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

2024/03/04 by Carbone, Lisa, Coelho, Terence, Murray, Scott H. +2
#17B22 #17B67 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2403.01687

Abstract

Let \mathfrakg be a symmetrizable Kac--Moody algebra. Given a root α and a real root β of \mathfrakg, 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→∞. 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) ≠ 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→∞. We also prove that dim \mathfrakg_α+β ≥ dim \mathfrakg_α+ dim \mathfrakg_β-1 whenever α≠ β with (α, β)<0.

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