Okounkov's conjecture via BPS Lie algebras

Abstract

Let $Q$ be an arbitrary finite quiver. We use nonabelian stable envelopes to relate representations of the Maulik-Okounkov Lie algebra $\mathfrak{g}^{MO}_Q$ to representations of the BPS Lie algebra associated to the tripled quiver $\tilde Q$ with its canonical potential. We use this comparison to provide an isomorphism between the Maulik-Okounkov Lie algebra and the BPS Lie algebra. Via this isomorphism we prove Okounkov's conjecture, equating the graded dimensions of the Lie algebra $\mathfrak{g}^{MO}_Q$ with the coefficients of Kac polynomials. Via general results regarding cohomological Hall algebras in dimensions two and three we furthermore give a complete description of $\mathfrak{g}^{MO}_Q$ as a generalised Kac-Moody Lie algebra with Cartan datum given by intersection cohomology of singular Nakajima quiver varieties, and prove a conjecture of Maulik and Okounkov, stating that their Lie algebra is obtained from a Lie algebra defined over the rationals, by extension of scalars. Finally, we explain how our results suggest the correct definition of critical stable envelopes in vanishing cycle cohomology.