabolic inductions and Jacquet modules for the general linear group $G$ over a non-Archimedean local field. Given the Zelevinsky data or Langlands data of an irreducible smooth representation $\pi$ of $G$ and an essentially square-integrable representation $\sigma$, we explicitly determine the Jacquet module of $\pi$ with respect to $\sigma$ and the socle of the normalized parabolic induction $\pi \times \sigma$. Our result builds on and extends some previous work of M\oe glin-Waldspurger, Jantzen, M\'inguez, and Lapid-M\'inguez, and also uses other methods such as sequences of derivatives and an exotic duality. As an application, we give a simple algorithm for computing the highest derivative multisegment and an algorithm for computing the Langlands parameter of the highest Bernstein-Zelevinsky derivatives.

math.RT↗
arXiv2024

Dual of the Geometric Lemma and the Second Adjointness Theorem for $p$-adic reductive groups

Let $P,Q$ be standard parabolic subgroups of a $p$-adic reductive group $G$. We study the smooth dual of the filtration on a parabolically induced module arising from the geometric lemma associated to the cosets $P\setminus G/Q$. We prove that the dual filtration coincides with the filtration associated to the cosets $P\setminus G/Q^-$ via the Bernstein-Casselman canonical pairing from the second adjointness of parabolic induction. This result generalizes a result of Bezrukavnikov-Kazhdan on the explicit description in the second adjointness. Along the way, we also study some group theoretic results.

math.RT↗
arXiv2024

Duality for generalized Gan-Gross-Prasad relevant pairs for $p$-adic $\mathrm{GL}_n$

The main goal of this article is to formulate a notion, called a generalized GGP relevant pair, governing the quotient branching law for $p$-adic general linear groups. Such notion relies on a commutation relation between derivatives (from Jacquet functors) and integrals (from parabolic inductions), for which we provide both representation-theoretic and combinatorial perspectives. Our main result proves a duality on those relevant pairs, which is compatible with a dual restriction in branching law.

math.RT↗
arXiv2024

On the product functor on inner forms of the general linear group over a non-Archimedean local field

Let $G_n$ be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation $σ$ of $G_n$. Lapid-Mínguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form $π\boxtimes σ$ when $π$ is a segment representation. We show that their criteria can be used to define a full subcategory of the category of smooth representation of some $G_m$, on which the parabolic induction functor $τ\mapsto τ\times σ$ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.

math.RT↗
arXiv2023

Some methods of computing first extensions between modules of graded Hecke algebras

In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen filtrations. Our approach involves the use of information from the Langlands classification as well as some deeper understanding on some structure of some modules. Such module arises from the image of a Knapp-Stein type intertwining operator and is a quotient of a generalized standard module. Along the way, we also deduce some results on the blocks for finite-dimensional modules of graded Hecke alebras. As an application, we compute the Ext-groups for irreducible modules in a block for the graded Hecke algebra of type $C_3$, assuming the truth of a version of Jantzen conjecture.

math.RT↗
arXiv2023

On the Lefschetz Principle for $\mathrm{GL}(n,\mathbb{C})$ and $\mathrm{GL}(m,\mathbb{Q}_p)$

We construct an exact functor from the category of Harish-Chandra modules of $\mathrm{GL}_n(\mathbb C)$ to the category of finite-dimensional modules of graded Hecke algebras of type A. We show that the functor preserves parabolically induced modules, standard modules, irreducible modules, unitary modules and Dirac series. We also use the functor to connect a Bernstein-Zelevinsky type functor for graded Hecke algebra side to the tensor product for $\mathrm{GL}_n(\mathbb C)$ side. Some applications are also discussed.

math.RT↗
arXiv2023

Ext-Multiplicity Theorem for Standard Representations of $(\mathrm{GL}_{n+1},\mathrm{GL}_n)$

Let $π_1$ be a standard representation of $\mathrm{GL}_{n+1}(F)$ and let $π_2$ be the smooth dual of a standard representation of $\mathrm{GL}_n(F)$. When $F$ is non-Archimedean, we prove that $\mathrm{Ext}^i_{\mathrm{GL}_n(F)}(π_1, π_2)$ is $\cong \mathbb C$ when $i=0$ and vanishes when $i \geq 1$. The main tool of the proof is a notion of left and right Bernstein-Zelevinsky filtrations. An immediate consequence of the result is to give a new proof on the multiplicity at most one theorem. Along the way, we also study an application of an Euler-Poincaré pairing formula of D. Prasad on the coefficients of Kazhdan-Lusztig polynomials. When $F$ is an Archimedean field, we use the left-right Bruhat-filtration to prove a multiplicity result for the equal rank Fourier-Jacobi models of standard principal series.

math.RT↗
arXiv2022

Quotient branching law for $p$-adic $(\mathrm{GL}_{n+1}, \mathrm{GL}_n)$ I: generalized Gan-Gross-Prasad relevant pairs

Let $G_n=\mathrm{GL}_n(F)$ be the general linear group over a non-Archimedean local field $F$. We formulate and prove a necessary and sufficient condition on determining when \[ \mathrm{Hom}_{G_n}(\pi, \pi') \neq 0 \] for irreducible smooth representations $\pi$ and $\pi'$ of $G_{n+1}$ and $G_n$ respectively. This resolves the problem of the quotient branching law. We also prove that any simple quotient of a Bernstein-Zelevinsky derivative of an irreducible representation can be constructed by a sequence of derivatives of essentially square-integrable representations. This result transferred to affine Hecke algebras of type A gives a generalization of the classical Pieri's rule of symmetric groups. One key new ingredient is a characterization of the layer in the Bernstein-Zelevinsky filtration that contributes to the branching law, obtained by the multiplicity one theorem for standard representations, which also gives a refinement of the branching law. Another key new ingredient is constructions of some branching laws and simple quotients of Bernstein-Zelevinsky derivatives by taking certain highest derivatives.

math.RT↗
arXiv2021

Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments I: reduction to combinatorics

Let $F$ be a local non-Archimedean field. A sequence of derivatives of generalized Steinberg representations can be used to construct simple quotients of Bernstein-Zelevinsky derivatives of irreducible representations of $\mathrm{GL}_n(F)$. In the first of a series of articles, we introduce a notion of a highest derivative multisegment, which in turn gives a combinatorial approach to study problems about those simple quotients. We also prove a double derivative result along the way.

math.RT↗
arXiv2020

Homological branching law for $(\mathrm{GL}_{n+1}(F), \mathrm{GL}_n(F))$: projectivity and indecomposability

Let $F$ be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main result shows that each Bernstein component of an irreducible smooth representation of $\mathrm{GL}_{n+1}(F)$ restricted to $\mathrm{GL}_n(F)$ is indecomposable. We also classify all irreducible representations which are projective when restricting from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main tool of our study is a notion of left and right derivatives, extending some previous work joint with Gordan Savin. As a by-product, we also determine the branching law in the opposite direction.

math.RT↗