2014
DOI: 10.1007/s00039-014-0276-5
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Local theta lifting of generalized Whittaker models associated to nilpotent orbits

Abstract: Let (G,G) be a reductive dual pair over a local field k of characteristic 0, and denote by V andṼ the standard modules of G andG, respectively. Consider the set Max Hom(V,Ṽ ) of full rank elements in Hom(V,Ṽ ), and the nilpotent orbit correspondence O ⊂ g and Θ(O) ⊂g induced by elements of Max Hom(V,Ṽ ) via the moment maps. Let (π, V ) be a smooth irreducible representation of G. We show that there is a correspondence of the generalized Whittaker models of π of type O and of Θ(π) of type Θ(O), where Θ(π) is th… Show more

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Cited by 16 publications
(31 citation statements)
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“…Note that ϕ is necessarily nilpotent and given by the Killing form pairing with a (unique) nilpotent element f = f ϕ ∈ g. Following [MW87] we attach to (S, ϕ) a certain smooth representation W S,ϕ of G called a degenerate Whittaker model for G. Two classes of Whittaker pairs and the corresponding models will be of special interest to us. If S is a neutral element for f ϕ (see Definition 2.2.2 below), then we will say that (S, ϕ) is a neutral pair and call W S,ϕ a neutral model or a generalized model (see [Kaw85,MW87,Ya86,GZ14]). The second class consists of Whittaker pairs (S, ϕ) where S is the neutral element of a principal sl 2 -triple in G; in this case f ϕ is necessarily a principal nilpotent element for a Levi subgroup of G, and we will say (S, ϕ) is a PL pair, and W S,ϕ is a PL model or a principal degenerate model (see [Zel80,MW87,BH03,GS13]).…”
Section: General Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…Note that ϕ is necessarily nilpotent and given by the Killing form pairing with a (unique) nilpotent element f = f ϕ ∈ g. Following [MW87] we attach to (S, ϕ) a certain smooth representation W S,ϕ of G called a degenerate Whittaker model for G. Two classes of Whittaker pairs and the corresponding models will be of special interest to us. If S is a neutral element for f ϕ (see Definition 2.2.2 below), then we will say that (S, ϕ) is a neutral pair and call W S,ϕ a neutral model or a generalized model (see [Kaw85,MW87,Ya86,GZ14]). The second class consists of Whittaker pairs (S, ϕ) where S is the neutral element of a principal sl 2 -triple in G; in this case f ϕ is necessarily a principal nilpotent element for a Levi subgroup of G, and we will say (S, ϕ) is a PL pair, and W S,ϕ is a PL model or a principal degenerate model (see [Zel80,MW87,BH03,GS13]).…”
Section: General Resultsmentioning
confidence: 99%
“…Recently, the behaviour of the wave-front set and the generalized Whittaker models under θ-correspondence was studied in [GZ14,LoMa15]. For F non-archimedean, Moeglin and Waldspurger [MW87] have established that WFC(π) completely controls the spaces of generalized Whittaker models of interest, namely, the set of maximal orbits in WFC(π) coincides 1 If F is archimedean then by admissible we mean admissible Fréchet representation of moderate growth.…”
Section: General Resultsmentioning
confidence: 99%
“…In [GZ14,LoMa15,Prz] it is shown that the Whittaker support and the wave-front set have the expected behaviour under θ-correspondence.…”
mentioning
confidence: 91%
“…Important earlier work on this double fiberation of moment maps include those of Kraft-Procesi [24] and Prezbinda [44]. More recent work connected to the theme of this article include [41,42,33,15,35].…”
Section: Introductionmentioning
confidence: 99%