Generalized and degenerate Whittaker models

Raul Gomez, Dmitry Gourevitch, Siddhartha Sahi

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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 GL(n)(F) this implies that a smooth admissible representation p has a generalized Whittaker model WO(pi) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wave-front set WF(p). Previously this was only known to hold for F non-archimedean and O maximal in WF(p), see Moeglin andWaldspurger [Modeles de Whittaker degeneres pour des groupes p-adiques, Math. Z. 196 (1987), 427-452]. We also express WO(pi) as an iteration of a version of the Bernstein-Zelevinsky derivatives [Bernstein and Zelevinsky, Induced representations of reductive p-adic groups. I., Ann. Sci. Ec. Norm. Super. (4) 10 (1977), 441-472; Aizenbud et al., Derivatives for representations of GL(n, R) and GL(n, C), Israel J. Math. 206 (2015), 1-38]. This enables us to extend to GL(n)(R) and GL(n)(C) several further results by Moeglin and Waldspurger on the dimension of WO(pi) and on the exactness of the generalized Whittaker functor.

Original languageEnglish
Pages (from-to)223-256
Number of pages34
JournalCompositio Mathematica
Volume153
Issue number2
DOIs
StatePublished - Feb 2017

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