Genericity of Representations of p-Adic Sp2n and Local Langlands Parameters

2011 ◽  
Vol 63 (5) ◽  
pp. 1107-1136 ◽  
Author(s):  
Baiying Liu
Keyword(s):  
Local Field ◽  
Symplectic Group ◽  
Rational Points ◽  
Descent Method ◽  
Irreducible Representations ◽  
Local Factors ◽  
Langlands Parameters ◽  
Repre Sentations

Abstract Let G be the F-rational points of the symplectic group Sp2n, where F is a non-Archimedean local field of characteristic 0. Cogdell, Kim, Piatetski-Shapiro, and Shahidi constructed local Lang- lands functorial lifting from irreducible generic representations of G to irreducible representations of GL2n+1(F). Jiang and Soudry constructed the descent map from irreducible supercuspidal repre- sentations of GL2n+1(F) to those of G, showing that the local Langlands functorial lifting from the irreducible supercuspidal generic representations is surjective. In this paper, based on above results, using the same descent method of studying SO2n+1 as Jiang and Soudry, we will show the rest of local Langlands functorial lifting is also surjective, and for any local Langlands parameter , we construct a representation such that and ¾ have the same twisted local factors. As one application, we prove the G-case of a conjecture of Gross-Prasad and Rallis, that is, a local Langlands parameter is generic, i.e., the representation attached to is generic, if and only if the adjoint L-function of is holomorphic at s = 1. As another application, we prove for each Arthur parameter , and the corresponding local Langlands parameter , the representation attached to is generic if and only if is tempered.

A Functional Equation for Degree two Local Factors

1985 ◽  
Vol 28 (3) ◽  
pp. 355-371
Author(s):  
Paul Gérardin ◽  
Wen-Ch'ing Winnie Li
Keyword(s):  
Functional Equation ◽  
Local Field ◽  
Fourier Transforms ◽  
Discrete Series ◽  
Irreducible Representations ◽  
Quaternion Group ◽  
Langlands Correspondence ◽  
Local Factors

AbstractWe show that the Fourier transforms of the admissible irreducible representations of the group GL2 over a nonarchimedian local field F are characterized by a functional equation (MF). We also prove that the functions satisfying (MF) and having at most one pole are exactly the Fourier transforms of the irreducible representations of the quaternion group H over F. The Jacquet-Langlands correspondence between irreducible representations of H and discrete series of GL2 then follows immediately from our criteria.

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 Conjectures and results about parabolic induction of representations of $${\text {GL}}_n(F)$$

2020 ◽  
Vol 222 (3) ◽  
pp. 695-747
Author(s):  
Erez Lapid ◽  
Alberto Mínguez
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Irreducible Representations ◽  
General Linear ◽  
Parabolic Induction ◽  
Geometric Conditions ◽  
Irreducible Components ◽  
Special Cases

Abstract In 1980 Zelevinsky introduced certain commuting varieties whose irreducible components classify complex, irreducible representations of the general linear group over a non-archimedean local field with a given supercuspidal support. We formulate geometric conditions for certain triples of such components and conjecture that these conditions are related to irreducibility of parabolic induction. The conditions are in the spirit of the Geiss–Leclerc–Schröer condition that occurs in the conjectural characterization of $$\square $$ □ -irreducible representations. We verify some special cases of the new conjecture and check that the geometric and representation-theoretic conditions are compatible in various ways.

scholarly journals Doubling zeta integrals and local factors for metaplectic groups

2012 ◽  
Vol 208 ◽  
pp. 67-95 ◽  
Author(s):  
Wee Teck Gan
Keyword(s):  
Irreducible Representations ◽  
Metaplectic Groups ◽  
Local Factors

AbstractWe develop the theory of the doubling zeta integral of Piatetski-Shapiro and Rallis for metaplectic groups Mp2n, and we use it to give precise definitions of the local γ-factors, L-factors, and ε-factors for irreducible representations of Mp2n × GL1, following the footsteps of Lapid and Rallis.

scholarly journals Stability of Asai local factors for GL(2)

2020 ◽  
pp. 1-18
Author(s):  
Yeongseong Jo ◽  
M. Krishnamurthy
Keyword(s):  
Local Field ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Quadratic Algebra ◽  
Local Factors ◽  
The Stability ◽  
Gamma Factor

Let [Formula: see text] be a non-archimedean local field of characteristic not equal to 2 and let [Formula: see text] be a quadratic algebra. We prove the stability of local factors attached to irreducible admissible (complex) representations of [Formula: see text] via the Rankin–Selberg method under highly ramified twists. This includes both the Asai as well as the Rankin–Selberg local factors attached to pairs. Our method relies on expressing the gamma factor as a Mellin transform using Bessel functions.

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.

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