Growth of root multiplicities along imaginary root strings in Kac--Moody algebras

Let $\mathfrak{g}$ be a symmetrizable Kac–Moody algebra. Given a root $\alpha$ and a real root $\beta$ of $\mathfrak{g}$, it is known that the $\beta$-string through $\alpha$, denoted $R_{\alpha }(\beta )$, is finite. Given an imaginary root $\beta$, we show that $R_{\alpha }(\beta )=\{\beta \}$ or $R_{\alpha }(\beta )$ is infinite. If $(\beta ,\beta )<0$, we also show that the multiplicity of the root $\alpha +n\beta$ grows at least exponentially as $n\to \mathrm{\infty }$. If $(\beta ,\beta )=(\alpha ,\beta )=0$, we show that $R_{\alpha }(\beta )$ is bi-infinite and the multiplicities of $\alpha +n\beta$ are bounded. If $(\beta ,\beta )=0$ and $(\alpha ,\beta )\mathrm{\ne }0$, we show that $R_{\alpha }(\beta )$ is semi-infinite and the muliplicity of $\alpha +n\beta$ or $\alpha -n\beta$ grows faster than every polynomial as $n\to \mathrm{\infty }$. We also prove that $\dim {\mathfrak{g}}_{\alpha +\beta }\ge \dim {\mathfrak{g}}_{\alpha }+\dim {\mathfrak{g}}_{\beta }-1$ whenever $\alpha \mathrm{\ne }\beta$ with $(\alpha ,\beta )<0$.