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 ${\mathfrak{g}}_Q^{MO}$ 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 ${\mathfrak{g}}_Q^{MO}$ 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 ${\mathfrak{g}}_Q^{MO}$ 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.

v4 79 pages, lots of corrections and minor improvements. v3, 77 pages, added discussion of critical stable envelopes, slightly strengthened results. v2, 73 pages, Theorem C slightly strengthened, lots of minor improvements. v1, 71 pages, comments very welcome

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