scholarly journals Determination of the intertwining operators for holomorphically induced representations of Hermitian symmetric pairs

1988 ◽  
Vol 131 (1) ◽  
pp. 39-50 ◽  
Author(s):  
Brian Boe ◽  
Thomas Enright ◽  
Brad Shelton
Keyword(s):  
Intertwining Operators ◽  
Induced Representations

scholarly journals Automatic discontinuity of intertwining operators

1982 ◽  
Vol 25 (1) ◽  
pp. 49-54 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Spaces ◽  
Linear Transformation ◽  
Linear Operators ◽  
Intertwining Operators ◽  
Continuous Linear ◽  
Automatic Continuity ◽  
Converse Problem ◽  
Simple Modification ◽  
Continuous Linear Operators

Throughout this paper, we suppose that T and R are continuous linear operators on the Banach spaces X and Y, respectively. One of the basic problems in the theory of automatic continuity is the determination of conditions under which a linear transformation S: X → Y which satisfies RS = ST is continuous or is discontinuous. Johnson and Sinclair [4], [6], [11; pp. 24–30] have given a variety of conditions on R and T which guarantee that all such S are automatically continuous. In this paper we consider the converse problem and find conditions on the range S(X) which guarantee that S is automatically discontinuous. The construction of such automatically discontinuous S is then accomplished by a simple modification of a technique of Sinclair's [10; pp. 260–261], [11; pp. 21–23].

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.

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.

The Archimedean Intertwining Operator for GLN

2019 ◽  
pp. 168-210
Author(s):  
Uwe Weselmann
Keyword(s):  
General Setting ◽  
Intertwining Operators ◽  
Induced Representations ◽  
Special Values ◽  
Intertwining Operator

This chapter generalizes an identity conjectured by G. Harder and proved by D. Zagier from the case of GL3(ℝ)-representations to the case of general GLN(ℝ)-representations. These are useful in applying the results of Harder and Anantharam Raghuram on quotients of special values of L-functions. To begin, the chapter provides the general setting for this analysis—the group theoretic data and the induced representations. It then discusses the intertwining operators as well as the J-admissible permutations. The chapter goes on to discuss representations and L-functions before turning to the main theorem on Archimedean intertwining operator. Finally, the chapter discusses some applications to cohomology.

Eisenstein Cohomology

2019 ◽  
pp. 71-83
Author(s):  
Günter Harder ◽  
A. Raghuram
Keyword(s):  
Eisenstein Series ◽  
Constant Term ◽  
Isotropic Subspace ◽  
Intertwining Operators ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Induced Representations ◽  
Rank One ◽  
Eisenstein Cohomology

This chapter provides the Eisenstein cohomology. It begins with the Poincaré duality and maximal isotropic subspace of boundary cohomology. Here, the chapter considers the compatibility of duality isomorphisms with the connecting homomorphism. It then states and proves the main result on rank-one Eisenstein cohomology. Thereafter, the chapter presents a theorem of Langlands: the constant term of an Eisenstein series. It draws some details from the Langlands–Shahidi method in this context. Induced representations are examined, as are standard intertwining operators. The chapter finally illustrates the Eisenstein series, the constant term of an Eisenstein series, and the holomorphy of the Eisenstein series at the point of evaluation.

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