scholarly journals On automorphic forms for the general linear group

1982 ◽  
Vol 12 (1) ◽  
pp. 123-144 ◽  
Author(s):  
Audrey Terras
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
Automorphic Forms ◽  
General Linear

scholarly journals Local Rankin–Selberg integrals for Speh representations

2020 ◽  
Vol 156 (5) ◽  
pp. 908-945 ◽  
Author(s):  
Erez M. Lapid ◽  
Zhengyu Mao
Keyword(s):  
Eisenstein Series ◽  
Linear Group ◽  
General Linear Group ◽  
Automorphic Forms ◽  
Global Field ◽  
General Linear ◽  
Whittaker Model ◽  
Selberg Integrals

We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.

Representations induced from the Zelevinsky segment and discrete series in the half-integral case

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ivan Matić
Keyword(s):  
Linear Group ◽  
Local Field ◽  
General Linear Group ◽  
Discrete Series ◽  
Series Representation ◽  
General Linear ◽  
Composition Series ◽  
Induced Representations ◽  
Discrete Series Representation

AbstractLet {G_{n}} denote either the group {\mathrm{SO}(2n+1,F)} or {\mathrm{Sp}(2n,F)} over a non-archimedean local field of characteristic different than two. We study parabolically induced representations of the form {\langle\Delta\rangle\rtimes\sigma}, where {\langle\Delta\rangle} denotes the Zelevinsky segment representation of the general linear group attached to the segment Δ, and σ denotes a discrete series representation of {G_{n}}. We determine the composition series of {\langle\Delta\rangle\rtimes\sigma} in the case when {\Delta=[\nu^{a}\rho,\nu^{b}\rho]} where a is half-integral.

scholarly journals Triple factorisations of the general linear group and their associated geometries

2015 ◽  
Vol 469 ◽  
pp. 169-203 ◽  
Author(s):  
Seyed Hassan Alavi ◽  
John Bamberg ◽  
Cheryl E. Praeger
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
General Linear

scholarly journals Kirillov models for distinguished representations

1989 ◽  
Vol 116 ◽  
pp. 89-110 ◽  
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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