Correspondance de Howe et front d'onde

Mœglin, Colette

doi:10.1006/aima.1997.1689

Cited by 11 publications

(5 citation statements)

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“…Then every nilpotent orbit O ⊂ g is in the image of Max Hom(V, Ṽ ) under the moment map ϕ, and for any smooth irreducible genuine representation (π, V ) of G, we have Θ(π) = 0 [26,33]. In this case and for k non-Archimedean, Theorem 1.1 is due to Moeglin [25]. Our approach, explained in the following paragraph, is in some sense more conceptual.…”

Section: Setmentioning

confidence: 96%

Local theta lifting of generalized Whittaker models associated to nilpotent orbits

Gomez

Zhu

2014

Geom. Funct. Anal.

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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 the full theta lift of π. When (G,G) is in the stable range with G the smaller member, every nilpotent orbit O ⊂ g is in the image of the moment map from Max Hom (V,Ṽ ). In this case, and for k non-Archimedean, the result has been previously obtained by Moeglin in a different approach.2000 Mathematics Subject Classification. 22E46 (Primary). A.1. Vector valued distributions 42 A.2. Transverse order of a V -distribution 42 A.3. Transverse jet bundle 43 References 43 1. Introduction and main resultLet k be a local field of characteristic 0, for which we fix a non-trivial unitary character ψ. Let G be a reductive group over k, g its Lie algebra, on which we fix an Ad G-invariant non-degenerateSet g j = {Z ∈ g | ad(H)Z = jZ} , for j ∈ Z. Then, from standard sl 2 -theory, we have a finite direct sum g = ⊕ j∈Z g j . Define the Lie subalgebras u = ⊕ j≤−2 g j , n = ⊕ j≤−1 g j , p = ⊕ j≤0 g j and m = g 0 . Let U , N , P , and M be the corresponding subgroups of G. Thus U = exp u, N = exp n,Since κ(X, [u, u]In particular M X is reductive. For the moment assume that g −1 = 0, or equivalently u n. In this case ad(X)| g −1 : g −1 −→ g 1 is an isomorphism, and we may define a symplectic structure on g −1 by setting, for all S, T ∈ g −1 .We may exhibit a canonical surjective group homomorphism from N to the associated Heisenberg group H which maps exp Z to κ(X, Z) in the center of H, for Z ∈ u. Then, according to the Stonevon Neumann theorem, there exists a unique, up to equivalence, smooth irreducible (unitarizable) representation (ρ χγ , S χγ ) of N such that U acts by the character χ γ . See Section 3.3 for details.Since M X preserves γ and thus the symplectic form κ −1 , it is well-known [51] that there exists a central cover of M X , to be denoted by M χγ , and a representation of a semi-direct product M χγ ⋉ N on S χγ which extends the representation ρ χγ of N . We refer to the representation of M χγ ⋉N on S χγ as the smooth oscillator-Heisenberg (or Weil) representation. If g −1 = 0, then we define M χγ to be just M X . For notational convenience, we also denote by (ρ χγ , S χγ ) the 1-dimensional representation of N = U given by the character χ γ . We may again view (ρ χγ , S χγ ) as a representation of M X ⋉ N , with the trivial M X action.In this article, a smooth representation of a reductive group over k will mean a smooth representation in the usual sense for k non-Archimedean, namely it is locally constant, and a Casselman-Wallach representation for k Archimedean. The reader may consult [50, Chapter 11] for mor...

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Section: Setmentioning

confidence: 96%

Local theta lifting of generalized Whittaker models associated to nilpotent orbits

Gomez

Zhu

2014

Geom. Funct. Anal.

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“…In this section we will discuss a refinement, due to Gomez and the author, of a previous work [15], which computes the generalized Whittaker models of the full theta lift in a general setting. Related earlier works on transition of models include those of Furusawa [9], Moeglin [37] and Ginzburg-Jiang-Soudry [14]. See also Gan's article in this volume [10].…”

Section: Correspondence Of Generalized Whittaker Modelsmentioning

confidence: 99%

Local theta correspondence and nilpotent invariants

Zhu¹

2019

Representations of Reductive Groups

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We consider two types of nilpotent invariants associated to smooth representations, namely generalized Whittaker models, and associated characters (in the case of a real reductive group). We survey some recent results on the behavior of these nilpotent invariants under local theta correspondence, and highlight the special role of a certain double fiberation of moment maps.2000 Mathematics Subject Classification. 22E46 (Primary).

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“…We will discuss both local and global aspects. As advanced topics we introduce the useful notion of wave-front set [312,[322][323][324] and discuss the method of Piatetski-Shapiro and Shalika [348,390]. General references are [66,183,244] and we also found the discussions in [171,202,247,275,317] very useful.…”

Section: Chapter 7 Whittaker Functions and Fourier Coefficientsmentioning

confidence: 99%

Eisenstein series and automorphic representations

Fleig¹,

Gustafsson²,

Kleinschmidt³

et al. 2015

Preprint

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Note to the readerThis book has grown out of our endeavour to understand the theory of automorphic representations and the structure of Fourier expansions of automorphic forms with a particular emphasis on adelic methods and Eisenstein series. Our intention is also to open a path of communication between mathematicians and physicists, in particular string theorists, interested in these topics. Most of the results presented herein exist already in the literature and we benefitted greatly from [66, 98, 183-185, 297, 389, 405, 406]; our exposition differs, however, at places from the standard one. A few new results and examples are included as well; in particular, we provide many techniques for working out aspects of the Fourier expansion of Eisenstein series. We would be very grateful to learn of any omissions and mistakes that we have made unintentionally.Sections that are more advanced or explore topics beyond the main focus of this treatise are marked with an asterisk. discussions with colleagues both from the mathematics community and from the string theory community. We are especially indebted to

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Correspondance de Howe et front d'onde

Cited by 11 publications

References 11 publications

Local theta lifting of generalized Whittaker models associated to nilpotent orbits

Local theta lifting of generalized Whittaker models associated to nilpotent orbits

Local theta correspondence and nilpotent invariants

Eisenstein series and automorphic representations

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