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

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

• $x \le w$ 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. Oct 5 '19 at 3:53
• @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 $m_{\lambda,\mu}\neq 0$? Oct 5 '19 at 9:24

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 $$w\in W$$.