Colored Vertex Models and Iwahori Whittaker Functions

Abstract

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.

References in corpus (11)

• Another proof of the alternating sign matrix conjecture
• Integrality of two variable Kostka functions
• Cylindric versions of specialised Macdonald functions and a deformed Verlinde algebra
• Between the stochastic six vertex model and Hall-Littlewood processes
• A Yang-Baxter equation for metaplectic ice
• Schubert puzzles and integrability I: invariant trilinear forms
• Metaplectic Ice for Cartan Type C
• Duality for metaplectic ice
• Metaplectic Demazure operators and Whittaker functions
• Colored five-vertex models and Demazure atoms
• Fusion procedure for the Yang-Baxter equation and Schur-Weyl duality

Cited by in corpus (7)

• Colored five-vertex models and Lascoux polynomials and atoms
• Frozen Pipes: Lattice Models for Grothendieck Polynomials
• Stochastic symplectic ice
• Colored five-vertex models and Demazure atoms
• Metaplectic Iwahori Whittaker functions and supersymmetric lattice models
• A class of partition functions associated with $E_{\tau ,\eta }(g{l}_3)$ by Izergin-Korepin analysis
• Quasi-solvable lattice models for ${\mathrm{Sp}}_{2n}$ and ${\mathrm{SO}}_{2n+1}$ Demazure atoms and characters