Parahoric Lie algebras and parasymmetric Macdonald polynomials

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.