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 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...

Generalized and degenerate Whittaker models

“…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]).…”

“…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.…”

Generalized and degenerate Whittaker quotients and Fourier coefficients

“…In [GZ14,LoMa15,Prz] it is shown that the Whittaker support and the wave-front set have the expected behaviour under θ-correspondence.…”

Local theta correspondence and nilpotent invariants

“…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].…”