For a complex reductive group $G$ and its Borel subgroup $B$, it seems to be well-known that the equivariant homology group $H^G_*(G/B\times G/B)$ forms a nil-Heck algebra $$NH=\Bbbk[y_i,\partial_{j}]_{{1\leq i\leq n}\atop{1\leq j\leq n-1}}\big/\left<\begin{array}{c} \partial_i\partial_{i+1}\partial_i=\partial_{i+1}\partial_{i}\partial_{i+1}\\ \partial_{i}\partial_j=\partial_j\partial_i, |i-j|\geq 2\\ \partial_i^2=0\end{array},\quad \begin{array}{c}y_j\partial_j=\partial_j y_{j+1}\\ y_{j+1}\partial_j=\partial_j y_{j}\\ y_j\partial_i=\partial_iy_j, |i-j|\geq 2 \end{array}\right>$$ under convolution with the Schubert cells $X_w$ corresponding to the symbol $\partial_w$. Besides, its action over the equivariant cohomology group $H_G^*(G/B)=H_T(pt)=\Bbbk[x_1,\ldots,x_n]$ is the Demazure operator.

But I did not find any reference for this fact even for the definition of convolution. I only saw the usual homology (Borel--Moore homology) version and the K-theory version in *Representation Theory and Complex Geometry* by Neil ChrissVictor Ginzburg. Besides, they refer without proofs. Maybe it can be defined by sheaf theory, but then how to compute with the Schubert cells? Since $H_G(G/B\times G/B)=H_T(G/T)$, it has Schubert cells.

In the cohomology case, we can define the convolution in a proper way to be $$H^*_G(B\times A)\times H^*_G(C\times B)\stackrel{p_1^*\otimes p_3^*}\longrightarrow H_G^*(C\times B\times A)\otimes H_G^*(C\times B\times A)\stackrel{\smile}\longrightarrow H_G^*(C\times B\times A)\stackrel{(p_2)_*}\longrightarrow H_G^*(C\times A)$$ The last map is the Gysin push forward when $B$ is smooth compact. The problem of homology is that there is no intersection product for $EG\times_G C\times B\times A$ since it is infinite dimensional. Moreover when I compute the convolution over equivariant cohomology, it does not give the a proper isomorphism $H_G^*(G/B\times G/B)\to NH$.

My question is, are there any references for the fact that $H^G_*(G/B\times G/B)\cong NH$ under convolution and references for the definition of convolution algebra in equivariant homology? Further I also wonder if there is an isomorphism from cohomology to $NH$?