Take $W = S_n$ for simplicity, though other Weyl groups work too. Let $r_i$ denote the $i$th simple reflection acting on ${\mathbb A}^n$, and $\partial_i = 1/(x_i - x_{i+1}) (Id - r_i)$ denote the corresponding divided difference operator.

It's easy to show that the operators $r_i + c \partial_i$ satisfy the Coxeter relations. I know I saw this in a Lascoux article, but there are so many that I'm hoping mathoverflow can tell me which one so I don't have to pore over the French, or can suggest some other canonical reference, the older the better.

Separately, I'd like to know if any author explicitly discusses these in the context of the Steinberg variety, where the $c$ should be the equivariant cohomology parameter corresponding to dilation of the cotangent bundle, I guess.