scholarly journals Dimension of the space of intertwining operators from degenerate principal series representations

2018 ◽  
Vol 24 (4) ◽  
pp. 3649-3662 ◽  
Author(s):  
Taito Tauchi
Keyword(s):  
Principal Series ◽  
Intertwining Operators ◽  
Series Representations ◽  
Degenerate Principal Series ◽  
Principal Series Representations

scholarly journals Analytic Continuation of Equivariant Distributions

2018 ◽  
Vol 2019 (23) ◽  
pp. 7160-7192 ◽  
Author(s):  
Dmitry Gourevitch ◽  
Siddhartha Sahi ◽  
Eitan Sayag
Keyword(s):  
Analytic Continuation ◽  
Reductive Group ◽  
Principal Series ◽  
Real Algebraic Varieties ◽  
Series Representations ◽  
Degenerate Principal Series ◽  
Special Case ◽  
Adic Case ◽  
Principal Series Representations ◽  
Whittaker Models

Abstract We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein’s theory of analytic continuation of holonomic distributions. We use this to construct H-equivariant functionals on principal series representations of G, where G is a real reductive group and H is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on principal series representations and of analytic continuation of standard intertwining operators. Finally, we extend our constructions to the p-adic case using a recent result of Hong and Sun.

EXPLICIT DOUBLING INTEGRALS FOR Sp2(F) AND $\widetilde{{{\rm Sp}}_{2}}(F)$ USING "GOOD TEST VECTORS"

2011 ◽  
Vol 07 (02) ◽  
pp. 449-527 ◽  
Author(s):  
CHRISTIAN ZORN
Keyword(s):  
Finite Extension ◽  
Principal Series ◽  
Good Test ◽  
Dual Pairs ◽  
Series Representations ◽  
Degenerate Principal Series ◽  
Doubling Method ◽  
Test Vectors ◽  
Metaplectic Cover ◽  
Principal Series Representations

In this paper, we offer some explicit computations of a formulation of the doubling method of Piatetski-Shapiro and Rallis for the groups Sp 2(F) (the rank 2 symplectic group) and its metaplectic cover [Formula: see text] for F a finite extension of ℚp with p ≠ 2. We determine a set of "good test vectors" for the irreducible constituents of unramified principal series representations for these groups as well as a set of "good theta test sections" in a family of degenerate principal series representations of Sp 4(F) and [Formula: see text]. Determining "good test data" that produces a non-vanishing doubling integral should indicate the existence of a non-vanishing theta lifts for dual pairs of the type ( Sp 2(F), O (V)) (respectively [Formula: see text]) where V is a quadratic space of an even (respectively odd) dimension.

Reducibility of the Principal Series for (F) over a p-adic Field

2010 ◽  
Vol 62 (4) ◽  
pp. 914-960 ◽  
Author(s):  
Christian Zorn
Keyword(s):  
Symplectic Group ◽  
Principal Series ◽  
Complete List ◽  
Intertwining Operators ◽  
Series Representations ◽  
Principal Series Representations

AbstractLet Gn = Spn(F) be the rank n symplectic group with entries in a nondyadic p-adic field F. We further let be the metaplectic extension of Gn by defined using the Leray cocycle. In this paper, we aim to demonstrate the complete list of reducibility points of the genuine principal series of . In most cases, we will use some techniques developed by Tadić that analyze the Jacquetmodules with respect to all of the parabolics containing a fixed Borel. The exceptional cases involve representations induced from unitary characters χ with χ2 = 1. Because such representations π are unitary, to show the irreducibility of π, it suffices to show that . We will accomplish this by examining the poles of certain intertwining operators associated to simple roots. Then some results of Shahidi and Ban decompose arbitrary intertwining operators into a composition of operators corresponding to the simple roots of . We will then be able to show that all such operators have poles at principal series representations induced from quadratic characters and therefore such operators do not extend to operators in for the π in question.

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