The nil-Hecke algebra of a Weyl group $W$ is a version of the Hecke algebra of $W$ where the square of each simple generator $T_i$ is set equal to zero. For each $w\in W$, one defines an element $T_w$ of the nil-Hecke algebra to be a product of simple generators $T_i$ corresponding to a reduced factorization of $w$. It is known that the resulting element $T_w$ is independent of the choice of reduced factorization of $w$ and, moreover, the elements $T_w, w\in W$, form a basis of the nil-Hecke algebra. Further, the nil-Hecke algebra acts naturally on the space of polynomials on the Cartan subalgebra by divided difference operators, aka BGG-Demazure operators.
I am interested in the basis element $T_w$, as well as its action on the space of polynomials, in the particular case where $w$ is a Coxeter element of $W$. Has that particular element appeared somewhere before, eg., when $W$ is the Symmetric group ?
