This question is a sequel to 66602

The Hecke algebra is the quotient of the group algebra of the braid group of type $A_n$ by quadratic relations and the affine Hecke algebra is the quotient of the group algebra of the braid group of type $B_n$ by quadratic relations. This is well known and the relations can be found in the references in the answer and comment to the above question.

This gives an inclusion of the Hecke algebra in the affine Hecke algebra and I am interested in the centraliser algebra. The obvious elements in the centraliser are the central elements in the affine Hecke algebra. This centre is (I don't have the reference) the algebra of symmetric polynomials in the $x_1,\ldots ,x_n$. The other obvious elements are the central elements in the Hecke algebra.

These are all the elements I can construct in the centraliser. Is there any construction which gives more elements in the centraliser? Of course I would really like to have a set of generators.

The motivation is to understand the $6j$ symbols of $GL(n)$ which was also the subject of my question 15800.