Colored Bosonic models and matrix coefficients

Abstract

We develop the theory of colored bosonic models (initiated by Borodin and Wheeler). We will show how a family of such models can be used to represent the values of Iwahori vectors in the “spherical model” of representations of $\mathrm{GL}_r (F)$, where $F$ is a nonarchimedean local field. Among our results are a monochrome factorization, which is the realization of the Boltzmann weights by fusion of simpler weights, a local lifting property relating the colored models with uncolored models, and an action of the Iwahori–Hecke algebra on the partition functions of a particular family of models by Demazure–Lusztig operators. As an application of the local lifting property we reprove a theorem of Korff evaluating the partition functions of the uncolored models in terms of Hall–Littlewood polynomials. Our results are very closely parallel to the theory of fermionic models representing Iwahori–Whittaker functions developed by Brubaker, Buciumas, Bump and Gustafsson, with many striking relationships between the two theories, confirming the philosophy that the spherical and Whittaker models of principal series representations are dual.