scholarly journals Towards a Goldberg–Shahidi pairing for classical groups

2018 ◽  
Vol 30 (2) ◽  
pp. 347-384
Author(s):  
Arnab Mitra ◽  
Steven Spallone
Keyword(s):  
Adjoint Action ◽  
Bilinear Pairing ◽  
Vector Spaces ◽  
Unipotent Radical ◽  
Levi Subgroup ◽  
Intertwining Operators ◽  
Maximal Parabolic Subgroup ◽  
Induced Representations ◽  
Matrix Coefficients ◽  
Symplectic Case

AbstractLet{G^{1}}be an orthogonal, symplectic or unitary group over a local field and let{P=MN}be a maximal parabolic subgroup. Then the Levi subgroupMis the product of a group of the same type as{G^{1}}and a general linear group, acting on vector spacesXandW, respectively. In this paper we decompose the unipotent radicalNofPunder the adjoint action ofM, assuming{\dim W\leq\dim X}, excluding only the symplectic case with{\dim W}odd. The result is a Weyl-type integration formula forNwith applications to the theory of intertwining operators for parabolically induced representations of{G^{1}}. Namely, one obtains a bilinear pairing on matrix coefficients, in the spirit of Goldberg–Shahidi, which detects the presence of poles of these operators at 0.

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.

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.

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 On the degrees of matrix coefficients of intertwining operators

2012 ◽  
Vol 260 (2) ◽  
pp. 433-456 ◽  
Author(s):  
Tobias Finis ◽  
Erez Lapid ◽  
Werner Müller
Keyword(s):  
Intertwining Operators ◽  
Matrix Coefficients

The P-radical classes in simple algebraic groups and finite groups of Lie type

2012 ◽  
Vol 15 (5) ◽  
Author(s):  
R. Lawther
Keyword(s):  
Finite Group ◽  
Algebraic Group ◽  
Parabolic Subgroup ◽  
Finite Groups ◽  
Algebraic Groups ◽  
Unipotent Radical ◽  
Maximal Parabolic Subgroup ◽  
Simple Algebraic Group ◽  
A Finite Group ◽  
Groups Of Lie Type

Abstract.Given either a simple algebraic group or a finite group of Lie type, of rank at least 2, and a maximal parabolic subgroup, we determine which non-trivial unipotent classes have the property that their intersection with the parabolic subgroup is contained within its unipotent radical. Such classes are rare; listing them provides a basis for inductive arguments.

MATRIX COEFFICIENTS OF REPRESENTATIONS OF SU(2,2): THE CASE OF PJ-PRINCIPAL SERIES

2004 ◽  
Vol 15 (10) ◽  
pp. 1033-1064
Author(s):  
HARUTAKA KOSEKI ◽  
TAKAYUKI ODA
Keyword(s):  
Discrete Series ◽  
Principal Series ◽  
Radial Part ◽  
Maximal Parabolic Subgroup ◽  
Parabolic Induction ◽  
Matrix Coefficients ◽  
Series Representations ◽  
The Matrix ◽  
Discrete Series Representations ◽  
Restricted Root System

The real Lie group G=SU(2,2) has a standard maximal parabolic subgroup PJ corresponding to the long simple roots in its restricted root system. We have an explicit formula of the holonomic system of the radial part of the matrix coefficients with corner K-type of the generalized principal series representations of G obtained by parabolic induction with respect to PJ from the discrete series representations of MJ. It is equivalent to the modified F2 system of Appell.

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