Lucien Hennecart (CNRS): The BPS sheaf for preprojective algebras and moduli of Higgs bundles

In the first part of the talk, I will introduce the BPS sheaf associated with the preprojective algebra of a quiver. This is a perverse sheaf on the moduli space of representations, endowed with a Lie algebra structure, which encodes the Kac polynomials of the quiver. Its structure is described in terms of generators and relations in joint work with Davison and Schlegel Mejia. This description is formulated in terms of a generalized Kac–Moody algebra.

In the second part, I will explain how this description makes it possible to determine the supports and local systems of the direct summands of the BPS sheaf. This description involves the geometry of hypertoric varieties in general, and can in certain cases be simplified. Applying these results to the Hitchin fibration, one recovers results due to de Cataldo, Heinloth, Migliorini, and also Mauri, concerning the effective decomposition theorem for the Hitchin morphism involving Ngo strings. This is all based on various discussions with Ben Davison.

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