  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.
