Return to 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).

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 $\mathbb C[W]\otimes \mathbb C[x_1,x_2,\dotsc,x_n]$ (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 Degenerate affine Hecke algebras and Yangians (for type A) and Lusztig's paper Cuspidal local systems and graded Hecke algebras, I, and the Lusztig paper that Stephen references, which is pure algebra. I think it also arises in some form in Cherednik's paper A new interpretation of Gelʹfand–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).

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).

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).

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).

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).