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Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{h}$ be a Cartan subalgebra. For any $\lambda\in \mathfrak{h}^{*}$ let $M(\lambda)$ and $L(\lambda)$ be the Verma module and the simple module of highest weight $\lambda$, respectively.

Let $W$ be the weight group of $\mathfrak{g}$. For $w\in W$, the shifted action on $\mathfrak{h}^{*}$ is defined as $w\cdot \lambda:=w(\lambda+\rho)-\rho$, where $\rho$ is the half sum of all positive roots.

For $x,w\in W$ Kazhdan and Lusztig introduced the Kazhdan-Lusztig polynomial $P_{x,w}(q)$. The Kazhdan-Lusztig conjecture, which was proved in the 1980's, claims that $$ \text{ch}(L(w\cdot (-2\rho)))=\sum_{x\leq w}(-1)^{l(x)-l(w)}P_{x,w}(1)\text{ch}(M(x\cdot (-2\rho))). $$ where $\leq$ is the Bruhat ordering of elements in $W$.

Now it is clear that ch$(M(\lambda))$ appears in the linear combination of $\text{ch}(L(w\cdot (-2\rho)))$ only when $\lambda=x\cdot (-2\rho)$ for some $x\leq w$.

I want to know if there is an "iff" criterion. More precisely, for $x\leq w$, when do we have $P_{x,w}(1)\neq 0$?

More generally, for arbitrary $L(\lambda)$, do we have an "iff" condition on which $\text{ch}(M(\mu))$ appears in the linear combination of $\text{ch}(L(\lambda))$?

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    $\begingroup$ $x \le w$ if and only if $P_{x,w}(1) \ne 0$. Indeed, it is easy to see by induction that the constant term of $P_{x,w}$ is 1 if and only if $x \le w$, and the fact that they have positive coefficients then implies the result. $\endgroup$ Oct 5 '19 at 3:53
  • $\begingroup$ @GeordieWilliamson Oh, I see. How about the general case. We know that $\text{ch}(L(\lambda))=\sum_{\mu}m_{\lambda,\mu}\text{ch}(M(\mu))$ and the coefficient $m_{\lambda,\mu}\neq 0$ only if $ \mu\leq \lambda$ and $\mu=w\cdot \lambda$ for some $w\in W$. The strongly linkage gives more restrictions. Do we have some "iff" condition that $m_{\lambda,\mu}\neq 0$? $\endgroup$ Oct 5 '19 at 9:24
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For any weight $\lambda$, the multiplicities $m_{\lambda,\mu}$ are given by the Kazhdan-Lusztig polynomials of a Coxeter group called the integral Weyl group, given by the elements $w$ such that $w\lambda-\lambda$ is in the root lattice. So, by Geordie's observation above, we have that $m_{\lambda,\mu}\neq 0$ if and only if $\mu \leq \lambda$ in the sense that $\lambda-\mu$ is a positive element of the root lattice, and $\mu=w\cdot \lambda$ for some $w\in W$.

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