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

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