DUALIZING INVOLUTIONS ON THE METAPLECTIC GL(2) à la TUPAN

Metaplectic Group ◽

Topological Argument ◽

Abstract Let F be a non-Archimedean local field of characteristic zero. Let G = GL(2, F) and $3\widetildeG = \widetilde{GL}(2,F)$ be the metaplectic group. Let τ be the standard involution on G. A well-known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we make some modifications and adapt a topological argument of Tupan to the metaplectic group $\widetildeG$ and give an elementary proof that any lift of the standard involution to $\widetildeG$ ; is also a dualizing involution.

Twisting of paramodular vectors

International Journal of Number Theory ◽

10.1142/s1793042114500146 ◽

2014 ◽

Vol 10 (04) ◽

pp. 1043-1065 ◽

Cited By ~ 2

Author(s):

Jennifer Johnson-Leung ◽

Brooks Roberts

Keyword(s):

Local Field ◽

Characteristic Zero ◽

Central Character ◽

Quadratic Character ◽

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.

Theta correspondence for p-adic dual pairs of type I

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0006 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Petar Bakić ◽

Marcela Hanzer

Keyword(s):

Local Field ◽

Type I ◽

Theta Correspondence ◽

Dual Pairs ◽

Langlands Parameters ◽

First Occurrence ◽

Unitary Dual ◽

Howe Correspondence ◽

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.

(GLn+1(F), GLn(F)) is a Gelfand pair for any local field F

Compositio Mathematica ◽

10.1112/s0010437x08003746 ◽

2008 ◽

Vol 144 (6) ◽

pp. 1504-1524 ◽

Cited By ~ 18

Author(s):

Avraham Aizenbud ◽

Dmitry Gourevitch ◽

Eitan Sayag

Keyword(s):

Local Field ◽

Invariant Distribution ◽

Gelfand Pair ◽

Smooth Manifolds ◽

Invariant Distributions ◽

AbstractLet F be an arbitrary local field. Consider the standard embedding $\mathrm {GL}_n(F) \hookrightarrow \mathrm {GL}_{n+1}(F)$ and the two-sided action of GLn(F)×GLn(F) on GLn+1(F). In this paper we show that any GLn(F)×GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), $\dim Hom_{\mathrm {GL}_n(F)}(E,\mathbb {C}) \leqslant 1$. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.

Globalization of Distinguished Supercuspidal Representations of GL(n)

Canadian Mathematical Bulletin ◽

10.4153/cmb-2002-025-x ◽

2002 ◽

Vol 45 (2) ◽

pp. 220-230 ◽

Cited By ~ 7

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.

The distinction problem for metaplectic case

International Journal of Number Theory ◽

10.1142/s1793042120500591 ◽

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]

Geometric Weil representation: local field case

Compositio Mathematica ◽

10.1112/s0010437x08003771 ◽

2009 ◽

Vol 145 (1) ◽

pp. 56-88 ◽

Cited By ~ 5

Author(s):

Vincent Lafforgue ◽

Sergey Lysenko

Keyword(s):

Local Field ◽

Dual Pair ◽

Weil Representation ◽

Closed Field ◽

Perverse Sheaves ◽

Algebraically Closed Field ◽

Field Case ◽

Metaplectic Group ◽

Geometric Langlands ◽

Langlands Functoriality

AbstractLet k be an algebraically closed field of characteristic greater than 2, and let F=k((t)) and G=𝕊p2d. In this paper we propose a geometric analog of the Weil representation of the metaplectic group $\widetilde G(F)$. This is a category of certain perverse sheaves on some stack, on which $\widetilde G(F)$ acts by functors. This construction will be used by Lysenko (in [Geometric theta-lifting for the dual pair S𝕆2m, 𝕊p2n, math.RT/0701170] and subsequent publications) for the proof of the geometric Langlands functoriality for some dual reductive pairs.

Zero-Separating Invariants for Linear Algebraic Groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091515000322 ◽

2015 ◽

Vol 59 (4) ◽

pp. 911-924 ◽

Cited By ~ 1

Author(s):

Jonathan Elmer ◽

Martin Kohls

Keyword(s):

Algebraic Group ◽

Characteristic Zero ◽

Elementary Proof ◽

Algebraic Groups ◽

Linear Algebraic Group ◽

Null Cone ◽

Closed Field ◽

Unipotent Group ◽

Linear Algebraic Groups ◽

Separating Invariants

AbstractAbstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.

LOCAL NEWFORMS AND FORMAL EXTERIOR SQUARE L-FUNCTIONS

International Journal of Number Theory ◽

10.1142/s179304211350070x ◽

2013 ◽

Vol 09 (08) ◽

pp. 1995-2010 ◽

Cited By ~ 3

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.

Epsilon factors of symplectic type characters in the wild case

Forum Mathematicum ◽

10.1515/forum-2020-0281 ◽

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.

Langlands lambda function for quadratic tamely ramified extensions

Journal of Algebra and Its Applications ◽

10.1142/s0219498819501329 ◽

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].