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1. A new approach to the generalized Springer correspondence

   https://doi.org/10.1090/tran/8890  

   Graham, William; Precup, Martha; Russell, Amber (June 2023, Transactions of the American Mathematical Society)

   The Springer resolution of the nilpotent cone is used to give a geometric construction of the irreducible representations of Weyl groups. Borho and MacPherson obtain the Springer correspondence by applying the decomposition theorem to the Springer resolution, establishing an injective map from the set of irreducible Weyl group representations to simple equivariant perverse sheaves on the nilpotent cone. In this manuscript, we consider a generalization of the Springer resolution using a variety defined by the first author. Our main result shows that in the type A case, applying the decomposition theorem to this map yields all simple perverse sheaves on the nilpotent cone with multiplicity as predicted by Lusztig’s generalized Springer correspondence. 

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2. The six functors for Zariski-constructible sheaves in rigid geometry

   https://doi.org/10.1112/S0010437X22007291  

   Bhatt, Bhargav; Hansen, David (February 2022, Compositio Mathematica)

   We prove a generic smoothness result in rigid analytic geometry over a characteristic zero non-archimedean field. The proof relies on a novel notion of generic points in rigid analytic geometry which are well adapted to ‘spreading out’ arguments, in analogy with the use of generic points in scheme theory. As an application, we develop a six-functor formalism for Zariski-constructible étale sheaves on characteristic zero rigid spaces. Among other things, this implies that characteristic zero rigid spaces support a well-behaved theory of perverse sheaves. 

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3. Hitchin fibrations, abelian surfaces, and the P=W conjecture

   https://doi.org/10.1090/jams/989  

   de Cataldo, Mark; Maulik, Davesh; Shen, Junliang (November 2021, Journal of the American Mathematical Society)

   We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role. 

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4. The purity locus of matrix Kloosterman sums

   https://doi.org/10.1090/tran/9149  

   Erdélyi, Márton; Sawin, Will; Tóth, Árpád (April 2024, Transactions of the American Mathematical Society)

   We construct a perverse sheaf related to the the matrix exponential sums investigated by Erdélyi and Tóth [Matrix Kloosterman sums, 2021, arXiv:2109.00762]. As this sheaf appears as a summand of certain tensor product of Kloosterman sheaves, we can establish the exact structure of the cohomology attached to the sums by relating it to the Springer correspondence and using the recursion formula of Erdélyi and Tóth. 

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5. Perverse Leray filtration and specialisation with applications to the Hitchin morphism

   https://doi.org/10.1017/S0305004121000293  

   DE CATALDO, MARK ANDREA (March 2022, Mathematical Proceedings of the Cambridge Philosophical Society)

   Abstract We initiate and develop a framework to handle the specialisation morphism as a filtered morphism for the perverse, and for the perverse Leray filtration, on the cohomology with constructible coefficients of varieties and morphisms parameterised by a curve. As an application, we use this framework to carry out a detailed study of filtered specialisation for the Hitchin morphisms associated with the compactification of Dolbeault moduli spaces in [de 2018]. 

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