The Kazhdan-Lusztig Conjecture (which is actually a theorem) gives the character of some irreducible modules of a (say) simple complex Lie algebra $\mathfrak{g}$ in terms of characters of Verma modules. More precisely, following the notations of *Representations of Semisimple Lie Algebras in the BGG Category O*, by Humphreys, we define for $w$ in the Weyl group


* $M\_w(\lambda)$ to be the Verma module with highest weight $w(\lambda + \rho) - \rho$
* $L\_w(\lambda)$ to be the irreducible module with the same highest weight


Then if we choose $\lambda = -2 \rho$, we have (see Conjecture 8.4 in Humphreys's book) $$\mathrm{ch}\, L\_w (\lambda) = \sum\limits\_{x \leq w} (-1)^{\ell (x,w)} P\_{x,w} (1) \, \mathrm{ch}\, M\_x (\lambda) \, . $$
In this formula, the polynomials $P\_{x,w}$ are the Kazhdan-Lusztig associated to the Weyl group, and we use the Bruhat ordering. 


Is this formula still valid for (antidominant) weights $\lambda \neq -2 \rho$ ?



Kazhdan-Lusztig theorem for composition factors of Verma modules
It's valid for integral weights of the form $\lambda-2\rho$, with $\lambda$ antidominant (these are the anti-dominant weights that have a dominant weight $w\_0\lambda$ in their orbit under the dot action). The proof is that if you tensor the simples $L\_w(-2\rho)$ and Vermas $M\_w(-2\rho)$ with the simple represenation of lowest weight $\lambda$, and then consider the summand where the center of the universal enveloping algebra acts with the expected character, you get the simples $L\_w(\lambda-2\rho)$ and Vermas $M\_w(\lambda-2\rho)$, on the nose.


If $\lambda$ is not integral, then the formula is wrong; also if we choose a weight that doesn't have a dominant weight in its dot orbit (a singular block), then the answers are different as well.


