Say everything is over $\mathbb C$, and I have an action $act: N \times X \to X$ of an affine algebraic group $N$ on a smooth variety, say with finitely many orbits. I'm trying to compute the $\mathcal D$-module direct image $act_+$ for sheaves on $N \times X$ of the form $\mathcal O_N \boxtimes \mathcal F$. Is there any useful tricks in general for these kinds of computations?
A fact that may be helpful is that $act$ is smooth and affine, so $act_+ = act_* ( \Omega_{(N \times X)/_{act}X}^\bullet \otimes_{\mathcal O_{N \times X}} - )[\dim N]$, where $\Omega_{(N \times X)/_{act}X}^\bullet$ is the relative de Rham complex. But $\Omega_{(N \times X)/_{act}X}^\bullet$ seems hard to compute as well.
If the above problem is too general, figuring out the following example will also help.
Let $X$ be the flag variety of $G = \mathrm{SL}(3,\mathbb C)$, $B$ a Borel subgroup of $G$, $N,T$ the unipotent radical of $B$ and a maximal torus in $B$. Let $act: N \times X \to X$ be the natural action. Take the set of positive roots in the root system of $(\mathfrak g,\mathfrak t)$ defined by $B$, and take the $\mathfrak{sl}_2$-triple $\mathfrak g_1$ of the non-simple roots, then the inclusion $\mathfrak g_1 \hookrightarrow \mathfrak g$ descends to a closed immersion $j$ of the flag variety $X_1 \cong \mathbb P^1$ of $\mathfrak g_1$ into $X$, whose image is contained in the union of the smallest and the largest Schubert cells in $X$. How to compute $act_+ (\mathcal O_N \boxtimes j_+ \mathcal O_{X_1})$?
Thanks in advance!
