When does the Kazhdan-Lusztig polynomial $P_{x,w}(q)$ not vanish at $q=1$?

Question

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 $\text{ch}(L(\lambda))$?

Answer 1

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 $w\in W$.