arXiv subjects

Yeansu Kim

Publications and source records attributed to Yeansu Kim.

11 recordsLinked to original sources

arXiv2025

Aubert duals of strongly positive representations for metaplectic groups

We determine the Aubert duals of strongly positive representations of the metaplectic group $\widetilde{Sp}(n)$ over a non-Archimedean local field $F$ of characteristic different from two. Using the classification of Matić and an explicit analysis of Jacquet modules, we describe these duals in terms of precise inducing data. Our results extend known descriptions for classical groups to the metaplectic groups case and clarify the role of Aubert duality for non-linear covering groups, providing a foundation for future applications to the study of unitary representations for those cases. Furthermore, We are able to show that the same method applies to odd general spin groups $GSpin(2n+1)$, yielding an explicit description of Aubert duals in that setting as well.

math.NT↗

arXiv2025

The local converse theorem for quasi-split $O_{2n}$ and $SO_{2n}$

Let $F$ be a non-archimedean local field of characteristic not equal to 2. In this paper, we prove the local converse theorem for quasi-split $Ø_{2n}(F)$ and $\SO_{2n}(F)$, via the description of the local theta correspondence between $Ø_{2n}(F)$ and $\Sp_{2n}(F)$. More precisely, as a main step, we explicitly describe the precise behavior of the $γ$-factors under the correspondence. Furthermore, we apply our results to prove the weak rigidity theorems for irreducible generic cuspidal automorphic representations of $Ø_{2n}(\A)$ and $\SO_{2n}(\mathbb{A})$, respectively, where $\A$ is a ring of adele of a global number field $L$.

math.NT↗

arXiv2025

$L$-packets and the generic Arthur packet conjectures for even unitary similitude groups

We establish the generic local Langlands correspondence by showing the equality of the Langlands-Shahidi $L$-functions and Artin $L$-functions in the case of even unitary similitude groups. As an application, we prove both weak and strong versions of the generic Arthur packet conjectures in the cases of even unitary similitude groups and even unitary groups. We further describe (not necessarily generic) $L$-packets for even unitary similitude groups and establish their expected properties, including Shahidi's conjecture, the finiteness of $L$-packets, and other related results.

math.NT↗

arXiv2024

Multiplicity One Theorem for General Spin Groups: The Archimedean Case

Let $\GSpin(V)$ (resp. $\GPin(V)$) be a general spin group (resp. a general Pin group) associated with a nondegenerate quadratic space $V$ of dimension $n$ over an Archimedean local field $F$. For a nondegenerate quadratic space $W$ of dimension $n-1$ over $F$, we also consider $\GSpin(W)$ and $\GPin(W)$. We prove the multiplicity-at-most-one theorem in the Archimedean case for a pair of groups ($\GSpin(V), \GSpin(W)$) and also for a pair of groups ($\GPin(V), \GPin(W)$); namely, we prove that the restriction to $\GSpin(W)$ (resp. $\GPin(W)$) of an irreducible Casselman-Wallach representation of $\GSpin(V)$ (resp. $\GPin(V)$) is multiplicity free.

math.RT↗

arXiv2022

Discrete series of odd general spin groups

We obtain a classification of discrete series representations of odd general spin groups, generalizing the Mœglin-Tadić classification for classical groups. Using mostly algebraic methods, available in both classical and the odd general spin case, we provide a simplified and shorter version of the classification of non-cuspidal discrete series in terms of the admissible triples. The presented classification can potentially be directly applied to many other reductive $p$-adic groups.

