A mixed hook-length formula for affine Hecke algebras

Abstract

Consider the affine Hecke algebra $H_l$ corresponding to the group $GL_l$ over a $p$-adic field with the residue field of cardinality $q$. Regard $H_l$ as an associative algebra over the field $C(q)$. Consider the $H_{l+m}$-module $W$ induced from the tensor product of the evaluation modules over the algebras $H_l$ and $H_m$. The module $W$ depends on two partitions $\lambda$ of $l$ and $\mu$ of $m$, and on two non-zero elements of the field $C(q)$. There is a canonical operator $J$ acting on $W$, it corresponds to the trigonometric $R$-matrix. The algebra $H_{l+m}$ contains the finite dimensional Hecke algebra of rank $l+m$ as a subalgebra, and the operator $J$ commutes with the action of this subalgebra on $W$. Under this action, $W$ decomposes into irreducible subspaces according to the Littlewood-Richardson rule. We compute the eigenvalues of $J$, corresponding to certain multiplicity-free irreducible components of $W$. In particular, we give a formula for the ratio of two eigenvalues of $J$, corresponding to the ``highest'' and the ``lowest'' components. As an application, we derive the well known $q$-analogue of the hook-length formula for the number of standard tableaux of shape $\lambda$.