Application II: Periods, Distinguished Representations and ( g , K ) $$(\mathfrak {g},K)$$ -cohomologies

Symmetry Breaking for Representations of Rank One Orthogonal Groups II - Lecture Notes in Mathematics ◽

10.1007/978-981-13-2901-2_12 ◽

2018 ◽

pp. 207-222

Author(s):

Toshiyuki Kobayashi ◽

Birgit Speh

Keyword(s):

Distinguished Representations

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Kirillov models for distinguished representations

Nagoya Mathematical Journal ◽

10.1017/s0027763000001707 ◽

1989 ◽

Vol 116 ◽

pp. 89-110 ◽

Cited By ~ 1

Author(s):

Courtney Moen

Keyword(s):

Linear Group ◽

General Linear Group ◽

Automorphic Forms ◽

Principal Series ◽

Weil Representation ◽

Local Factors ◽

Distinguished Representations ◽

Group A ◽

Local Representations ◽

Covering Groups

In the theory of automorphic forms on covering groups of the general linear group, a central role is played by certain local representations which have unique Whittaker models. A representation with this property is called distinguished. In the case of the 2-sheeted cover of GL2, these representations arise as the the local components of generalizations of the classical θ-function. They have been studied thoroughly in [GPS]. The Weil representation provides these representations with a very nice realization, and the local factors attached to these representations can be computed using this realization. It has been shown [KP] that only in the case of a certain 3-sheeted cover do we find other principal series of covering groups of GL2 which have a unique Whittaker model. It is natural to ask if these distinguished representations also have a realization analgous to the Weil representation.

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Gamma Factors, Root Numbers, and Distinction

Canadian Journal of Mathematics ◽

10.4153/cjm-2017-011-6 ◽

2018 ◽

Vol 70 (3) ◽

pp. 683-701 ◽

Cited By ~ 1

Author(s):

Nadir Matringe ◽

Omer Offen

Keyword(s):

Linear Group ◽

General Linear Group ◽

Quadratic Extension ◽

Root Number ◽

Converse Problem ◽

Root Numbers ◽

Special Values ◽

Distinguished Representations ◽

Local Invariants ◽

Gamma Factor

AbstractWe study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of p-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

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