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) $$\mathrm{ch}\, L_w (\lambda) = \sum\limits_{x \leq w} (-1)^{\ell (x,w)} P_{x,w} (1) \, \mathrm{ch}\, M_w (\lambda) \, . $$ 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 $\lambda \neq -2 \rho$ ?

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 $\lambda \neq -2 \rho$ ?