Equivariant (co)homology of flag manifolds, convolution algebra and nil hecke algebra?

Question

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 $NH$?

Answer 1

I did more computation recently, and I got what I desired.

Firstly, to be exact, it should be the cohomology group rather than homology group, and presentation in the question is wrong, it should be $$\Bbbk\left<X_i,\partial_j\right>_{{1\leq i\leq n}\atop{1\leq j\leq n-1}}\bigg/ \left<\begin{array}{c} \partial_i\partial_{i-1}\partial_i=\partial_{i-1}\partial_i\partial_{i-1},\\ |i-j|\geq 2, \quad \partial_i\partial_j=\partial_j\partial_i,\\ \partial_i^2=0. \end{array}\begin{array}{c} X_iX_j=X_jX_i,\\ \partial_iX_j-X_{s_i(j)}\partial_i\\ =\delta_{i,j}-\delta_{i+1,j}. \end{array}\right>$$ I was mislead by Kumar's definition (Kac-Moody Groups, their Flag Variety and Representation Theory) of Nil-Hecke ring and the definition of convolution in homology.

• To prove this, one can first do it in nonequivairant case, the $G$-orbits of $G/B\times G/B$ are one-to-one correspondent to $B$-orbit of $G/B$, i.e. Schubert cells.
• The Poincar'e duality of each, say $\partial_w$, with respect to the Schuber cell $BwB/B$, acts on $H^*(G/B)$ by the Demazure operator $\partial_w$. To check this, it suffices to do the intersection product, where they all intersects transversally.
• The $X_i=X_i\partial_e$, where $H^*(G/B)$ acts on $H^*(G/B\times G/B)$ by the first projection acts on $H^*(G/B)$ by left multiplication $X_i$.
• Now relation is easy to check, by a standard topological argument, it is an isomorphism (for example, Harish–Leray). Actually, $H^*(G/B\times G/B)$ is actually the subalgebra generated by left mutiplications and Demazure operators in $\operatorname{End}_{\Bbbk}(H^*(G/B))$.
• To deal with equivariant case, we first do it in $T$-equivariant case, it is harmless since $H_G^*(X)\to H_T^*(X)$ is always injective ($\operatorname{char} \Bbbk=0$).
• There no longer exists Poincar'e duality, but the pairing of cells also gives a well-defined cohomology class. The computation of the result of paring of cells in nonequivariant case can be directly move to equivariant case. As a result, so it also acts as Demazure operator.
• The rest is completely the same to nonequivariant case. Actually, $H_G^*(G/B\times G/B)$ is actually the subalgebra generated by left mutiplications and Demazure operators in $\operatorname{End}_{H_G^*(pt)}(H^*(G/B))$. The actions are all $H_G^*(pt)$ map by the associativity of convolution $$H_G^*(G/B\times G/B)\stackrel{\displaystyle\curvearrowright}{\phantom{\square}} H_G^*(G/B\times pt)\stackrel{\displaystyle\curvearrowleft}{\phantom{\square}} H_G^*(pt\times pt). $$

---

The last two points are wrong. The real reason is, over characteristic zero case, $H_G(G/B)$ is known to be free of rank $\dim H(G/B)$ over $H_G(pt)$. So bthe convolution algebra is eaxctly of rank $\dim H(G/B)^2$ over $H_G(pt)$. This is the main point.