Rank One Reducibility for Metaplectic Groups via Theta Correspondence

2011 ◽  
Vol 63 (3) ◽  
pp. 591-615 ◽  
Author(s):  
Marcela Hanzer ◽  
Goran Muić
Keyword(s):  
Parabolic Subgroups ◽  
Orthogonal Groups ◽  
Metaplectic Groups ◽  
Theta Correspondence ◽  
Rank One ◽  
Maximal Parabolic Subgroups ◽  
Cuspidal Representations

Abstract We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous representations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients.

scholarly journals Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

scholarly journals Eisenstein Series Arising from Jordan Algebras

2019 ◽  
Vol 72 (1) ◽  
pp. 183-201 ◽  
Author(s):  
Marcela Hanzer ◽  
Gordan Savin
Keyword(s):  
Eisenstein Series ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Algebra Structure ◽  
Parabolic Subgroups ◽  
Automorphic Representations ◽  
Unipotent Radicals ◽  
Maximal Parabolic Subgroups

AbstractWe describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Reducibility for Non-Connected p-Adic Groups, With G° Of Prime Index

1995 ◽  
Vol 47 (2) ◽  
pp. 344-363 ◽  
Author(s):  
David Goldberg
Keyword(s):  
Cyclic Group ◽  
Prime Order ◽  
Discrete Series ◽  
Parabolic Subgroups ◽  
Orthogonal Groups ◽  
Definition Of

AbstractWe determine the structure of representations induced from discrete series of parabolic subgroups of quasi-split p-adic groups G with G/G° a cyclic group of prime order. We attach to each such representation an R-group which extends the definition of the Knapp-Stein R-group. We show that this R-group has the properties predicted by Arthur. We apply our results to the case of Orthogonal groups.

Poles ofL-functions and theta liftings for orthogonal groups

2009 ◽  
Vol 8 (4) ◽  
pp. 693-741 ◽  
Author(s):  
David Ginzburg ◽  
Dihua Jiang ◽  
David Soudry
Keyword(s):  
Eisenstein Series ◽  
Orthogonal Group ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Metaplectic Groups ◽  
Automorphic Representations ◽  
Domain Of Holomorphy ◽  
First Occurrence ◽  
Cuspidal Automorphic Representations

AbstractIn this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standardL-functions.

PARABOLIC SUBGROUPS AND CUSPIDAL REPRESENTATIONS OF FINITE MONOIDS

1991 ◽  
Vol 01 (01) ◽  
pp. 33-47 ◽  
Author(s):  
JAN OKNIŃSKI ◽  
MOHAN S. PUTCHA
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Special Class ◽  
Semigroup Algebra ◽  
Parabolic Subgroups ◽  
Finite Monoids ◽  
Representations Of Finite Groups ◽  
Symmetric Inverse Semigroup ◽  
Complex Semigroup ◽  
Cuspidal Representations

This paper is mostly concerned with arbitrary finite monoids M with the complex semigroup algebra [Formula: see text] semisimple. Using the 1942 work of Clifford, we develop for these monoids a theory of cuspidal representations. Harish-Chandra's philosophy of cuspidal representations of finite groups can then be derived with an appropriate specialization. For [Formula: see text], we use Solomon's Hecke algebra to obtain a correspondence between the 'simple' representations of [Formula: see text] and the representations of the symmetric inverse semigroup. We also prove a semisimplicity theorem for a special class of finite monoids of the type which was earlier used by the authors to prove the semisimplicity of [Formula: see text].

scholarly journals Symmetry breaking for representations of rank one orthogonal groups

2015 ◽  
Vol 238 (1126) ◽  
pp. 0-0 ◽  
Author(s):  
Toshiyuki Kobayashi ◽  
Birgit Speh
Keyword(s):  
Symmetry Breaking ◽  
Orthogonal Groups ◽  
Rank One

scholarly journals On (2, 3, 7)-generation of maximal parabolic subgroups

2001 ◽  
Vol 71 (2) ◽  
pp. 187-200 ◽  
Author(s):  
L. Di Martino ◽  
M. C. Tamburini
Keyword(s):  
Parabolic Subgroups ◽  
Finitely Generated ◽  
Maximal Parabolic Subgroups

AbstractLet R be a ring with 1 and En (R) be the subgroup of GLn(R) generated by the matrices I + reij, r ∈ R, i ≠ j. We prove that the subgroup of consisting of the matrices of shape , where and , is (2, 3, 7)-generated whenever R is finitely generated and n, are large enough.

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