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 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.

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 Young modules for symmetric groups

2001 ◽  
Vol 71 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Karin Erdmann
Keyword(s):  
Representation Theory ◽  
Finite Groups ◽  
Symmetric Groups ◽  
General Linear ◽  
Linear Groups ◽  
Schur Algebras ◽  
General Linear Groups ◽  
Green Correspondence

AbstractLet K be a field of characteristic p. The permutation modules associated to partitions of n, usually denoted as Mλ, play a central role not only for symmetric groups but also for general linear groups, via Schur algebras. The indecomposable direct summands of these Mλ were parametrized by James; they are now known as Young modules; and Klyachko and Grabmeier developed a ‘Green correspondence’ for Young modules. The original parametrization used Schur algebras; and James remarked that he did not know a proof using only the representation theory of symmetric groups. We will give such proof, and we will at the same time also prove the correspondence result, by using only the Brauer construction, which is valid for arbitrary finite groups.

scholarly journals MODULAR DECOMPOSITION NUMBERS OF CYCLOTOMIC HECKE AND DIAGRAMMATIC CHEREDNIK ALGEBRAS: A PATH THEORETIC APPROACH

2018 ◽  
Vol 6 ◽  
Author(s):  
C. BOWMAN ◽  
A. G. COX
Keyword(s):  
Representation Theory ◽  
Symmetric Groups ◽  
Upper Bounds ◽  
Linear Groups ◽  
Theoretic Approach ◽  
Strong Linkage ◽  
Cherednik Algebras ◽  
Arbitrary Characteristic ◽  
General Linear Groups ◽  
Decomposition Numbers

We introduce a path theoretic framework for understanding the representation theory of (quantum) symmetric and general linear groups and their higher-level generalizations over fields of arbitrary characteristic. Our first main result is a ‘super-strong linkage principle’ which provides degree-wise upper bounds for graded decomposition numbers (this is new even in the case of symmetric groups). Next, we generalize the notion of homomorphisms between Weyl/Specht modules which are ‘generically’ placed (within the associated alcove geometries) to cyclotomic Hecke and diagrammatic Cherednik algebras. Finally, we provide evidence for a higher-level analogue of the classical Lusztig conjecture over fields of sufficiently large characteristic.

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 Estimating deep Littlewood-Richardson Coefficients

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Hariharan Narayanan
Keyword(s):  
Representation Theory ◽  
Approximation Scheme ◽  
General Linear ◽  
Linear Groups ◽  
General Linear Groups ◽  
International Audience ◽  
Structure Constants ◽  
Strongly Polynomial

International audience Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups $(GL_n)$. The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for Littlewood-Richardson coefficients corresponding to indices sufficiently far from the boundary of the Littlewood Richardson cone. 2. A proof of approximate log-concavity of the above mentioned class of Littlewood-Richardson coefficients. Coefficients de Littlewood Richardson sont des constantes de structure apparaissant dans la théorie de la représentation des groupes linéaires généraux $(GL_n)$. Les principaux résultats de cette étude sont les suivants: 1. Un schéma d’approximation polynomiale randomisée fortement pour des coefficients de Littlewood-Richardson correspondant aux indices suffisamment loin de la limite du cône Littlewood Richardson. 2. Une preuve de l’approximatif log-concavité de la classe de coefficients de Littlewood-Richardson mentionné ci-dessus.

scholarly journals DECOMPOSITION THEOREMS FOR AUTOMORPHISM GROUPS OF TREES

2020 ◽  
pp. 1-9
Author(s):  
MAX CARTER ◽  
GEORGE A. WILLIS
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Continuous Homomorphism ◽  
Local Fields ◽  
General Linear ◽  
Linear Groups ◽  
Closed Range ◽  
Regular Tree ◽  
General Linear Groups ◽  
Decomposition Theorems

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

scholarly journals Branching laws for classical groups: the non-tempered case

2020 ◽  
Vol 156 (11) ◽  
pp. 2298-2367
Author(s):  
Wee Teck Gan ◽  
Benedict H. Gross ◽  
Dipendra Prasad
Keyword(s):  
Representation Theory ◽  
Local Fields ◽  
Classical Groups ◽  
Global Fields ◽  
Root Numbers ◽  
Branching Laws ◽  
Local Root ◽  
Local Root Numbers

This paper generalizes the Gan–Gross–Prasad (GGP) conjectures that were earlier formulated for tempered or more generally generic L-packets to Arthur packets, especially for the non-generic L-packets arising from Arthur parameters. The paper introduces the key notion of a relevant pair of Arthur parameters that governs the branching laws for ${{\rm GL}}_n$ and all classical groups over both local fields and global fields. It plays a role for all the branching problems studied in Gan et al. [Symplectic local root numbers, central critical L-values and restriction problems in the representation theory of classical groups. Sur les conjectures de Gross et Prasad. I, Astérisque 346 (2012), 1–109] including Bessel models and Fourier–Jacobi models.

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