On the Tempered Spectrum of Quasi-Split Classical Groups II

2001 ◽  
Vol 53 (2) ◽  
pp. 244-277 ◽  
Author(s):  
David Goldberg ◽  
Freydoon Shahidi
Keyword(s):  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Intertwining Operators ◽  
Classical Groups ◽  
Maximal Parabolic Subgroup ◽  
Supercuspidal Representation

AbstractWe determine the poles of the standard intertwining operators for a maximal parabolic subgroup of the quasi-split unitary group defined by a quadratic extension E/F of p-adic fields of characteristic zero. We study the case where the Levi component M ≃ GLn(E) × Um(F), with n ≡ m (mod 2). This, along with earlier work, determines the poles of the local Rankin-Selberg product L-function L(s, t′ × τ), with t′ an irreducible unitary supercuspidal representation of GLn(E) and τ a generic irreducible unitary supercuspidal representation of Um(F). The results are interpreted using the theory of twisted endoscopy.

scholarly journals Residues of intertwining operators for SO*6 as character identities

2010 ◽  
Vol 146 (3) ◽  
pp. 772-794 ◽  
Author(s):  
Freydoon Shahidi ◽  
Steven Spallone
Keyword(s):  
Quadratic Extension ◽  
The Other ◽  
Levi Subgroup ◽  
Intertwining Operators ◽  
Central Character ◽  
Supercuspidal Representation ◽  
Induced Representations ◽  
Intertwining Operator ◽  
Other Hand ◽  
Hilbert’S Theorem 90

AbstractWe show that the residue at s=0 of the standard intertwining operator attached to a supercuspidal representation π⊗χ of the Levi subgroup GL2(F)×E1 of the quasisplit group SO*6(F) defined by a quadratic extension E/F of p-adic fields is proportional to the pairing of the characters of these representations considered on the graph of the norm map of Kottwitz–Shelstad. Here π is self-dual, and the norm is simply that of Hilbert’s theorem 90. The pairing can be carried over to a pairing between the character on E1 and the character on E× defining the representation of GL2(F) when the central character of the representation is quadratic, but non-trivial, through the character identities of Labesse–Langlands. If the quadratic extension defining the representation on GL2(F) is different from E the residue is then zero. On the other hand when the central character is trivial the residue is never zero. The results agree completely with the theory of twisted endoscopy and L-functions and determines fully the reducibility of corresponding induced representations for all s.

Prehomogeneity on Quasi-Split Classical Groups and Poles of Intertwining Operators

2009 ◽  
Vol 61 (3) ◽  
pp. 691-707 ◽  
Author(s):  
Xiaoxiang Yu
Keyword(s):  
Parabolic Subgroup ◽  
Classical Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Intertwining Operators ◽  
Classical Groups ◽  
Intertwining Operator ◽  
Necessary And Sufficient ◽  
Tempered Representation

Abstract.Suppose that P = MN is amaximal parabolic subgroup of a quasisplit, connected, reductive classical group G defined over a non-Archimedean field and A is the standard intertwining operator attached to a tempered representation of G induced from M . In this paper we determine all the cases in which Lie(N ) is prehomogeneous under Ad(m) when N is non-abelian, and give necessary and sufficient conditions for A to have a pole at 0.

Centralizers and Twisted Centralizers: Application to Intertwining Operators

2006 ◽  
Vol 58 (3) ◽  
pp. 643-672
Author(s):  
Xiaoxiang Yu
Keyword(s):  
Parabolic Subgroup ◽  
Case Analysis ◽  
Unipotent Radical ◽  
Intertwining Operators ◽  
Maximal Parabolic Subgroup

AbstractThe equality of the centralizer and twisted centralizer is proved based on a case-by-case analysis when the unipotent radical of a maximal parabolic subgroup is abelian. Then this result is used to determine the poles of intertwining operators.

COMPUTATION OF A TWISTED CHARACTER OF A SMALL REPRESENTATION OF GL(3, E)

2012 ◽  
Vol 08 (05) ◽  
pp. 1153-1230
Author(s):  
YUVAL Z. FLICKER ◽  
DMITRII ZINOVIEV
Keyword(s):  
Conjugacy Class ◽  
Parabolic Subgroup ◽  
Quadratic Extension ◽  
Character Formula ◽  
The Other ◽  
Trivial Representation ◽  
Maximal Parabolic Subgroup ◽  
Local Means ◽  
Stable Class ◽  
Endoscopic Group

Let E/F be a quadratic extension of p-adic fields, p ≠ 2. Let [Formula: see text] be the involution of E over F. The representation π of GL (3, E) normalizedly induced from the trivial representation of the maximal parabolic subgroup is invariant under the involution [Formula: see text]. We compute — by purely local means — the σ-twisted character [Formula: see text] of π. We show that it is σ-unstable, namely its value at one σ-regular-elliptic conjugacy class within a stable such class is equal to negative its value at the other such conjugacy class within the stable class, or zero when the σ-regular-elliptic stable conjugacy class consists of a single such conjugacy class. Further, we relate this twisted character to the twisted endoscopic lifting from the trivial representation of the "unstable" twisted endoscopic group U (2, E/F) of GL (3, E). In particular π is σ-elliptic, that is, [Formula: see text] is not identically zero on the σ-elliptic set.

On Residues of Intertwining Operators in Cases with Prehomogeneous Nilradical

2017 ◽  
Vol 69 (5) ◽  
pp. 1169-1200
Author(s):  
Sandeep Varma
Keyword(s):  
Parabolic Subgroup ◽  
Reductive Group ◽  
Intertwining Operators ◽  
Maximal Parabolic Subgroup ◽  
Supercuspidal Representations ◽  
Levi Decomposition ◽  
Endoscopic Group

AbstractLet P = M N be a Levi decomposition of a maximal parabolic subgroup of a connected reductive group G over a p-adic field F. Assume that there exists w0 ∊ G(F) that normalizes M and conjugates P to an opposite parabolic subgroup. When N has a Zariski dense Int M-orbit, F. Shahidi and X. Yu described a certain distribution D on M(F), such that, for irreducible unitary supercuspidal representations π of M(F) with is irreducible if and only if D( f )≠ 0 for some pseudocoefficient f of π. Since this irreducibility is conjecturally related to π arising via transfer from certain twisted endoscopic groups of M, it is of interest to realize D as endoscopic transfer from a simpler distribution on a twisted endoscopic group H of M. This has been done in many situations where N is abelian. Here we handle the standard examples in cases where N is nonabelian but admit a Zariski dense Int M-orbit.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

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 Geometry of Neumann subgroups

1989 ◽  
Vol 47 (3) ◽  
pp. 350-367 ◽  
Author(s):  
Ravi S. Kulkarni
Keyword(s):  
Parabolic Subgroup ◽  
Fuchsian Group ◽  
Hyperbolic Plane ◽  
Modular Group ◽  
Natural Extension ◽  
Geometric Method ◽  
General Context ◽  
Open Disk ◽  
Finitely Generated ◽  
Maximal Parabolic Subgroup

AbstractA Neumann subgroup of the classical modular group is by definition a complement of a maximal parabolic subgroup. Recently Neumann subgroups have been studied in a series of papers by Brenner and Lyndon. There is a natural extension of the notion of a Neumann subgroup in the context of any finitely generated Fuchsian group Γ acting on the hyperbolic plane H such that Γ/H is homeomorphic to an open disk. Using a new geometric method we extend the work of Brenner and Lyndon in this more general context.

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