Quiver varieties and fusion products for \(\mathfrak{sl}_2\) (Q1414675)

A geometric realization of integrable highest weight representations \(V_{\lambda}\) of a Kac-Moody algebra \(g\) in the homology of a certain Lagrangian subvariety \({\mathcal L}(\lambda )\) of a symplectic variety \({\mathcal M}(\lambda )\) (called the quiver variety) constructed from the Dynkin diagram of \(g\) was done by Nakajima in a series of papers. He also realized the tensor product \(V_{\lambda}\otimes V_{\mu}\) as the homology of a variety \({\mathcal L}(\lambda ,\mu )\subset {\mathcal M}(\lambda +\mu )\). In the paper under review, the authors are concerned to the question whether for simple \(g\), a similar construction can produce the fusion tensor products \(V_{\lambda}\otimes_l V_{\mu}\), certain truncations of \(V_{\lambda}\otimes V_{\mu}\). The question is answered affirmatively for \(g=sl_2\), where \(V_{\lambda}\otimes_l V_{\mu}\) is realized as the homology of a natural subvariety of \({\mathcal L}(\lambda ,\mu )\). The case of a tensor product of finitely many \(sl_2\)-modules is also considered. Combinatorial descriptions of the irreducible components are given.