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Register a new userA Yang-Baxter equation for metaplectic ice
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We will give new applications of quantum groups to the study of spherical Whittaker functions on the metaplectic $n$-fold cover of $GL(r,F)$, where $F$ is a nonarchimedean local field. Earlier Brubaker, Bump, Friedberg, Chinta and Gunnells had shown that these Whittaker functions can be identified with the partition functions of statistical mechanical systems. They postulated that a Yang-Baxter equation underlies the properties of these Whittaker functions. We confirm this, and identify the corresponding Yang-Baxter equation with that of the quantum affine Lie superalgebra $U_{sqrt{v}}(widehat{mathfrak{gl}}(1|n))$, modified by Drinfeld twisting to introduce Gauss sums. (The deformation parameter $v$ is specialized to the inverse of the residue field cardinality.) For principal series representations of metaplectic groups, the Whittaker models are not unique. The scattering matrix for the standard intertwining operators is vector valued. For a simple reflection, it was computed by Kazhdan and Patterson, who applied it to generalized theta series. We will show that the scattering matrix on the space of Whittaker functions for a simple reflection coincides with the twisted $R$-matrix of the quantum group $U_{sqrt{v}}(widehat{mathfrak{gl}}(n))$. This is a piece of the twisted $R$-matrix for $U_{sqrt{v}}(widehat{mathfrak{gl}}(1|n))$, mentioned above.
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We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of $p$-adic groups and $R$-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on $p$-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplect
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 stabilize the full Arthur-Selberg trace formula for the metaplectic covering of symplectic groups over a number field. This provides a decomposition of the invariant trace formula for metaplectic groups, which encodes information about the genuine $L^2$-automorphic spectrum, into a linear combination of stable trace formulas of products of split odd orthogonal groups via endoscopic transfer. By adapting the strategies of Arthur and Moeglin-Waldspurger from the linear case, the proof is built on a long induction process that mixes up local and global, geometric and spectral data. As a by-product, we also stabilize the local trace formula for metaplectic groups over any local field of characteristic zero.
We interpret values of spherical Whittaker functions on metaplectic covers of the general linear group over a nonarchimedean local field as partition functions of two different solvable lattice models. We prove the equality of these two partition functions by showing the commutativity of transfer matrices associated to different models via the Yang-Baxter equation.
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