A mixed hook-length formula for affine Hecke algebras

Let Hl be the affine Hecke algebra corresponding to the group GLl over a p-adic field with residue field of cardinality q. We will regard Hl as an associative algebra over the field C(q). Consider the Hl+m-module W induced from the tensor product of the evaluation modules over the algebras Hl and Hm. The module W depends on two partitions λ of l and μ 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 Hl+m contains the finite dimensional Hecke algebra Hl+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 λ.