Mixed hook-length formula for degenerate a fine Hecke algebras

Abstract

Take the degenerate a fine Hecke algebra H _l+m corresponding to the group GL _{l +m} over a p-adic field.Consider the H _l+m -module W induced from the tensor product of the evaluation modules over the algebras H _l x and H _m .The module W depends on two partitions λ of l and μ of m, and on two complex numbers.There is a canonical operator J acting in W, it corresponds to the Yang R-matrix.The algebra H _l+m contains the symmetric group algebra ℂ S _{l +m} as a subalgebra, and J commutes with the action of this subalgebra in 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 maximal and minimal irreducible components. As an application of our results,we derive the well-known hook-length formula for the dimension of the irreducible ℂ S _l -module corresponding to λ.