Let $G$ be a semi-simple complex algebraic group with lie algebra $\mathfrak{g}$. For a fix Borel subgroup $B$ let $X=G/B$ be the flag variety. Let $i_l,i_r:X \to X \times X=Z$ denote the inclusion of $X$ the left(right) inclusion of $X$ into $X \times X$.

In Borho-Brylinski, Differential operators on homogeneous spaces. III paper they prove in proposition 3.6 that via the functors $i_l^*$ ($i_r^*$) respectively there is an equivalence of categories:

$$Coh(\mathcal{D}_{X \times X},G) \cong Coh(\mathcal{D}_{X},B).$$

Let $\mathcal{I} \in Coh(\mathcal{D}_{X \times X},G) $ and let $I=\Gamma(X \times X,\mathcal{I}$). My question is how can one compute $\Gamma(X,i_l^* \mathcal{I})$ given $I$ ?

I should mention that I am looking for a purely geometric way; I want to avoid using the Beilinson-Bernstein theorem together with the Bernstein-Gelfand theorem.

In their paper, Borho-Brylinski prove that $\Gamma(X,i_l^*\mathcal{I})=M(0)^{\vee} \otimes_{U(\mathfrak{g}_r)}I$, but I do not seem to understand some of the claims in section 3.8. I would be very grateful if someone could provide a big picture idea of what is actually going on or reference a paper where this done in greater detail.

Many thanks.


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