Parabolic Kazhdan–Lusztig polynomials, plethysm and generalized Hall–Littlewood functions for classical types

Abstract

We use power sums plethysm operators to introduce H functions which interpolate between the Weyl characters and the Hall–Littlewood functions Q ′ corresponding to classical Lie groups. The coefficients of these functions on the basis of Weyl characters are parabolic Kazhdan–Lusztig polynomials and thus, by works of Kashiwara and Tanisaki, are nonnegative. We prove that they can be regarded as quantizations of branching coefficients obtained by restriction to certain subgroups of Levi type. The H functions associated to linear groups coincide with the polynomials introduced by Lascoux, Leclerc and Thibon in [A. Lascoux, B. Leclerc, J.Y. Thibon, Ribbon tableaux, Hall Littelwood functions, quantum affine algebras, J. Math. Phys. 38 (1996) 1041–1068].