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Lusztig varieties and Macdonald polynomials

Arun Ram

Abstract

This paper uses Lusztig varieties to give central elements of the Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the finite Chevalley group $GL_n(\mathbb{F}_q)$. We explain how these central elements are related to Macdonald polynomials and how this provides a framework for generalizing integral form and modified Macdonald polynomials to Lie types other than $GL_n$. The key steps are to recognize (a) that counting points in Lusztig varieties is equivalent to computing traces on the Hecke algebras, (b) that traces on the Hecke algebra determine elements of the center of the Hecke algebra, (c) that the Geck-Rouquier basis elements of the center of the Hecke algebra produce an `expansion matrix', (d) that the parabolic subalgebras of the Hecke algebra produce a `contraction matrix' and (e) that the combination `expansion-contraction' is the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials.

Lusztig varieties and Macdonald polynomials

Abstract

This paper uses Lusztig varieties to give central elements of the Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the finite Chevalley group . We explain how these central elements are related to Macdonald polynomials and how this provides a framework for generalizing integral form and modified Macdonald polynomials to Lie types other than . The key steps are to recognize (a) that counting points in Lusztig varieties is equivalent to computing traces on the Hecke algebras, (b) that traces on the Hecke algebra determine elements of the center of the Hecke algebra, (c) that the Geck-Rouquier basis elements of the center of the Hecke algebra produce an `expansion matrix', (d) that the parabolic subalgebras of the Hecke algebra produce a `contraction matrix' and (e) that the combination `expansion-contraction' is the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials.
Paper Structure (25 sections, 7 theorems, 130 equations)

This paper contains 25 sections, 7 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Motivation for this paper
  3. Plan of the paper
  4. The path to generalization to Lie types other than $GL_n$
  5. Acknowledgments
  6. Hecke algebras and $\mathbf{1}_B^G$
  7. The $(G,H)$-bimodule $\mathbf{1}_B^G$
  8. Bases of the center of $H$
  9. Conjugacy classes, Schubert cells and Lusztig varieties
  10. Parabolic Springer fibers
  11. Counting in $GL_n(\mathbb{F}_q)$
  12. Macdonald polynomials
  13. Counting points of Lusztig varieties in $GL_n(\mathbb{F}_q)$
  14. Modified Hall-Littlewoods and parabolic Springer fibers
  15. Expansion-contraction
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let $g\in G$ and let $\mathcal{C}_g$ be the conjugacy class of $g$. Let $w\in W$. Then Using notations as in mincentH and GRbasis, the element of $Z(H)$ given by acts on $\mathbf{1}_B^G$ the same way as the element of $Z(\mathbb{C} G)$ given by $\frac{\vert G/B\vert}{\vert \mathcal{C}_g\vert} C_g$, where

Theorems & Definitions (12)

  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Proposition 2.3
  • proof
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 4.1
  • proof
  • ...and 2 more