# Nonsymmetric $q$-Cauchy identity and representations of the Iwahori algebra

## TL;DR

This paper connects specialized nonsymmetric Macdonald polynomials to the representation theory of the Iwahori algebra, providing a filtration of function spaces and conjecturing a generalization to all simple Lie algebras.

## Contribution

It introduces a new filtration of function spaces linked to Iwahori algebra modules and proves it for SL_n, extending the van der Kallen filtration.

## Key findings

- Filtration of function spaces with graded pieces isomorphic to tensor products of Weyl modules.
- Characters of graded pieces match terms of the specialized Mimachi-Noumi formula.
- Conjecture of analogous filtration for all simple Lie algebras, proven for SL_n.

## Abstract

The $t=0$ specialization of the Mimachi-Noumi Cauchy-type identity rewrites certain infinite product in terms of specialized nonsymmetric Macdonald polynomials of type $GL_n$. We interpret the infinite product as a character of the space of functions on a certain matrix space. We show that the space of functions admits a filtration such that the graded pieces are isomorphic to the tensor products of certain generalized global Weyl modules of the Iwahori algebra. We identify the characters of the graded pieces with the terms of the specialized Mimachi-Noumi formula. We conjecture the existence of an analogous filtration on the space of functions on the Iwahori group for all simple Lie algebras and prove the conjecture for $SL_n$. Our construction can be seen as a current algebra extension of the van der Kallen filtration on functions on a Borel subgroup.