scholarly journals Endo-parameters for p-adic classical groups

2020 ◽  
Author(s):  
Robert Kurinczuk ◽  
Daniel Skodlerack ◽  
Shaun Stevens
Keyword(s):  
Local Field ◽  
Classical Group ◽  
Linear Groups ◽  
Closed Field ◽  
Classical Groups ◽  
Langlands Correspondence ◽  
General Linear Groups ◽  
Langlands Parameters ◽  
Local Langlands Correspondence ◽  
Residual Characteristic

Abstract For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field $${\mathbf {C}}$$ C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal $${\mathbf {C}}$$ C -representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.

ON THE LOCAL LANGLANDS CORRESPONDENCE FOR SPLIT CLASSICAL GROUPS OVER LOCAL FUNCTION FIELDS

2015 ◽  
Vol 16 (5) ◽  
pp. 987-1074 ◽  
Author(s):  
Radhika Ganapathy ◽  
Sandeep Varma
Keyword(s):  
Representation Theory ◽  
Function Fields ◽  
Local Fields ◽  
Linear Groups ◽  
Classical Groups ◽  
Local Function ◽  
Langlands Correspondence ◽  
General Linear Groups ◽  
Residue Characteristic ◽  
Local Langlands Correspondence

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

scholarly journals THE -MODULAR LOCAL LANGLANDS CORRESPONDENCE AND LOCAL CONSTANTS

2019 ◽  
pp. 1-51 ◽  
Author(s):  
R. Kurinczuk ◽  
N. Matringe
Keyword(s):  
Prime Number ◽  
Local Field ◽  
Equivalence Classes ◽  
Modular Representations ◽  
Langlands Correspondence ◽  
Weil Group ◽  
Local Langlands Correspondence ◽  
Residual Characteristic

Let  $F$ be a non-archimedean local field of residual characteristic  $p$ , $\ell \neq p$ be a prime number, and  $\text{W}_{F}$ the Weil group of  $F$ . We classify equivalence classes of  $\text{W}_{F}$ -semisimple Deligne  $\ell$ -modular representations of  $\text{W}_{F}$ in terms of irreducible  $\ell$ -modular representations of  $\text{W}_{F}$ , and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the  $\ell$ -modular local Langlands correspondence which satisfies a preservation of local constants statement for pairs of generic representations.

scholarly journals Local Langlands Correspondence for Regular Supercuspidal Representations of GL(n)

2020 ◽  
Author(s):  
Masao Oi ◽  
Kazuki Tokimoto
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
General Linear Groups ◽  
The Difference ◽  
Local Langlands Correspondence

Abstract In this paper, we prove the coincidence of Kaletha’s recent construction of the local Langlands correspondence for regular supercuspidal representations with Harris–Taylor’s one in the case of general linear groups. The keys are Bushnell–Henniart’s essentially tame local Langlands correspondence and Tam’s result on Bushnell–Henniart’s rectifiers. By combining them, our problem is reduced to an elementary root-theoretic computation on the difference between Kaletha’s and Tam’s $\chi $-data.

scholarly journals Cuspidal ℓ-modular representations of p-adic classical groups

2020 ◽  
Vol 2020 (764) ◽  
pp. 23-69 ◽  
Author(s):  
Robert Kurinczuk ◽  
Shaun Stevens
Keyword(s):  
Local Field ◽  
Classical Group ◽  
Irreducible Representations ◽  
Closed Field ◽  
Modular Representations ◽  
Algebraically Closed Field ◽  
Fundamental Step ◽  
Residual Characteristic ◽  
Cuspidal Representations

AbstractFor a classical group over a non-archimedean local field of odd residual characteristic p, we construct all cuspidal representations over an arbitrary algebraically closed field of characteristic different from p, as representations induced from a cuspidal type. We also give a fundamental step towards the classification of cuspidal representations, identifying when certain cuspidal types induce to equivalent representations; this result is new even in the case of complex representations. Finally, we prove that the representations induced from more general types are quasi-projective, a crucial tool for extending the results here to arbitrary irreducible representations.

scholarly journals Sur les -blocs de niveau zéro des groupes -adiques

2018 ◽  
Vol 154 (7) ◽  
pp. 1473-1507
Author(s):  
Thomas Lanard
Keyword(s):  
Abelian Category ◽  
Langlands Correspondence ◽  
Parabolic Induction ◽  
Langlands Parameters ◽  
Tits Building ◽  
Local Langlands Correspondence

Let $G$ be a $p$-adic group that splits over an unramified extension. We decompose $\text{Rep}_{\unicode[STIX]{x1D6EC}}^{0}(G)$, the abelian category of smooth level $0$ representations of $G$ with coefficients in $\unicode[STIX]{x1D6EC}=\overline{\mathbb{Q}}_{\ell }$ or $\overline{\mathbb{Z}}_{\ell }$, into a product of subcategories indexed by inertial Langlands parameters. We construct these categories via systems of idempotents on the Bruhat–Tits building and Deligne–Lusztig theory. Then, we prove compatibilities with parabolic induction and restriction functors and the local Langlands correspondence.

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.

PARITY OF THE LANGLANDS PARAMETERS OF CONJUGATE SELF-DUAL REPRESENTATIONS OF AND THE LOCAL JACQUET–LANGLANDS CORRESPONDENCE

2019 ◽  
Vol 19 (6) ◽  
pp. 2017-2043
Author(s):  
Yoichi Mieda
Keyword(s):  
Local Field ◽  
The Self ◽  
Supercuspidal Representation ◽  
Langlands Correspondence ◽  
Dual Representations ◽  
Langlands Parameters ◽  
Dual Case

We determine the parity of the Langlands parameter of a conjugate self-dual supercuspidal representation of $\text{GL}(n)$ over a non-archimedean local field by means of the local Jacquet–Langlands correspondence. It gives a partial generalization of a previous result on the self-dual case by Prasad and Ramakrishnan.

scholarly journals Unitary dual of GL(n) at archimedean places and global Jacquet–Langlands correspondence

2010 ◽  
Vol 146 (5) ◽  
pp. 1115-1164 ◽  
Author(s):  
A. I. Badulescu ◽  
D. Renard
Keyword(s):  
General Linear ◽  
Linear Groups ◽  
Automorphic Representations ◽  
Langlands Correspondence ◽  
General Linear Groups ◽  
Multiplicity One ◽  
Unitary Dual ◽  
Strong Multiplicity ◽  
Local Results

AbstractIn a paper by Badulescu [Global Jacquet–Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), 383–438], results on the global Jacquet–Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over ℝ, ℂ or ℍ which are of independent interest.

Conjugacy classes of small sizes in the linear and unitary groups

2013 ◽  
Vol 16 (6) ◽  
Author(s):  
Shawn T. Burkett ◽  
Hung Ngoc Nguyen
Keyword(s):  
Conjugacy Classes ◽  
Unitary Groups ◽  
General Linear ◽  
Linear Groups ◽  
Classical Groups ◽  
General Linear Groups ◽  
Finite Classical Groups

Abstract.Using the classical results of G. E. Wall on the parametrization and sizes of (conjugacy) classes of finite classical groups, we present some gap results for the class sizes of the general linear groups and general unitary groups as well as their variations. In particular, we identify the classes in

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.

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