math.RT↗

arXiv2020

Possible probability and irreducibility of balanced non-transitive dice

We construct irreducible balanced non-transitive sets of $n$-sided dice for any positive integer $n$, which was raised in \cite[Question 5.2]{SS17}. One main tool of the construction is to study so-called fair sets of dice. Furthermore, we also study the distribution of the probabilities of balanced non-transitive sets of dice. For a lower bound, we show that the probability could be arbitrarily close to $\frac{1}{2}$ and for a upper bound, we construct a balanced non-transitive set of dice whose probability is $\frac{1}{2} + \frac{13-\sqrt{153}}{24} \approx \frac{1}{2} + \frac{1}{9.12}.$

math.PR↗

arXiv2020

The Bernstein projector determined by a weak associate class of good cosets

Let $G$ be a reductive group over a $p$-adic field $F$ of characteristic zero, with $p \gg 0$. In [Kim04], J.-L. Kim studied an equivalence relation called weak associativity on the set of unrefined minimal $K$-types for $G$ in the sense of A. Moy and G. Prasad. Following [Kim04], we attach to the set $\overline{\mathfrak s}$ of good $K$-types in a weak associate class of positive-depth unrefined minimal $K$-types a $G(F)$-invariant open and closed subset $\mathfrak g(F)_{\overline{\mathfrak s}}$ of the Lie algebra $\mathfrak g(F)$ of $G(F)$, and a subset $\tilde G_{\overline{\mathfrak s}}$ of the admissible dual $\tilde G$ of $G(F)$ consisting of those representations containing an unrefined minimal $K$-type that belongs to $\overline{\mathfrak s}$. Then $\tilde G_{\overline{\mathfrak s}}$ is the union of finitely many Bernstein components for $G$, so that we can consider the Bernstein projector $E_{\overline{\mathfrak s}}$ that it determines. We show that $E_{\overline{\mathfrak s}}$ vanishes outside the Moy--Prasad $G(F)$-domain $G(F)_r \subset G(F)$, and reformulate a result of Kim as saying that the restriction of $E_{\overline{\mathfrak s}}$ to $G(F)_r$, pushed forward via the logarithm to the Moy--Prasad $G(F)$-domain $\mathfrak g(F)_r \subset \mathfrak g(F)$, agrees on $\mathfrak g(F)_r$ with the inverse Fourier transform of the characteristic function of $\mathfrak g(F)_{\overline{\mathfrak s}}$. This is a variant of one of the descriptions given by R. Bezrukavnikov, D. Kazhdan and Y. Varshavsky in arXiv:1504.01353 for the depth-$r$ Bernstein projector.

math.RT↗

arXiv2019

Degenerate principal series in the general case

Let $G_n$ denote either the group $SO(2n+1, F)$, $Sp(2n, F)$, or $GSpin(2n+1, F)$ over a non-archimedean local field of characteristic different than two. We determine all composition factors of degenerate principal series of $G_n$, using methods based on the Aubert involution and known results on irreducible subquotiens of the generalized principal series of particular type.

math.RT↗

arXiv2019

Classification of Strongly Positive Representations of Even General Unitary Groups

We explicitly construct the structure of Jacquet modules of parabolically induced representations of even unitary groups and even general unitary groups over a $p$-adic field $F$ of characteristic different than two. As an application, we obtain a classification of strongly positive discrete series representations of those groups.

math.RT↗

arXiv2017

Langlands-Shahidi $L$-functions for $GSpin$ groups and the generic Arthur packet conjecture

We prove that $L$-functions from Langlands-Shahidi method in the case of $GSpin$ groups over a non-Archimedean local field of characteristic zero are Artin $L$-functions through the local Langlands correspondence. It has an application on the proof of a weak version of the generic Arthur packet conjecture. Furthermore, we study and describe a local $L$-packet that contains a generic member in the case of $GSpin$ groups. Using this description of a local $L$-packet, we strengthen a weak version of the generic Arthur packet conjecture in those cases (local version of the generalized Ramanujan conjecture).

math.NT↗

arXiv2015

On the generic local Langlands correspondence for GSpin groups

In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of the first author which allow to reduce the problem to supercuspidal representations of Levi subgroups of $GSpin$, by constructing Langlands parameters for general generic irreducible admissible representations of $GSpin $ from the one for generic irreducible supercuspidal representations of its Levi subgroups.

math.NT↗