arXiv subjects

Kwangho Choiy

Publications and source records attributed to Kwangho Choiy.

14 recordsLinked to original sources

arXiv2025

Distinguished representations for $\rm{SL}(n,F)$

Let $F$ be a finite field, and let $\mathbb{E}$ be either a quadratic field extension $E/F$ or the split algebra $F \oplus F$. We study distinguished representations of $\rm{SL}_{2n}(F)$ by the subgroup $H_{\flat} := \rm{SL}_{2n}(F) \cap \rm{GL}_{n}(\mathbb{E})$, which is a variation of the work of Anandavardhanan and Prasad on distinguished representations of $\rm{SL}_{n}(\mathbb{E})$ by the subgroup $\rm{SL}_n(F)$. This is in a similar framework of our earlier work of a $p$-adic non-split variation of Anandavardhanan-Prasad over finite fields. We give a formula for the dimension of the complex vector space $\rm{Hom}_{H_{\flat}}(π_{\flat}, 1)$ in terms of certain characters of $F^{\times}$, where $π_{\flat}$ is an irreducible representation which is also distinguished by $H_{\flat}$.

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arXiv2025

Component groups for non-supercuspidal $L$-parameters for $p$-adic $\mathrm{SL}_3$

We explicitly classify all the component groups associated to the non-supercuspidal, tempered $L$-parameters of $\mathrm{SL}_3(F)$ for a $p$-adic field $F$ of characteristic $0$ by direct case-by-case computations in $\mathrm{PGL}_3(\mathbb{C})$, following earlier work for $\mathrm{SL}_2$ by Labesse-Langlands and Shelstad.

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arXiv2025

Distinguished Representations for $\rm{SL}_n(D)$ where $D$ is a quaternion division algebra over a $p$-adic field

Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\rm{SL}_{n}^{*}(E) = {\rm{GL}}_{n}(E) \cap \rm{SL}_{n}(D)$. We study distinguished representations of $\rm{SL}_{n}(D)$ by the subgroup $\rm{SL}_{n}^{*}(E)$. Let $π$ be an irreducible admissible representation of $\rm{SL}_{n}(D)$ which is distinguished by $\rm{SL}_{n}^{*}(E)$. We give a multiplicity formula, i.e. a formula for the dimension of the $\mathbb{C}$-vector space ${\rm{Hom}}_{\rm{SL}_{n}^{*}(E)} (π, \mathbbm{1})$, where $\mathbbm{1}$ denotes the trivial representation of $\rm{SL}_{n}^{*}(E)$. This work is a non-split inner form analog of a work by Anandavardhanan-Prasad which gives a multiplicity formula for $\rm{SL}_{n}(F)$-distinguished irreducible admissible representation of $\rm{SL}_{n}(E)$.

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arXiv2023

Representations of the $p$-Adic $GSpin_4$ and $GSpin_6$ and the Adjoint L-Function

We prove a conjecture of B. Gross and D. Prasad about determination of generic $L$-packets in terms of the analytic properties of the adjoint $L$-function for $p$-adic general even spin groups of semi-simple ranks 2 and 3. We also explicitly write the adjoint $L$-function for each $L$-packet in terms of the local Langlands $L$-functions for the general linear groups.

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arXiv2022

Weakly unramified representations, finite morphisms, and Knapp-Stein $R$-groups

We transfer Knapp-Stein $R$-groups for unitary weakly unramified characters between a $p$-adic quasi-split group and its non-quasi-split inner forms, and provide the structure of those $R$-groups for a general connected reductive group over a $p$-adic field. This work supports previous studies on the behavior of $R$-groups between inner forms, and extends Keys' classification for unitary unramified cases of simply-connected, almost simple, semi-simple groups.

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arXiv2022

On compatibility in restriction of Arthur's conjecture for $R$-groups

We study the compatibility of Arthur's conjecture for $R$-groups in the restriction of discrete series representations from Levi subgroups of a $p$-adic group to those of its closed subgroup having the same derived group. The compatibility is conditional as it holds under the conjectural local Langlands correspondence in general. This work is applied to several cases and further suggests a uniform way to verify the Arthur's conjecture for $R$-groups in the setting.

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arXiv2018

Generalized pseudo-coefficients of discrete series of $p$-adic groups

Let $G$ be a connected reductive group over a $p$-adic field $F$ of characteristic 0 and let $M$ be an $F$-Levi subgroup of $G.$ Given a discrete series representation $σ$ of $M(F),$ we prove that there exists a locally constant and compactly supported function on $M(F),$ which generalizes a pseudo-coefficient of $σ.$ This function satisfies similar properties to the pseudo-coefficient, and its lifting to $G(F)$ is applied to the Plancherel formula.

