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Peter-Weyl theorem for Iwahori groups and highest weight categories

Evgeny Feigin, Anton Khoroshkin, Ievgen Makedonskyi, Daniel Orr

TL;DR

The paper develops a categorical framework to analyze the algebra of functions on the Iwahori subgroup of an affine Kac–Moody group, proving a Peter–Weyl-type theorem in this setting. By identifying standard and costandard objects with generalized Weyl modules and expressing their characters via specialized nonsymmetric Macdonald polynomials, it shows that the category of Iwahori representations is stratified and yields a filtrational decomposition of the function bimodule gr k[𝕀] as a sum of tensor products ∇_λ ⊗_{A_λ} (Δ_λ^∨)^o. This leads to a reciprocal Macdonald-type identity expressing indecomposable projective characters as nonnegative combinations of standard characters, with BGG reciprocity providing the positivity. The work unifies pro-algebraic group structures, Macdonald polynomial theory, and Iwahori module theory to produce explicit character formulas and structural filtrations, extending previous A-type results to general types and offering a categorical explanation for Macdonald identities in this setting.

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.

Peter-Weyl theorem for Iwahori groups and highest weight categories

TL;DR

The paper develops a categorical framework to analyze the algebra of functions on the Iwahori subgroup of an affine Kac–Moody group, proving a Peter–Weyl-type theorem in this setting. By identifying standard and costandard objects with generalized Weyl modules and expressing their characters via specialized nonsymmetric Macdonald polynomials, it shows that the category of Iwahori representations is stratified and yields a filtrational decomposition of the function bimodule gr k[𝕀] as a sum of tensor products ∇_λ ⊗_{A_λ} (Δ_λ^∨)^o. This leads to a reciprocal Macdonald-type identity expressing indecomposable projective characters as nonnegative combinations of standard characters, with BGG reciprocity providing the positivity. The work unifies pro-algebraic group structures, Macdonald polynomial theory, and Iwahori module theory to produce explicit character formulas and structural filtrations, extending previous A-type results to general types and offering a categorical explanation for Macdonald identities in this setting.

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.
Paper Structure (27 sections, 46 theorems, 143 equations)

This paper contains 27 sections, 46 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Highest weight categories and stratified categories
  3. Ordering irreducibles and Serre subcategories
  4. Stratified categories
  5. Stratified categories with enough projectives
  6. Stratified categories with enough injectives
  7. Structure of bimodule of functions
  8. Reductive Lie algebras
  9. Pro-algebraic groups
  10. Algebras of functions on pro-algebraic groups
  11. Highest weight algebras
  12. Filtrations and bimodules
  13. Nonsymmetric Macdonald polynomials
  14. Affine Weyl groups
  15. Nonsymmetric Macdonald polynomials
  16. ...and 12 more sections

Key Result

Theorem A

(Theorem FT) If the category ${\mathfrak C}$ of graded representations of a Lie algebra ${\mathfrak g}$ (satisfying technical assumptions from section progroup) is stratified, then the algebra of functions $\Bbbk[G]$ on the corresponding simply-connected algebraic group $G$ admits a filtration of $G

Theorems & Definitions (112)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 1.1
  • Remark 1.3
  • Remark 1.6
  • Definition 1.8
  • Lemma 1.10
  • proof
  • Remark 1.12
  • ...and 102 more