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Abstract We construct a family of solvable lattice models whose partition functions include ‐adic Whittaker functions for general linear groups from two very different sources: from Iwahori‐fixed vectors and from metaplectic covers. Interpolating between them by Drinfeld twisting, we uncover unexpected relationships between Iwahori and metaplectic Whittaker functions. This leads to new Demazure operator recurrence relations for spherical metaplectic Whittaker functions. In prior work of the authors it was shown that the row transfer matrices of certain lattice models for spherical metaplectic Whittaker functions could be represented as ‘half‐vertex operators’ operating on the ‐Fock space of Kashiwara, Miwa and Stern. In this paper the same is shown for all the members of this more general family of lattice models including the one representing Iwahori Whittaker functions. 

We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$$ -- {\it biaxial double} $$(\beta,q)$$-{\it Grothendieck polynomials} -- which specialize at $$q=0$ and $v=1$ to double $$\beta$$-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $$n$$ pairs of variables is a Drinfeld twist of the $$U_q(\widehat{\mathfrak{sl}}_{n+1})$$ $$R$$-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $$\beta$$-Grothendieck polynomials and dual double $$\beta$$-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for $$\beta$$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $$\beta$$-Grothendieck polynomials, and prove a new branching rule for double $$\beta$$-Grothendieck polynomials.