Unitary completions of locally algebraic and locally analytic principal series of p-adic GL2

2016 ◽  
Vol 12 (07) ◽  
pp. 1765-1781
Author(s):  
Keenan Kidwell
Keyword(s):  
Classical Result ◽  
Finite Extension ◽  
Principal Series ◽  
Topological Isomorphism ◽  
Tempered Distributions ◽  
Series Representations ◽  
Critical Slope ◽  
Local Convex ◽  
The Relationship ◽  
Analytic Induction

We generalize a result of Emerton on the relationship between unitary completions of locally [Formula: see text]-analytic and locally [Formula: see text]-algebraic principal series representations induced from certain locally [Formula: see text]-algebraic characters of the diagonal torus of [Formula: see text], where [Formula: see text] is a finite extension of [Formula: see text]. Namely, under a non-critical slope hypothesis on the character being induced, the map on universal unitary completions arising from the inclusion of the locally algebraic induction into the locally analytic induction is a topological isomorphism. (Emerton proved this result for [Formula: see text].) The main ingredients in carrying out a “several-variable” version of Emerton’s argument are the description of the local convex space of locally [Formula: see text]-analytic functions on the group [Formula: see text] in terms of the embeddings of [Formula: see text] into our [Formula: see text]-adic coefficient field, and a generalization by Breuil of the classical result of Amice-Vélu and Vishik on “tempered distributions” on [Formula: see text].

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.

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.

scholarly journals Casselman’s Basis of Iwahori Vectors and Kazhdan–Lusztig Polynomials

2019 ◽  
Vol 71 (6) ◽  
pp. 1351-1366
Author(s):  
Daniel Bump ◽  
Maki Nakasuji
Keyword(s):  
Representation Theory ◽  
Functional Equations ◽  
Transition Matrix ◽  
Principal Series ◽  
Surprising Result ◽  
Transition Matrices ◽  
Natural Basis ◽  
Series Representations ◽  
The Matrix ◽  
Lusztig Polynomials

AbstractA problem in representation theory of $p$-adic groups is the computation of the Casselman basis of Iwahori fixed vectors in the spherical principal series representations, which are dual to the intertwining integrals. We shall express the transition matrix $(m_{u,v})$ of the Casselman basis to another natural basis in terms of certain polynomials that are deformations of the Kazhdan–Lusztig R-polynomials. As an application we will obtain certain new functional equations for these transition matrices under the algebraic involution sending the residue cardinality $q$ to $q^{-1}$. We will also obtain a new proof of a surprising result of Nakasuji and Naruse that relates the matrix $(m_{u,v})$ to its inverse.

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