scholarly journals Spectrum Generating Operators and Intertwining Operators for Representations Induced from a Maximal Parabolic Subgroup

1996 ◽  
Vol 135 (1) ◽  
pp. 163-205 ◽  
Author(s):  
Thomas Branson ◽  
Gestur Òlafsson ◽  
Bent Ørsted
Keyword(s):  
Parabolic Subgroup ◽  
Intertwining Operators ◽  
Maximal Parabolic Subgroup ◽  
Generating Operators

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.

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 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.

scholarly journals Higher-rank zeta functions andSLn-zeta functions for curves

2020 ◽  
Vol 117 (12) ◽  
pp. 6398-6408
Author(s):  
Lin Weng ◽  
Don Zagier
Keyword(s):  
Zeta Function ◽  
Parabolic Subgroup ◽  
Vector Bundles ◽  
Reductive Group ◽  
Zeta Functions ◽  
Smooth Projective Curve ◽  
Maximal Parabolic Subgroup ◽  
Higher Rank ◽  
Semistable Vector Bundles

In earlier papers L.W. introduced two sequences of higher-rank zeta functions associated to a smooth projective curve over a finite field, both of them generalizing the Artin zeta function of the curve. One of these zeta functions is defined geometrically in terms of semistable vector bundles of rank n over the curve and the other one group-theoretically in terms of certain periods associated to the curve and to a split reductive group G and its maximal parabolic subgroup P. It was conjectured that these two zeta functions coincide in the special case whenG=SLnand P is the parabolic subgroup consisting of matrices whose final row vanishes except for its last entry. In this paper we prove this equality by giving an explicit inductive calculation of the group-theoretically defined zeta functions in terms of the original Artin zeta function (corresponding ton=1) and then verifying that the result obtained agrees with the inductive determination of the geometrically defined zeta functions found by Sergey Mozgovoy and Markus Reineke in 2014.

scholarly journals Branching laws for small unitary representations of GL(n, ℂ)

2014 ◽  
Vol 25 (06) ◽  
pp. 1450052
Author(s):  
Jan Möllers ◽  
Benjamin Schwarz
Keyword(s):  
Parabolic Subgroup ◽  
Principal Series ◽  
Unitary Representations ◽  
Symmetric Pair ◽  
Maximal Parabolic Subgroup ◽  
Infinite Dimensional ◽  
Branching Laws ◽  
Series Representations ◽  
Unitary Principal Series ◽  
Kirillov Dimension

The unitary principal series representations of G = GL (n, ℂ) induced from a character of the maximal parabolic subgroup P = ( GL (1, ℂ) × GL (n - 1, ℂ)) ⋉ ℂn-1 attain the minimal Gelfand–Kirillov dimension among all infinite-dimensional unitary representations of G. We find the explicit branching laws for the restriction of these representations to all reductive subgroups H of G such that (G, H) forms a symmetric pair.

scholarly journals The action of a maximal parabolic subgroup on the transpositions of the Baby Monster

1994 ◽  
Vol 37 (1) ◽  
pp. 185-189 ◽  
Author(s):  
Robert A. Wilson
Keyword(s):  
Simple Group ◽  
Parabolic Subgroup ◽  
Split Extension ◽  
Maximal Parabolic Subgroup ◽  
Technical Result

We prove a technical result required by Ivanov and Shpectorov in their construction of a non-split extension of 34371 by the Baby Monster simple group.

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.

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