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Generalized and degenerate Whittaker models

Part of: Lie algebras and Lie superalgebras Linear algebraic groups and related topics Lie groups

Published online by Cambridge University Press:  06 February 2017

Raul Gomez ,
Dmitry Gourevitch  and
Siddhartha Sahi
Raul Gomez
Affiliation:
Department of Mathematics, 593 Malott Hall, Cornell University, Ithaca, NY 14853, USA email [email protected]
Dmitry Gourevitch
Affiliation:
The Incumbent of Dr. A. Edward Friedmann Career Development Chair in Mathematics, Faculty of Mathematics and Computer Science, Weizmann Institute of Science, PO Box 26, Rehovot 76100, Israel email [email protected]
Siddhartha Sahi
Affiliation:
Department of Mathematics, Rutgers University, Hill Center - Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA email [email protected]
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Abstract

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.

Keywords

reductive groupWhittaker–Fourier coefficientBernstein–Zelevinsky derivativewave-front setoscillator representationHeisenberg group

MSC classification

Primary: 20G05: Representation theory 20G20: Linear algebraic groups over the reals, the complexes, the quaternions 20G25: Linear algebraic groups over local fields and their integers 20G30: Linear algebraic groups over global fields and their integers 20G35: Linear algebraic groups over adèles and other rings and schemes
Secondary: 22E27: Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.) 22E46: Semisimple Lie groups and their representations 22E50: Representations of Lie and linear algebraic groups over local fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings 17B08: Coadjoint orbits; nilpotent varieties
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 2 , February 2017 , pp. 223 - 256
Copyright
© The Authors 2017 

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