Let $G$ be a reductive complex algebraic group, $P$ a parabolic subgroup, $\mathbb{C}_{\lambda}$ a one-dimensional representation of $P$ and $\mathcal{L}_{\lambda}$ the corresponding line bundle on $G/P$.

$G/P$ admits an open affine cover by $B$-orbits $U_{w}$, where $w$ is indexed by $W_{G}/W_{P}$, where $W_G$ (resp. $W_{P}$) denotes the Weyl group of $G$ (resp. $P$).

Write $\mathfrak{g}$ for the Lie algebra of $G$. My question is, what does the $\mathfrak{g}$-module $\Gamma(U_{w_1}\cap\dots\cap U_{w_k},\mathcal{L})$ look like? Here of course I'm simply taking an arbitrary intersection of the open sets in our cover.

Can we say anything about these modules at all? References would be appreciated.

Let $G$ be a reductive complex algebraic group, $P$ a parabolic subgroup, $\mathbb{C}_{\lambda}$ a one-dimensional representation of $P$ and $\mathcal{L}_{\lambda}$ the corresponding line bundle on $G/P$.

Let $T$ be the maximal torus in $P$; then $T$ admits finitely many fixed points on $G/P$, labeled by $W_G/W_P$, where $W_G$ (resp. $W_{P}$) denotes the Weyl group of $G$ (resp. $P$). At each fixed point there is a Borel subgroup containing $T$ which has an open orbit there; write $U_w$ for this open orbit.

Write $\mathfrak{g}$ for the Lie algebra of $G$. My question is, what does the $\mathfrak{g}$-module $\Gamma(U_{w_1}\cap\dots\cap U_{w_k},\mathcal{L})$ look like? Here of course I'm simply taking an arbitrary intersection of the open sets in our cover.

Can we say anything about these modules at all? References would be appreciated.