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We propose an explicit conjectural lift of the Neron-Severi Lie algebra of ahyperk\"ahler variety $X$ of $K3^{[2]}$-type to the Chow ring ofcorrespondences ${\rm CH}^\ast(X \times X)$ in terms of a canonical lift of theBeauville-Bogomolov class obtained by Markman. We give evidence for thisconjecture in the case of the Hilbert scheme of two points of a $K3$ surfaceand in the case of the Fano variety of lines of a very general cubic fourfold.Moreover, we show that the Fourier decomposition of the Chow ring of $X$ ofShen and Vial agrees with the eigenspace decomposition of a canonical lift ofthe grading operator.

The Chow ring of hyperk\"ahler varieties of $K3^{[2]}$-type via Lefschetz actions