## Abstract

In the basic representation of[formula]realized via the algebra of symmetric functions, we compare the canonical basis with the basis of Macdonald polynomials wheret=q^{2}. We show that the Macdonald polynomials are invariant with respect to the bar involution defined abstractly on the representations of quantum groups. We also prove that the Macdonald scalar product coincides with the abstract Kashiwara form. This implies, in particular, that the Macdonald polynomials form an intermediate basis between the canonical basis and the dual canonical basis, and the coefficients of the transition matrix are necessarily bar invariant. We also verify that the Macdonald polynomials (after a natural rescaling) form a sublattice in the canonical basis lattice which is invariant under the divided powers action. The transition matrix with respect to this rescaling is integral and we conjecture its positivity. For levelk, we expect a similar relation between the canonical basis and Macdonald polynomials withq^{2}=t^{k}.

Original language | English |
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Pages (from-to) | 95-127 |

Number of pages | 33 |

Journal | Advances in Mathematics |

Volume | 140 |

Issue number | 1 |

DOIs | |

State | Published - 1 Dec 1998 |

Externally published | Yes |

### Bibliographical note

Funding Information:We thank H. Garland and I. Grojnowski for their active interest and participation at different stages of this work. I. Frenkel is supported in part by NSF grants DMS-9400908, DMS-9700765. N. Jing is supported in part by NSA grant MDA 904-97-1-0062.

### Funding

We thank H. Garland and I. Grojnowski for their active interest and participation at different stages of this work. I. Frenkel is supported in part by NSF grants DMS-9400908, DMS-9700765. N. Jing is supported in part by NSA grant MDA 904-97-1-0062.

Funders | Funder number |
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National Science Foundation | DMS-9700765, DMS-9400908 |

Directorate for Mathematical and Physical Sciences | 9700765, 9400908 |

National Security Agency | MDA 904-97-1-0062 |