Peter-Weyl theorem for Iwahori groups and highest weight categories

TL;DR

This paper establishes a Peter-Weyl theorem for Iwahori groups by connecting representation categories with Macdonald polynomials, revealing a highest weight structure and new identities.

Contribution

It introduces a generalized highest weight structure for Iwahori group representations and links standard objects to Macdonald polynomials, extending classical harmonic analysis results.

Findings

Identification of standard and costandard objects with Weyl modules

Expression of characters via specialized nonsymmetric Macdonald polynomials

Establishment of a stratified highest weight structure

Abstract

We study the algebra of functions on the Iwahori group via the category of graded bounded representations of its Lie algebra. In particular, we identify the standard and costandard objects in this category with certain generalized Weyl modules. Using this identification we express the characters of the standard and costandard objects in terms of specialized nonsymmetric Macdonald polynomials. We also prove that our category of interest admits a generalized highest weight structure (known as stratified structure). We show, more generally, that such a structure on a category of representations of a Lie algebra implies the Peter-Weyl type theorem for the corresponding algebraic group. In the Iwahori case, standard filtrations of indecomposable projective objects correspond to new ``reciprocal'' Macdonald-type identities.