Let $G$ be a reductive Lie group over $\mathbb{C}$, and write $\mathfrak{g}$ for its Lie algebra. Let $T\subseteq B\subseteq G$ be a maximal torus and Borel subgroup, where $\operatorname{Lie}B=\mathfrak{b}$.

I am looking for an full abelian subcategory $\mathcal{C}$ of the category of all $(\mathfrak{g},T)$-modules (meaning $\mathfrak{g}$-modules integrating to an action of $T$) which is not too big, in the sense that it contains the category $\mathcal{F}$ of finite-dimensional modules, and the canonical map $$ K_0(\mathcal{F})\to K_0(\mathcal{C}) $$ is injective, and where $K_0({-})$ denotes the Grothendieck group of the given category.

I want something bigger than the category $\mathcal{O}$ of course as well; in particular the modules I want $\mathcal{C}$ to contain are the algebras of functions on arbitrary intersections of open 'big cells' of partial flag varities. By these open big cells, I mean for each $T$-fixed point on $G/P$, an appropriate conjugate of $B$ will admit an open orbit at that point, and its orbit I am calling a big cell.

It is true that functions on these intersections have a finite-dimensional $\mathfrak{b}$-eigenspaces; however taking $\mathcal{C}$ to be those modules with finite-dimensional $\mathfrak{b}$-eigenspaces doesn't seem to give an abelian category, a priori (issue arises with quotients).

Would appreciate any thoughts/questions/references!