Résumé
Pacific J. Math. 329 (2024) 1-38 We provide a new general scheme for the geometric quantisation of$\operatorname{Sp}(1)$ -symmetric hyper-Kähler manifolds, considering Hilbert spaces of holomorphic sections with respect to the complex structures in the hyper-Kähler 2-sphere. Under properness of an associated moment map, or other finiteness assumptions, we construct unitary quantum (super) representations of central extensions of certain subgroups of Riemannian isometries preserving the 2-sphere, and we study their decomposition in irreducible components. We apply this quantisation scheme to hyper-Kähler vector spaces, the Taub--NUT metric on$\mathbb{R}^4$ , moduli spaces of framed$\operatorname{SU}(r)$ -instantons on$\mathbb{R}^4$ , and partly to the Atiyah--Hitchin manifold of magnetic monopoles in$\mathbb{R}^3$