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arXiv2018

On multiplicity in restriction of tempered representations of $p$-adic groups

We establish an equality between two multiplicities: one in the restriction of tempered representations of a $p$-adic group to its closed subgroup with the same derived group; and one occurring in their corresponding component groups in Langlands dual sides, so-called $\mathcal{S}$-groups, under working hypotheses about the tempered local Langlands conjecture and the internal structure of tempered $L$-packets. This provides a formula of the multiplicity for $p$-adic groups by means of dimensions of irreducible representations of their $\mathcal{S}$-groups.

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arXiv2016

Local Langlands Conjecture for $p$-adic $GSpin_4,$ $GSpin_6,$ and their inner forms

We establish the local Langlands conjecture for small rank general spin groups $GSpin_4$ and $GSpin_6$ as well as their inner forms. We construct appropriate $L$-packets and prove that these $L$-packets satisfy the properties expected of them to the extent that the corresponding local factors are available. We are also able to determine the exact sizes of the $L$-packets in many cases.

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arXiv2016

Behavior of $R$-groups for $p$-adic inner forms of quasi-split special unitary groups

We study $R$-groups for $p$-adic inner forms of quasi-split special unitary groups. We prove Arthur's conjecture, the isomorphism between the Knapp-Stein $R$-group and the Langlands-Arthur $R$-group, for quasi-split special unitary groups and their inner forms. Furthermore, we investigate the invariance of the Knapp-Stein $R$-group within $L$-packets and between inner forms. This work is applied to transferring known results in the second-named author's earlier work for quasi-split special unitary groups to their non quasi-split inner forms.

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arXiv2015

The local Langlands conjecture for the $p$-adic inner form of Sp(4)

This paper proves the local Langlands conjecture for the non quasi-split inner form Sp(1,1) of Sp(4) over a p-adic field of characteristic 0, by studying the restriction of representations from the non quasi-split inner form GSp(1,1) of GSp(4) to Sp(1,1). The L-packets for Sp(1,1) are constructed based on the earlier work on the local Langlands correspondence for GSp(1,1) by Gan and Tantono. To parameterize them in terms of so-called S-groups, we establish and utilize the local Langlands correspondence for reductive dual groups which participate in the theta correspondence with Sp(1,1) and GSp(1,1). An interesting phenomenon arises when two distinct members in an L-packet of GSp(1,1) are restricted to Sp(1,1).

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arXiv2013

Invariance of R-groups between p-adic inner forms of quasi-split classical groups

We study the reducibility of parabolically induced representations of non-split inner forms of quasi-split classical groups. The isomorphism of Arthur R-groups, endoscopic R-groups and Knapp-Stein R-groups is established, as well as showing these R-groups are isomorphic to the corresponding ones for the quasi-split form. This shows R-groups are an invariant of the L-packets. The results are applied to classify the elliptic spectrum.

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arXiv2013

Transfer of Plancherel Measures for Unitary Supercuspidal Representations between p-adic Inner Forms

Let $F$ be a $p$-adic field of characteristic 0, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $Π_{i=1}^{r} SL_{n_i} \subseteq M \subseteq Π_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under \textit{the local Jacquet-Langlands type correspondence} between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Mui{ć} and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.

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arXiv2012

Transfer of R-groups between p-adic inner forms of SL_n

We study the Knapp-Stein $R$--groups for inner forms of the split group $SL_n(F),$ with $F$ a $p$--adic field of characteristic zero. Thus, we consider the groups $SL_m(D),$ with $D$ a central division algebra over $F$ of dimension $d^2,$ and $m=n/d.$ We use the generalized Jacquet-Langlands correspondence and results of the first named author to describe the zeros of Plancherel measures. Combined with a study of the behavior of the stabilizer of representations by elements of the Weyl group we are able to determine the Knapp-Stein $R$--groups in terms of those for $SL_n(F).$ We show the $R$--group for the inner form embeds as a subgroup of the $R$--group for the split form, and we characterize the quotient. We are further able to show the Knapp-Stein $R$--group is isomorphic to the Arthur, or Endoscopic $R$--group as predicted by Arthur. Finally, we give some results on multiplicities and actions of Weyl groups on $L$--packets.

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