scholarly journals Local Langlands correspondence for unitary groups via theta lifts

2021 ◽  
Vol 25 (30) ◽  
pp. 861-896
Author(s):  
Rui Chen ◽  
Jialiang Zou
Keyword(s):  
Unitary Groups ◽  
Irreducible Representations ◽  
Theta Correspondence ◽  
Langlands Correspondence ◽  
Alternative Approach ◽  
Local Langlands Correspondence

Using the theta correspondence, we extend the classification of irreducible representations of quasi-split unitary groups (the so-called local Langlands correspondence, which is due to Mok) to non quasi-split unitary groups. We also prove that our classification satisfies some good properties, which characterize it uniquely. In particular, this paper provides an alternative approach to the works of Kaletha-Mínguez-Shin-White and Mœglin-Renard.

scholarly journals Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence

2012 ◽  
Vol 148 (6) ◽  
pp. 1655-1694 ◽  
Author(s):  
Wee Teck Gan ◽  
Gordan Savin
Keyword(s):  
Irreducible Representations ◽  
Metaplectic Groups ◽  
Theta Correspondence ◽  
Langlands Correspondence ◽  
Local Langlands Correspondence ◽  
Shimura Correspondence

AbstractUsing theta correspondence, we classify the irreducible representations of Mp2n in terms of the irreducible representations of SO2n+1 and determine many properties of this classification. This is a local Shimura correspondence which extends the well-known results of Waldspurger for n=1.

scholarly journals Local Langlands correspondence and ramification for Carayol representations

2019 ◽  
Vol 155 (10) ◽  
pp. 1959-2038
Author(s):  
Colin J. Bushnell ◽  
Guy Henniart
Keyword(s):  
Langlands Correspondence ◽  
Smooth Complex ◽  
Compact Field ◽  
Weil Group ◽  
Detailed Interpretation ◽  
Simple Character ◽  
Local Langlands Correspondence ◽  
Class Of Functions ◽  
Residual Characteristic

Let $F$ be a non-Archimedean locally compact field of residual characteristic $p$ with Weil group ${\mathcal{W}}_{F}$. Let $\unicode[STIX]{x1D70E}$ be an irreducible smooth complex representation of ${\mathcal{W}}_{F}$, realized as the Langlands parameter of an irreducible cuspidal representation $\unicode[STIX]{x1D70B}$ of a general linear group over $F$. In an earlier paper we showed that the ramification structure of $\unicode[STIX]{x1D70E}$ is determined by the fine structure of the endo-class $\unicode[STIX]{x1D6E9}$ of the simple character contained in $\unicode[STIX]{x1D70B}$, in the sense of Bushnell and Kutzko. The connection is made via the Herbrand function $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ of $\unicode[STIX]{x1D6E9}$. In this paper we concentrate on the fundamental Carayol case in which $\unicode[STIX]{x1D70E}$ is totally wildly ramified with Swan exponent not divisible by $p$. We show that, for such $\unicode[STIX]{x1D70E}$, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6E9}}$ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions $\unicode[STIX]{x1D6F9}_{\unicode[STIX]{x1D6EF}}$, as $\unicode[STIX]{x1D6EF}$ varies over the set of totally wild endo-classes of Carayol type. In a separate argument, we derive a complete description of the restriction of $\unicode[STIX]{x1D70E}$ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.

scholarly journals Epipelagic Langlands parameters and L-packets for unitary groups

2020 ◽  
Vol 16 (07) ◽  
pp. 1449-1491
Author(s):  
Tony Feng ◽  
Niccolò Ronchetti ◽  
Cheng-Chiang Tsai
Keyword(s):  
Invariant Theory ◽  
Unitary Groups ◽  
General Construction ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
Langlands Parameters ◽  
New Construction ◽  
Local Langlands Correspondence ◽  
Invent Math

Reeder and Yu have recently given a new construction of a class of supercuspidal representations called epipelagic representations [M. Reeder and J.-K. Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc. 27(2) (2014) 437–477, MR 3164986]. We explicitly calculate the Local Langlands Correspondence for certain families of epipelagic representations of unitary groups, following the general construction of Kaletha [Epipelagic [Formula: see text]-packets and rectifying characters, Invent. Math. 202(1) (2015) 1–89, MR 3402796]. The interesting feature of our computation is that we find simplifications within [Formula: see text]-packets of the two novel invariants introduced in the above-mentioned paper of Kaletha, the toral invariant and the admissible L-embedding.

CNN-based Classification of Degraded Images

2020 ◽  
Vol 2020 (10) ◽  
pp. 28-1-28-7 ◽  
Author(s):  
Kazuki Endo ◽  
Masayuki Tanaka ◽  
Masatoshi Okutomi
Keyword(s):  
Neural Networks ◽  
Image Restoration ◽  
Image Classification ◽  
Convolutional Neural Networks ◽  
Deep Convolutional Neural Networks ◽  
Alternative Approach ◽  
Degraded Image ◽  
Degraded Images ◽  
Straightforward Approach

Classification of degraded images is very important in practice because images are usually degraded by compression, noise, blurring, etc. Nevertheless, most of the research in image classification only focuses on clean images without any degradation. Some papers have already proposed deep convolutional neural networks composed of an image restoration network and a classification network to classify degraded images. This paper proposes an alternative approach in which we use a degraded image and an additional degradation parameter for classification. The proposed classification network has two inputs which are the degraded image and the degradation parameter. The estimation network of degradation parameters is also incorporated if degradation parameters of degraded images are unknown. The experimental results showed that the proposed method outperforms a straightforward approach where the classification network is trained with degraded images only.

scholarly journals On the Notion of Conductor in the Local Geometric Langlands Correspondence

2017 ◽  
Vol 69 (1) ◽  
pp. 107-129
Author(s):  
Masoud Kamgarpour
Keyword(s):  
Irreducible Representation ◽  
Galois Representation ◽  
Loop Group ◽  
Langlands Correspondence ◽  
Categorical Representation ◽  
Swan Conductor ◽  
Geometric Langlands ◽  
Meromorphic Connection ◽  
Local Langlands Correspondence ◽  
Geometric Analogue

AbstractUnder the local Langlands correspondence, the conductor of an irreducible representation of Gln(F) is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

scholarly journals REFINING THE CLASSIFICATION OF THE IRREPS OF THE 1D N-EXTENDED SUPERSYMMETRY

2008 ◽  
Vol 23 (01) ◽  
pp. 37-51 ◽  
Author(s):  
ZHANNA KUZNETSOVA ◽  
FRANCESCO TOPPAN
Keyword(s):  
Quantum Mechanics ◽  
Extended Supersymmetry ◽  
Supersymmetric Quantum Mechanics ◽  
Irreducible Representations

The linear finite irreducible representations of the algebra of the 1D N-Extended Supersymmetric Quantum Mechanics are discussed in terms of their "connectivity" (a symbol encoding information on the graphs associated to the irreps). The classification of the irreducible representations with the same fields content and different connectivity is presented up to N ≤ 8.

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