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Register a new userGeneralized and degenerate Whittaker quotients and Fourier coefficients
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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?
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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.
Let $G$ be a reductive group over a local field $F$ of characteristic zero, Archimedean or not. Let $X$ be a $G$-space. In this paper we study the existence of generalized Whittaker quotients for the space of Schwartz functions on $X$, considered as a representation of $G$. We show that the set of nilpotent elements of the dual space to the Lie algebra such that the corresponding generalized Whittaker quotient does not vanish contains the nilpotent part of the image of the moment map, and lies in the closure of this image. This generalizes recent results of Prasad and Sakellaridis. Applying our theorems to symmetric pairs $(G,H)$ we show that there exists an infinite-dimensional $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is non-compact. For quasi-split $G$ we also extend to the Archimedean case the theorem of Prasad stating that there exists a generic $H$-distinguished representation of $G$ if and only if the real reductive group corresponding to the pair $(G,H)$ is quasi-split. In the non-Archimedean case our result also gives bounds on the wave-front sets of distinguished representations.
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.
We show that spherical Whittaker functions on an $n$-fold cover of the general linear group arise naturally from the quantum Fock space representation of $U_q(widehat{mathfrak{sl}}(n))$ introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering solvable lattice models known as `metaplectic ice whose partition functions are metaplectic Whittaker functions. First, we show that a certain Hecke action on metaplectic Whittaker coinvariants agrees (up to twisting) with a Hecke action of Ginzburg, Reshetikhin, and Vasserot. This allows us to expand the framework of KMS by Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are necessary for connections to metaplectic forms. Our main theorem interprets the row transfer matrices of this ice model as `half vertex operators on quantum Fock space that intertwine with the action of $U_q(widehat{mathfrak{sl}}(n))$. In the process, we introduce new symmetric functions termed textit{metaplectic symmetric functions} and explain how they relate to Whittaker functions on an $n$-fold metaplectic cover of GL$_r$. These resemble textit{LLT polynomials} introduced by Lascoux, Leclerc and Thibon; in fact the metaplectic symmetric functions are (up to twisting) specializations of textit{supersymmetric LLT polynomials} defined by Lam. Indeed Lam constructed families of symmetric functions from Heisenberg algebra actions on the Fock space commuting with the $U_q(widehat{mathfrak{sl}}(n))$-action. We explain that half vertex operators agree with Lams construction and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials can be related to vertex operators on the $q$-Fock space, only metaplectic symmetric functions are connected to solvable lattice models.
We study the dimension of the space of Whittaker functionals for depth zero representations of covering groups. In particular, we determine such dimensions for arbitrary Brylinski-Deligne coverings of the general linear group. The results in the paper are motivated by and compatible with the work of Howard and the second author, and earlier work by Blondel.
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