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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 $mathrm{GL}_n(F)$ this implies that a smooth admissible representation $pi$ has a generalized Whittaker model $mathcal{W}_{mathcal{O}}(pi)$ corresponding to a nilpotent coadjoint orbit $mathcal{O}$ if and only if $mathcal{O}$ lies in the (closure of) the wave-front set $mathrm{WF}(pi)$. Previously this was only known to hold for $F$ non-archimedean and $mathcal{O}$ maximal in $mathrm{WF}(pi)$, see [MW87]. We also express $mathcal{W}_{mathcal{O}}(pi)$ as an iteration of a version of the Bernstein-Zelevinsky derivatives [BZ77,AGS15a]. This enables us to extend to $mathrm{GL_n}(mathbb{R})$ and $mathrm{GL_n}(mathbb{C})$ several further results from [MW87] on the dimension of $mathcal{W}_{mathcal{O}}(pi)$ and on the exactness of the generalized Whittaker functor.
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The study of Whittaker models for representations of reductive groups over local and global fields has become a central tool in representation theory and the theory of automorphic forms. However, only generic representations have Whittaker models. In order to encompass other representations, one attaches a degenerate (or a generalized) Whittaker model $W_{mathcal{O}}$, or a Fourier coefficient in the global case, to any nilpotent orbit $mathcal{O}$. In this note we survey some classical and some recent work in this direction - for Archimedean, p-adic and global fields. The main results concern the existence of models. For a representation $pi$, call the set of maximal orbits $mathcal{O}$ with $W_{mathcal{O}}$ that includes $pi$ the Whittaker support of $pi$. The two main questions discussed in this note are: (1) What kind of orbits can appear in the Whittaker support of a representation? (2) How does the Whittaker support of a given representation $pi$ relate to other invariants of $pi$, such as its wave-front set?
We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group $G$ over a nonarchimedean local field $F$. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case $G=mathrm{GL}_r$ we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here `solvable means that the models have a family of Yang-Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang-Baxter equations come from a Drinfeld twist of the quantum group $U_q(widehat{mathfrak{gl}}(r|1))$, which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.
Let X be a smooth projective curve over an algebraically closed field of characteristic >2. Let Bun_{Mp_2} be the stack of metaplectic bundles on X of rank 2. In this paper we study the derived category of genuine l-adic sheaves on Bun_{Mp_2} in the framework of the quantum geometric Langlands. We describe the corresponding Whittaker category, develop the theory of geometric Eisenstein series and calculate the most non-degenerate Fourier coefficients of these Eisenstein series. The existing constructions of automorphic sheaves for GL_n are based on using Whittaker sheaves. Our calculations lead to a conjectural characterization of the Whittaker sheaf for Mp_2, though its existence is not clear. We also formulate a conjectural relation between the quantum Langlands functors and the theta-lifting functors for the dual pair (Mp_2, PGL_2).
In this paper we consider Iwahori Whittaker functions on $n$-fold metaplectic covers $widetilde{G}$ of $mathbf{G}(F)$ with $mathbf{G}$ a split reductive group over a non-archimedean local field $F$. For every element $phi$ of a basis of Iwahori Whittaker functions, and for every $ginwidetilde{G}$, we evaluate $phi(g)$ by recurrence relations over the Weyl group using vector Demazure-Whittaker operators. Specializing to the case of $mathbf{G} = mathbf{GL}_r$, we exhibit a solvable lattice model whose partition function equals $phi(g)$. These models are of a new type associated with the quantum affine super group $U_q(widehat{mathfrak{gl}}(r|n))$. The recurrence relations on the representation theory side then correspond to solutions to Yang-Baxter equations for the lattice models. Remarkably, there is a bijection between the boundary data specifying the partition function and the data determining all values of the Whittaker functions.
Let G be a reductive group (over an algebraically closed field) equipped with the metaplectic data. In this paper we study the corresponding twisted Whittaker category for G. We construct and study a functor from the latter category to the corresponding category of factorizable sheaves. It plays the role of the restriction functor from the category of representations of the big quantum group to those of the graded small quantum group. We also prove an analog in our setting of the Lusztig-Steinberg tensor product theorem for quantum groups describing the semi-simple part of the Whittaker category as a module over the Hecke algebra.
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