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 $\lambda \neq -2 \rho$ ?

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    $\begingroup$ If I understand your question correctly, the answer is yes (apply a translation functor to reduce the case of general integral antidominant $\lambda$ to any individual such weight.) $\endgroup$ – dhy Aug 2 '17 at 15:00
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    $\begingroup$ Actually, the notations aren't quite what I used in 8.4, since (as in the 1979 paper of Kazhdan-Lusztig) I emphasized the 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
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    $\begingroup$ P.S. A small typographical correction: it's $M_x$ rather than $M_w$ in the formula. $\endgroup$ – Jim Humphreys Aug 2 '17 at 17:28
  • $\begingroup$ Thank you for the useful comments, I agree that I adapted somewhat the notations to formulate my question, and maybe I altered the correctness of the textbook (about which I have no doubt !). I was aware that the answer should be found in the translation functors of chapter 7, but I confess I only read it superficially (I don't have much background in category theory), and didn't find the relevant statement. $\endgroup$ – Antoine Aug 3 '17 at 11:09

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.

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    $\begingroup$ Concerning non-integral weights, see my comments and references in 8.8. Though Kazhdan-Lusztig apparently didn't go into this, the non-integral case was later filled in by Soergel and generalized in the work of Kashiwara-Tanisaki for Kac-Moody Lie algebras. $\endgroup$ – Jim Humphreys Aug 2 '17 at 17:24
  • $\begingroup$ Thanks for the answer! You say the answer is different for singular blocks, can you say more about it (or give a reference)? $\endgroup$ – Antoine Aug 3 '17 at 11:13
  • $\begingroup$ What definition of dominant are you using here ? Is it $\Lambda^+$ (the $\mathbb{Z}^+$ span of the fundamental weights) or $\Lambda^+ - \rho$ ? $\endgroup$ – Antoine Aug 11 '17 at 15:33
  • $\begingroup$ I really mean dominant (all coroots are positive) and anti-dominant (all are negative). $\endgroup$ – Ben Webster Aug 13 '17 at 23:20

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