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1. Cocharge and Skewing Formulas for Δ-Springer Modules and the Delta Conjecture

   https://doi.org/10.1093/imrn/rnae090  

   Gillespie, Maria; Griffin, Sean T (May 2024, International Mathematics Research Notices)

   We prove that $$\omega \Delta ^{\prime}_{e_{k}}e_{n}|_{t=0}$$, the symmetric function in the Delta Conjecture at $t=0$, is a skewing operator applied to a Hall-Littlewood polynomial, and generalize this formula to the Frobenius series of all $$\Delta $$-Springer modules. We use this to give an explicit Schur expansion in terms of the Lascoux-Schützenberger cocharge statistic on a new combinatorial object that we call a battery-powered tableau. Our proof is geometric, and shows that the $$\Delta $$-Springer varieties of Levinson, Woo, and the second author are generalized Springer fibers coming from the partial resolutions of the nilpotent cone due to Borho and MacPherson. We also give alternative combinatorial proofs of our Schur expansion for several special cases, and give conjectural skewing formulas for the $$t$$ and $$t^{2}$$ coefficients of $$\omega \Delta ^{\prime}_{e_{k}}e_{n}$$. 

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2. Rigidity properties of the cotangent complex

   https://doi.org/10.1090/jams/1000  

   Briggs, Benjamin; Iyengar, Srikanth (January 2023, Journal of the American Mathematical Society)

   This work concerns a map φ : R → S \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if D n ( S / R ; − ) = 0 \mathrm {D}_n(S/R;-)=0 for some n ≥ 1 n\ge 1 , then D i ( S / R ; − ) = 0 \mathrm {D}_i(S/R;-)=0 for all i ≥ 2 i\ge 2 and φ {\varphi } is locally complete intersection. This extends Avramov’s theorem that draws the same conclusion assuming D n ( S / R ; − ) \mathrm {D}_n(S/R;-) vanishes for all n ≫ 0 n\gg 0 , confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of φ \varphi . 

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3. Perfect even modules and the even filtration

   https://doi.org/10.4171/JEMS/1669  

   Pstrągowski, Piotr (July 2025, Journal of the European Mathematical Society)

   Inspired by the work of Hahn–Raksit–Wilson, we introduce a variant of the even filtration which is naturally defined on\mathbf{E}_{1}-rings and their modules. We show that our variant satisfies flat descent and so agrees with the Hahn–Raksit–Wilson filtration on ring spectra of arithmetic interest, showing that various “motivic” filtrations are in fact invariants of the\mathbf{E}_{1}-structure alone. We prove that our filtration can be calculated via appropriate resolutions in modules and apply it to the study of even cohomology of connective\mathbf{E}_{1}-rings, proving vanishing above the Milnor line, base-change formulas, and explicitly calculating cohomology in low weights. 

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4. Quiver varieties and symmetric pairs

   https://doi.org/10.1090/ert/522  

   Li, Yiqiang (January 2019, Representation theory)

   We study fixed-point loci of Nakajima varieties under symplectomorphisms and their antisymplectic cousins, which are compositions of a diagram isomorphism, a reflection functor, and a transpose defined by certain bilinear forms. These subvarieties provide a natural home for geometric representation theory of symmetric pairs. In particular, the cohomology of a Steinberg-type variety of the symplectic fixed-point subvarieties is conjecturally related to the universal enveloping algebra of the subalgebra in a symmetric pair. The latter symplectic subvarieties are further used to geometrically construct an action of a twisted Yangian on a torus equivariant cohomology of Nakajima varieties. In the type A case, these subvarieties provide a quiver model for partial Springer resolutions of nilpotent Slodowy slices of classical groups and associated symmetric spaces, which leads to a rectangular symmetry and a refinement of Kraft-Procesi row/column removal reductions. 

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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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