Globalization of Distinguished Supercuspidal Representations of GL(n)

2002 ◽  
Vol 45 (2) ◽  
pp. 220-230 ◽  
Author(s):  
Jeffrey Hakim ◽  
Fiona Murnaghan
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Cuspidal Representation ◽  
Supercuspidal Representation ◽  
Local Component ◽  
Supercuspidal Representations

AbstractAn irreducible supercuspidal representation π of G = GL(n, F), where F is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup H of G and a quasicharacter χ of H if HomH(π, χ) ≠ 0. There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to GL(n). Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

scholarly journals The distinction problem for metaplectic case

2020 ◽  
Vol 16 (06) ◽  
pp. 1161-1183
Author(s):  
Hengfei Lu
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Quaternion Algebra ◽  
Quadratic Field ◽  
Field Extension ◽  
Similar Strategy

We use the theta lifts between [Formula: see text] and [Formula: see text] to study the distinction problems for the pair [Formula: see text] where [Formula: see text] is a quadratic field extension over a nonarchimedean local field [Formula: see text] of characteristic zero and [Formula: see text] is a quaternion algebra. With a similar strategy, we give a conjectural formula for the multiplicity of distinction problem related to the pair [Formula: see text]

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 Twisting of paramodular vectors

2014 ◽  
Vol 10 (04) ◽  
pp. 1043-1065 ◽  
Author(s):  
Jennifer Johnson-Leung ◽  
Brooks Roberts
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Central Character ◽  
Admissible Representation ◽  
Quadratic Character ◽  
Irreducible Admissible Representation

Let F be a non-Archimedean local field of characteristic zero, let (π, V) be an irreducible, admissible representation of GSp (4, F) with trivial central character, and let χ be a quadratic character of F× with conductor c(χ) > 1. We define a twisting operator Tχ from paramodular vectors for π of level n to paramodular vectors for χ ⊗ π of level max (n + 2c(χ), 4c(χ)), and prove that this operator has properties analogous to the well-known GL(2) twisting operator.

scholarly journals LOCAL NEWFORMS AND FORMAL EXTERIOR SQUARE L-FUNCTIONS

2013 ◽  
Vol 09 (08) ◽  
pp. 1995-2010 ◽  
Author(s):  
MICHITAKA MIYAUCHI ◽  
TAKUYA YAMAUCHI
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Principal Series ◽  
Whittaker Function ◽  
Series Representations ◽  
Principal Series Representations ◽  
Exterior Square

Let F be a non-archimedean local field of characteristic zero. Jacquet and Shalika attached a family of zeta integrals to unitary irreducible generic representations π of GL n(F). In this paper, we show that the Jacquet–Shalika integral attains a certain L-function, the so-called formal exterior square L-function, when the Whittaker function is associated to a newform for π. By considerations on the Galois side, formal exterior square L-functions are equal to exterior square L-functions for some principal series representations.

scholarly journals THE COHOMOLOGY OF UNRAMIFIED RAPOPORT–ZINK SPACES OF EL-TYPE AND HARRIS'S CONJECTURE

2021 ◽  
pp. 1-56
Author(s):  
Alexander Bertoloni Meli
Keyword(s):  
Levi Subgroup ◽  
Admissible Representation ◽  
Supercuspidal Representation ◽  
Langlands Correspondence ◽  
Supercuspidal Representations ◽  
Grothendieck Groups ◽  
Local Langlands Correspondence

Abstract We study the l-adic cohomology of unramified Rapoport–Zink spaces of EL-type. These spaces were used in Harris and Taylor's proof of the local Langlands correspondence for $\mathrm {GL_n}$ and to show local–global compatibilities of the Langlands correspondence. In this paper we consider certain morphisms $\mathrm {Mant}_{b, \mu }$ of Grothendieck groups of representations constructed from the cohomology of these spaces, as studied by Harris and Taylor, Mantovan, Fargues, Shin and others. Due to earlier work of Fargues and Shin we have a description of $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ a supercuspidal representation. In this paper, we give a conjectural formula for $\mathrm {Mant}_{b, \mu }(\rho )$ for $\rho $ an admissible representation and prove it when $\rho $ is essentially square-integrable. Our proof works for general $\rho $ conditionally on a conjecture appearing in Shin's work. We show that our description agrees with a conjecture of Harris in the case of parabolic inductions of supercuspidal representations of a Levi subgroup.

scholarly journals Epsilon factors of symplectic type characters in the wild case

2021 ◽  
Vol 33 (2) ◽  
pp. 569-577
Author(s):  
Sazzad Ali Biswas
Keyword(s):  
Local Field ◽  
Characteristic Zero ◽  
Quadratic Extension ◽  
Multiplicative Character ◽  
In The Wild ◽  
Epsilon Factors

Abstract By work of John Tate we can associate an epsilon factor with every multiplicative character of a local field. In this paper, we determine the explicit signs of the epsilon factors for symplectic type characters of K × {K^{\times}} , where K / F {K/F} is a wildly ramified quadratic extension of a non-Archimedean local field F of characteristic zero.

scholarly journals Base change for ramified unitary groups: The strongly ramified case

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Corinne Blondel ◽  
Geo Kam-Fai Tam
Keyword(s):  
Unitary Group ◽  
Base Change ◽  
Supercuspidal Representation ◽  
Supercuspidal Representations ◽  
Parabolic Induction ◽  
Level Zero ◽  
Simple Character ◽  
The Given ◽  
Special Case ◽  
Residual Characteristic

Abstract We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

scholarly journals On the construction of tame supercuspidal representations

2021 ◽  
Vol 157 (12) ◽  
pp. 2733-2746
Author(s):  
Jessica Fintzen
Keyword(s):  
Local Field ◽  
Reductive Group ◽  
Field Extension ◽  
Supercuspidal Representations ◽  
Smooth Complex ◽  
Residual Characteristic

Let $F$ be a non-archimedean local field of residual characteristic $p \neq 2$ . Let $G$ be a (connected) reductive group over $F$ that splits over a tamely ramified field extension of $F$ . We revisit Yu's construction of smooth complex representations of $G(F)$ from a slightly different perspective and provide a proof that the resulting representations are supercuspidal. We also provide a counterexample to Proposition 14.1 and Theorem 14.2 in Yu [Construction of tame supercuspidal representations, J. Amer. Math. Soc. 14 (2001), 579–622], whose proofs relied on a typo in a reference.

scholarly journals On Types for Unramified p-Adic Unitary Groups

2008 ◽  
Vol 60 (5) ◽  
pp. 1067-1107 ◽  
Author(s):  
Kazutoshi Kariyama
Keyword(s):  
Local Field ◽  
Unitary Group ◽  
Symplectic Group ◽  
Unitary Groups ◽  
Simple Type ◽  
Supercuspidal Representation ◽  
Fixed Field ◽  
Residue Characteristic

AbstractLet F be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field F0, and let G be a symplectic group over F or an unramified unitary group over F0. Following the methods of Bushnell–Kutzko for GL(N, F), we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in G. In particular, we obtain an irreducible supercuspidal representation of G like GL(N, F).

scholarly journals Langlands lambda function for quadratic tamely ramified extensions

2019 ◽  
Vol 18 (07) ◽  
pp. 1950132
Author(s):  
Sazzad Ali Biswas
Keyword(s):  
Explicit Formula ◽  
Local Field ◽  
Characteristic Zero

Let [Formula: see text] be a quadratic tamely ramified extension of a non-Archimedean local field [Formula: see text] of characteristic zero. In this paper, we give an explicit formula for Langlands’ lambda function [Formula: see text].

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