langlands parameters
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2021 ◽  
Vol 25 (29) ◽  
pp. 844-860
Author(s):  
Lucas Mason-Brown

In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the K \mathbf {K} -types, associated varieties, and Langlands parameters of all such representations.


Author(s):  
Petar Bakić ◽  
Marcela Hanzer

Abstract We describe explicitly the Howe correspondence for the symplectic-orthogonal and unitary dual pairs over a nonarchimedean local field of characteristic zero. More specifically, for every irreducible admissible representation of these groups, we find its first occurrence index in the theta correspondence and we describe, in terms of their Langlands parameters, the small theta lifts on all levels.


Author(s):  
Jerrod Manford Smith

Let [Formula: see text] be a non-Archimedean local field of characteristic zero. Let [Formula: see text] be the [Formula: see text]-adic symmetric space [Formula: see text], where [Formula: see text] and [Formula: see text]. We verify a conjecture of Sakellaridis and Venkatesh on the Langlands parameters of certain representations in the discrete spectrum of [Formula: see text].


Author(s):  
Robert Kurinczuk ◽  
Daniel Skodlerack ◽  
Shaun Stevens

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.


2020 ◽  
Vol 16 (07) ◽  
pp. 1449-1491
Author(s):  
Tony Feng ◽  
Niccolò Ronchetti ◽  
Cheng-Chiang Tsai

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.


2019 ◽  
Vol 19 (6) ◽  
pp. 2017-2043
Author(s):  
Yoichi Mieda

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.


2018 ◽  
Vol 154 (7) ◽  
pp. 1473-1507
Author(s):  
Thomas Lanard

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.


2018 ◽  
Vol 157 (1-2) ◽  
pp. 121-192 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Ahmed Moussaoui ◽  
Maarten Solleveld

2017 ◽  
Vol 357 (2) ◽  
pp. 775-789 ◽  
Author(s):  
Martin T. Luu ◽  
Matej Penciak

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