Equal-color quiver-flag correspondence is nilHecke

The data. For this quiver $E={⨁}_{h:i\to j}\mathrm{Hom}(V_i,V_j)=\{0\}$, so $X_{\nu }=F_{\nu }=\mathrm{Flag}({\mathbb{C}}^2)={\mathbb{P}}^1$ with $d_{\nu }={\dim }_{\mathbb{C}}{X}_{\nu }=1$ and $Z_{\nu ,\nu }=X_{\nu }{\times }_E{X}_{\nu }={\mathbb{P}}^1\times {\mathbb{P}}^1$. The convolution algebra is therefore the single corner $H^{\mathrm{B}\mathrm{M},G}({\mathbb{P}}^1\times {\mathbb{P}}^1;k)$, a free module over the polynomial ring $k[x_1,x_2]$ generated by the diagonal class and the elementary class, by [F3] with $S_2=\{1,s_1\}$.

The normalized elementary action. The relative-position-$s_1$ stratum in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ is the complement of the diagonal, and it is dense, so its closure is all of ${\mathbb{P}}^1\times {\mathbb{P}}^1$; at the two torus-fixed points of a ${\mathbb{P}}^1$ fiber the tangent Euler classes are $x_1-x_2$ and $x_2-x_1$. Hence the standard fundamental class acts by $(s_{1}f-f)\mathrm{/}(x_1-x_2)$, and the normalized class ${\tau }_1=-{\left[C\right]}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}}$ acts by the displayed divided difference $(f-s_{1}f)\mathrm{/}(x_1-x_2)$, which is polynomial because $f-s_{1}f$ is antisymmetric in $x_1,x_2$; this is the finite cellwise localization computation over a characteristic-zero field, using the cellular model of ${\mathbb{P}}^1$.

Direct evaluation gives ${\tau }_1^2=0$ because $D_{1}f$ is symmetric and hence killed by another divided difference; it also gives ${\tau }_1{x}_1-x_2{\tau }_1=1$ and ${\tau }_1{x}_2-x_1{\tau }_1=-1$. The degree is $-2$. In the abstract nil-Hecke presentation, these mixed relations move dots left and the square relation removes every repeated crossing, so it is spanned by $f+g{\tau }_1$ for $f,g\in k[x_1,x_2]$. The two polynomial summands map independently and onto the geometric corner by [F3] and step 1.1. Thus this presentation is isomorphic to the corner, which proves the asserted nil-Hecke identification.