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1. Perverse Sheaves and the Cohomology of Regular Hessenberg Varieties

   https://doi.org/10.1007/s00031-022-09755-3  

   Bălibanu, Ana; Crooks, Peter (September 2022, Transformation Groups)

   We use the Springer correspondence to give a partial characterization of the irre- ducible representations which appear in the Tymoczko dot action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the “Kahler package.” 

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2. 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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3. Coherent Springer theory and the categorical Deligne-Langlands correspondence

   https://doi.org/10.1007/s00222-023-01224-2  

   Ben-Zvi, David; Chen, Harrison; Helm, David; Nadler, David (February 2024, Inventiones mathematicae)

   Abstract Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant$$K$$ $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields$$F$$ $F$with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from$$K$$ $K$-theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, thecoherent Springer sheaf. As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of$$\mathrm{GL}_{n}(F)$$ ${\mathrm{GL}}_n\left(F\right)$into coherent sheaves on the stack of Langlands parameters. 

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4. Springer correspondence for the split symmetric pair in type

   https://doi.org/10.1112/S0010437X18007443  

   Chen, Tsao-Hsien; Vilonen, Kari; Xue, Ting (November 2018, Compositio Mathematica)

   In this paper we establish Springer correspondence for the symmetric pair $$(\text{SL}(N),\text{SO}(N))$$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$ . Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry. 

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5. Co-𝑡-structures on derived categories of coherent sheaves and the cohomology of tilting modules

   https://doi.org/10.1090/ert/655  

   Achar, Pramod; Hardesty, William (February 2024, Representation Theory of the American Mathematical Society)

   We construct a co- $t$-structure on the derived category of coherent sheaves on the nilpotent cone $\mathcal{N}$of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the “exotic parity objects” (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597–657]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co- $t$-structure on $\mathcal{N}$. We also demonstrate how the various parabolic co- $t$-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type $A$for $p>h$. 

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