Canonical Basis and Macdonald Polynomials

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=q2. 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 withq2=tk.