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 GLn(F) this implies that a smooth admissible representation π has a generalized Whittaker model WO(π) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wavefront set WF(π). Previously this was only known to hold for F nonarchimedean and O maximal in WF(π), see [MW87]. We also express WO(π) as an iteration of a version of the Bernstein-Zelevinsky derivatives [BZ77,AGS15a]. This enables us to extend to GLn(R) and GLn(C) several further results from [MW87] on the dimension of WO(π) and on the exactness of the generalized Whittaker functor.

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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...

This paper treats two topics: matrices with sign patterns and Jacobians of certain mappings. The main topic is counting the number of plus and minus coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg. We also give a graph-theoretic test for determining when the Jacobian of a chemical reaction dynamics has a sign pattern.Comment: 25 page