An Iwahori-Whittaker model for the Satake category

Models of Representations and Langlands Functoriality

Canadian Journal of Mathematics ◽

10.4153/s0008414x19000014 ◽

2019 ◽

Vol 72 (3) ◽

pp. 676-707 ◽

Cited By ~ 1

Author(s):

Arnab Mitra ◽

Eitan Sayag

Keyword(s):

Base Change ◽

General Linear ◽

Linear Groups ◽

General Linear Groups ◽

Whittaker Model ◽

Automorphic Induction ◽

Ladder Representations ◽

Langlands Functoriality ◽

Whittaker Models

AbstractIn this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.

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ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972708000701 ◽

2008 ◽

Vol 78 (2) ◽

pp. 261-284 ◽

Cited By ~ 3

Author(s):

XIN TANG ◽

YUNGE XU

Keyword(s):

Quantum Groups ◽

Irreducible Representations ◽

Whittaker Model ◽

Finite Dimensional ◽

Completely Reducible ◽

The Relationship ◽

Natural Families

AbstractWe construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

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Uniqueness of Whittaker model for the metaplectic group

Pacific Journal of Mathematics ◽

10.2140/pjm.2007.232.453 ◽

2007 ◽

Vol 232 (2) ◽

pp. 453-469 ◽

Cited By ~ 16

Author(s):

Dani Szpruch

Keyword(s):

Metaplectic Group ◽

Whittaker Model

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Generalized and degenerate Whittaker models

Compositio Mathematica ◽

10.1112/s0010437x16007788 ◽

2017 ◽

Vol 153 (2) ◽

pp. 223-256 ◽

Cited By ~ 10

Author(s):

Raul Gomez ◽

Dmitry Gourevitch ◽

Siddhartha Sahi

Keyword(s):

Fourier Coefficients ◽

Coadjoint Orbit ◽

Local Fields ◽

Admissible Representation ◽

Reductive Groups ◽

Automorphic Representations ◽

Wave Front Set ◽

Induced Representations ◽

Whittaker Model ◽

Whittaker Models

We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker–Fourier coefficients of automorphic representations. For $\text{GL}_{n}(\mathbb{F})$ this implies that a smooth admissible representation $\unicode[STIX]{x1D70B}$ has a generalized Whittaker model ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ corresponding to a nilpotent coadjoint orbit ${\mathcal{O}}$ if and only if ${\mathcal{O}}$ lies in the (closure of) the wave-front set $\operatorname{WF}(\unicode[STIX]{x1D70B})$. Previously this was only known to hold for $\mathbb{F}$ non-archimedean and ${\mathcal{O}}$ maximal in $\operatorname{WF}(\unicode[STIX]{x1D70B})$, see Moeglin and Waldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427–452]. We also express ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ as an iteration of a version of the Bernstein–Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), 441–472; Aizenbud et al., Derivatives for representations of$\text{GL}(n,\mathbb{R})$and$\text{GL}(n,\mathbb{C})$, Israel J. Math. 206 (2015), 1–38]. This enables us to extend to $\text{GL}_{n}(\mathbb{R})$ and $\text{GL}_{n}(\mathbb{C})$ several further results by Moeglin and Waldspurger on the dimension of ${\mathcal{W}}_{{\mathcal{O}}}(\unicode[STIX]{x1D70B})$ and on the exactness of the generalized Whittaker functor.

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Local Rankin–Selberg integrals for Speh representations

Compositio Mathematica ◽

10.1112/s0010437x2000706x ◽

2020 ◽

Vol 156 (5) ◽

pp. 908-945 ◽

Cited By ~ 1

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.

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