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$\imath$Schur duality and Kazhdan-Lusztig basis expanded

Yaolong Shen, Weiqiang Wang

Abstract

Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an $\imath$Schur duality between an $\imath$quantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wang's quasi-split $\imath$Schur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $\imath$canonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the $\imath$Schur duality.

$\imath$Schur duality and Kazhdan-Lusztig basis expanded

Abstract

Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. We formulate an Schur duality between an quantum group of type AIII (allowing black nodes in its Satake diagram) and a Hecke algebra of type B acting on a tensor space, providing a common generalization of Jimbo-Schur duality and Bao-Wang's quasi-split Schur duality. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the canonical basis on the tensor space. An inversion formula for quasi-parabolic KL polynomials is established via the Schur duality.

Paper Structure

This paper contains 33 sections, 41 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Type B Kazhdan-Lusztig, expanded
  3. $\imath$Schur duality
  4. Compatible canonical bases
  5. An inversion formula
  6. Related works
  7. Further directions
  8. Organization
  9. Modules over Hecke algebra of type B
  10. Weyl group and Hecke algebra of type B
  11. A tensor module of $\mathscr H_{B_d}$
  12. Quasi-permutation modules
  13. Canonical bases on quasi-permutation modules
  14. Basic properties of $W_d$
  15. Minimal length representatives
  16. ...and 18 more sections

Key Result

Lemma \oldthetheorem

There is a right action of the Hecke algebra $\mathscr{H}_{B_d}$ on $\mathbb V^{\otimes d}$ as follows:

Theorems & Definitions (80)

  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Remark \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • ...and 70 more