# Reference for representation of Weyl group using r_alpha + c partial_alpha

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.

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## 1 Answer

The difference operators (as you presumably know) are defined in a paper of Bernstein-Gelfand-Gelfand on Schubert cells etc. which is probably roughly as old as you can get. The fact that the $r_i + c\partial_i$ satisfy the Coxeter relations is implied by (equivalent to, pretty much) the fact that the graded/degenerate affine Hecke algebra is isomorphic as a vector space to $\mathbb C[W]\otimes \mathbb C[x_1,x_2,\ldots,x_n]$ (the operators give the action of the $\mathbb C[W]$ subalgebra on the polynomial representation of the degenerate affine Hecke algebra). The first references for the degenerate affine Hecke algebra are Drinfeld's paper on Yangians and degenerate affine Hecke algebras (for type A) and Lusztig's paper on cuspidal local systems and graded Hecke algebras (part I), and the Lusztig paper that Stephen references, which is pure algebra. I think it also arises in some form in Cherednik's paper on Gelfand-Tzetlin bases.

The connection to the equivariant cohomology of the Steinberg is examined in Lusztig's cuspidal local systems papers (it is the easiest "Springer theory" case, i.e. where you don't have to worry about cuspidal local systems).

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Better answer. It's embarrassing that I didn't know the Steinberg variety showed up in Lusztig's 1st paper. It's even more embarrassing that I forgot to mention BGG (though they don't study exactly the operators Allen was asking about). Also: would you mind adding links? –  GS Jan 14 '10 at 20:00
Added links to Drinfeld's and Lusztig's, I could find a link to Cherednik's, but the paper is "A new approach to Gelfand-Tzetlin bases" in Duke. You're right, BGG don't study the operators Allen talks about, I just added that for historical context, as it's as early a reference as I know to the difference operators. –  Kevin McGerty Jan 14 '10 at 22:39
Thanks for the links; I couldn't find Cherednik's paper for free either. –  GS Jan 14 '10 at 23:09