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 Lam's 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.

中文翻译：

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顶点算子、可解格模型和 Metaplectic Whittaker 函数

我们表明，一般线性群的 $n$-fold 覆盖上的球形 Whittaker 函数自然产生于 Kashiwara, Miwa 引入的 $U_q(\widehat{\mathfrak{sl}}(n))$ 的量子 Fock 空间表示和斯特恩 (KMS)。我们通过重新考虑称为“metaplectic ice”的可解晶格模型来达到这种联系，其分配函数是 metaplectic Whittaker 函数。首先，我们证明了对 metaplectic Whittaker 共变量的特定 Hecke 作用（直到扭曲）与 Ginzburg、Reshetikhin 和 Vasserot 的 Hecke 作用一致。这使我们能够通过 Drinfeld 扭曲来扩展 KMS 的框架，将高斯和引入到量子楔中，这是连接到 metaplectic 形式所必需的。我们的主要定理将这个冰模型的行转移矩阵解释为“一半” 量子 Fock 空间上的顶点算子与 $U_q(\widehat{\mathfrak{sl}}(n))$ 的作用交织在一起。在这个过程中，我们引入了新的对称函数，称为 \textit{metaplectic 对称函数}，并解释它们与 GL$_r$ 的 $n$-fold metaplectic 覆盖上的 Whittaker 函数的关系。这些类似于 Lascoux、Leclerc 和 Thibon 引入的 \textit{LLT 多项式}；事实上，metaplectic 对称函数是（直至扭曲）由 Lam 定义的 \textit{超对称 LLT 多项式} 的特化。事实上，Lam 根据与 $U_q(\widehat{\mathfrak{sl}}(n))$-action 交换的 Fock 空间上的 Heisenberg 代数动作构建了对称函数族。我们解释说半顶点算子同意 Lam' s 的构造和这种解释允许元反射对称函数和 Whittaker 函数的许多新恒等式，包括柯西恒等式。虽然 metaplectic 对称函数和 LLT 多项式都可以与 $q$-Fock 空间上的顶点算子相关，但只有 metaplectic 对称函数连接到可解格模型。