Strong multiplicity one theorems for affine Hecke algebras of type A

Given an irreducible module for the affine Hecke algebra H-n of type A, we consider its restriction to Hn-1. We prove that the socle of restriction is multiplicity free and moreover that the summands lie in distinct blocks. This is true regardless of the characteristic of the field or of the order of the parameter q in the definition of H-n. The result generalises and implies the classical "branching rules" that describe the restriction of an irreducible representation of the symmetric group S-n to Sn-1.