If one wants to understand representations of $\mathfrak{g}$ (a finite dimensional semisimple Lie algebra) of weight $\lambda$, the happiest you could be is if $\lambda+\rho$ is (integral) regular dominant, i.e. it's an element of the weight lattice whose product $\langle \lambda+\rho,\check{\alpha}\rangle$ with every simple coroot $\check{\alpha}$ is a negative integer. In this case there's an equivalence between the category of these representations and twisted $\mathscr{D}$ modules on the flag variety $U(\mathfrak{g})\_\lambda\text{-mod}\simeq \mathscr{D}\_\lambda\text{-mod}$. See [the book](http://www.math.ubc.ca/%7Ecautis/dmodules/hottaetal.pdf) by Hotta, Toshiyuki, Tanisaki, section 11.2.


If $\lambda+\rho$ is just (integral) regular (nothing in $W$ stabilises it), there is an equivalence of *derived* categories. I've not worked with this, but presumably along with translation functors (tensoring with $\mathscr{O}(\mu)$ to make the twist of the $\mathscr{D}$ module dominant) this allows you to use standard Beilinson-Bernstein above to get a handle on such modules.


My question is: what is currently known when $\lambda+\rho$ is *non-integral*? If it splits up into rational and irrational case (where all $\langle \lambda+\rho,\check{\alpha}\rangle\in\mathbf{Q}$ or otherwise), I am more interested in the rational case.



Beilinson-Bernstein for nonintegral levels
Integrality doesn't really seem to play a role in the statement of Beilinson-Bernstein as far as I can tell. For a general weight $\lambda \in \mathfrak h^\ast$ it makes sense to talk about the notions of dominant and regular.


If $\lambda$ is regular, then there is a derived localization:


$D(D\_{G/B}^\lambda-mod) \simeq D(U\_\lambda -mod)$


If $\lambda$ is both regular and dominant, there is an abelian category localization:


$D\_{G/B}^\lambda-mod \simeq U\_\lambda-mod$


There are no integrality assumptions in the statement of Theorem 3.3.1 in the paper "A Proof of the Jantzen Conjecture", for example.


