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 $NH=\Bbbk[y,\partial]$ under convolution with the Schubert cells $X_w$ corresponding to the symbol $\partial_w$.

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 the 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 between cohomology?