  The main goal of this paper is to categorify the specialized parasymmetric
(intermediate) Macdonald polynomials. These polynomials depend on a parabolic
subalgebra of a simple Lie algebra and generalize the symmetric and
nonsymmetric Macdonald polynomials.
  To achieve this we introduce cyclic modules of the parahoric subalgebras of
the affine Kac-Moody Lie algebras such that their characters coincide with the
specializations of the parasymmetric polynomials at zero and infinity. These
cyclic modules are proved to coincide with standard and costandard objects in
certain categories of representations of parahoric algebras. We show that the
categories in question are stratified, i.e. they are graded highest weight
categories. As a consequence, we derive an analog of the Peter-Weyl theorem
describing the bi-module of functions on the parahoric and parabolic Lie groups
via the mentioned above standard and costandard modules.
