Branching rules, Kostka-Foulkes polynomials and $q$-multiplicities in tensor product for the root systems $B\_{n},C\_{n}$ and $D\_{n}$

Abstract

The Kostka-Foulkes polynomials $K$ related to a root system $\phi$ can be defined as alternated sums running over the Weyl group associated to $\phi .$ By restricting these sums over the elements of the symmetric group when $$ is of type $B,C$ or $D$, we obtain again a class $\widetilde{K}$ of Kostka-Foulkes polynomials. When $\phi$ is of type $C$ or $D$ there exists a duality beetween these polynomials and some natural $q$-multiplicities $U$ in tensor product \cite{lec}. In this paper we first establish identities for the $\widetilde{K}$ which implies in particular that they can be decomposed as sums of Kostka-Foulkes polynomials related to the root system of type $A$ with nonnegative integer coefficients. Moreover these coefficients are branching rule coefficients. This allows us to clarify the connection beetween the $q$-multiplicities $U$ and the polynomials defined by Shimozono and Zabrocki in \cite{SZ}. Finally we establish that the $q$-multiplicities $U$ defined for the tensor powers of the vector representation coincide up to a power of $q$ with the one dimension sum $X$ introduced in \cite{Ok} This shows that in this case the one dimension sums $$ are affine Kazhdan-Lusztig polynomials