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 $\lambda \neq -2 \rho$ ?

principal block(block containing the trivial module) for which $M_w, L_x$ are shorthand labels. Aside from that, the idea is to use Jantzen's translation functors between regular blocks, while translating into upper closures of Weyl chambers to get singular weights. See my Chapter 7 and also the details in 8.8. Though Chapter 8 is only a "survey", I tried to state things correctly. See my AMS bookpage or my homepage for corrections and let me know of others. $\endgroup$ – Jim Humphreys Aug 2 '17 at 17:12