Derived Springer Correspondence for Reduced Motives

• Derived Springer correspondence for reduced motives unifies geometric representation theory, motive theory, and Hecke algebra modules through categorical equivalences.
• Reduced motives eliminate higher motivic cohomology to isolate representation-theoretically significant data using weight structures and purity conditions.
• The framework leverages convolution algebras and categorical lifts to establish integral, modular, and K-theoretic formulations that bridge algebraic geometry with the local Langlands program.

The derived Springer correspondence for reduced motives is an advanced synthesis of geometric representation theory, the theory of motives, and categorical methods applied to the paper of nilpotent cones, generalized flag varieties, and Hecke algebras. It extends the classical Springer correspondence—originally relating Weyl group representations to the topology of Springer fibers—by using derived and motivic categories, often with integral or K-theoretic coefficients, and explicitly incorporates reductions such as weight structures and purity conditions. The theory provides new categorical and motivic interpretations for modules over Hecke algebras, modular and integral settings, and forms a foundational aspect in bridging algebraic geometry, arithmetic, and the local Langlands program.

1. Foundational Principles: Motives, Reductions, and the Springer Setting

Classical motives, such as Grothendieck Chow motives and Voevodsky's triangulated categories of motives, are used to capture refined invariants of algebraic varieties beyond ordinary cohomology. In the context of varieties with group actions (notably, nilpotent cones or flag varieties related to reductive algebraic groups), these frameworks are extended to “equivariant motives” and, more recently, “reduced motives.” The term reduced motive (as in (Eberhardt et al., 2022, Landim, 22 Sep 2025)) refers to a quotient of the motivic category where the higher motivic cohomology of the base scheme has been removed, isolating representation-theoretically meaningful data and eliminating extension classes between Tate objects that are trivial from the viewpoint of characters of the Weyl group or Hecke algebra.

The “Springer setup”—which encompasses the nilpotent cone $\mathcal{N}$, the Springer resolution $\widetilde{\mathcal{N}} \to \mathcal{N}$, the associated flag or partial flag varieties, and convolution diagrams such as Steinberg varieties—provides the geometric setting. Essential to the theory are the following:

• Springer Fibers: Motives associated with Springer fibers are shown to be pure Tate (Eberhardt, 2018), admitting affine pavings and direct decompositions into Tate motives $1(n)$.
• Equivariant and Reduced Motives: Categories of equivariant Springer motives are constructed (as triangulated subcategories) and reduced using tensor products over the base Tate motive category, yielding integral and modular formalisms (Eberhardt et al., 2022).
• Block Decompositions and Categorical Lifts: Block decompositions from Lusztig’s generalized Springer correspondence are categorified via the construction of perverse, mixed, or tilting $t$-structures, further refined in the derived (or dg-) category (Rider et al., 2017, Achar et al., 2016, Eberhardt, 2018).

2. Derived and Motivic Equivalences: The Main Theorems

Recent work establishes canonical equivalences between triangulated (or differential graded) categories of Springer motives and derived module categories over Hecke-type algebras.

• Fundamental Equivalence (Eberhardt, 2018, Eberhardt et al., 2021, Eberhardt, 23 Jan 2024, Landim, 22 Sep 2025):

$$D^b(\mathcal{M}_{\mathcal{N}}^G) \cong D^b(\mathrm{grMod}\text{--}\mathcal{H})$$

where $D^b(\mathcal{M}_{\mathcal{N}}^G)$ is the bounded derived (or perfect) category of $G$-equivariant (reduced) Springer motives on the nilpotent cone, and $\mathrm{grMod}\text{--}\mathcal{H}$ denotes graded modules over the graded affine Hecke algebra $\mathcal{H}$ or (in quiver settings) the KLR/quiver Hecke algebra.

Formality and Weight Structures: The existence of affine pavings for Springer fibers and quiver flag varieties ensures the purity (pure Tate property) required for formality, so that higher Ext groups vanish outside degree zero shifts (Eberhardt, 2018, Eberhardt et al., 2021). This is then formalized via a weight structure $(w \leq 0, w \geq 0)$, allowing a weight complex functor $t$ to provide an equivalence with the homotopy or perfect category of modules over the extension algebra (Eberhardt et al., 2021, Eberhardt et al., 2022).

• Integral and K-theoretic Versions:

The equivalence extends integrally (Landim, 22 Sep 2025) and in $K$-theory (Eberhardt, 23 Jan 2024):

$$D(\widetilde{\mathcal{N}}) \simeq \operatorname{Rep}(H_q(W))$$

where $H_q(W)$ is a $q$-deformation of the Weyl group algebra, and $D(\widetilde{\mathcal{N}})$ is a derived category of reduced $K$-motives on the Springer resolution or related moduli stacks, equipped with a six functor formalism.

3. Generalized Springer Correspondence and Modular/Integral Aspects

The classical Springer correspondence assigns irreducible representations of the Weyl group to components in the cohomology of Springer fibers. The generalized (and modular) versions relate blocks in equivariant derived or perverse categories to representations of relative Weyl groups, possibly twisted by cocycles (Lusztig, 2016, Graham et al., 2020, Psaromiligkos et al., 25 Mar 2025, Juteau, 2014, Juteau et al., 2014). In motivic and reduced settings:

• Cuspidal Data and Block Decomposition: The derived category decomposes into blocks indexed by cuspidal data (triples $(L, \mathcal{O}_L, C)$), each equivalent to dg-modules over a graded Hecke algebra attached to the relative Weyl group and symmetric algebra of the center (Rider et al., 2017, Eberhardt, 2018).
• Twisted Group Algebras: In the modular setting for disconnected or non-connected groups, induction series partition simple perverse sheaves, and parameterization proceeds through twisted group algebras $k[W_t, \nu]$ where $\nu$ is a $2$-cocycle (Psaromiligkos et al., 25 Mar 2025).
• Compatibility with Integral Reductions and Weight Filtrations: The weight and $t$-structure formalism on reduced motives explains the appearance of parity sheaves and semisimplified Hodge motives in characteristic $0$ and positive characteristic. The reduction functor eliminates extensions among Tate objects, focusing on representation-theoretically visible pieces (Eberhardt et al., 2022).

4. Convolution Algebras, Steinberg Varieties, and Hecke Actions

The connection with Hecke algebras and quiver analogues is achieved through convolution structures on Steinberg varieties and flag varieties:

• Motivic Extension Algebras: The extension algebra

$$E = \bigoplus_{i,j} \operatorname{Hom}_{DM_G(N, \mathbb{Q})}\left(\mu_i ! (\mathbb{Q}_{\widetilde{N}_i}), \mu_j ! (\mathbb{Q}_{\widetilde{N}_j})(n)[2n]\right)$$

is identified with an equivariant Chow group of components of Steinberg varieties and is isomorphic to Lusztig's graded affine Hecke algebra (or, in the quiver case, KLR/Schur variants) (Eberhardt et al., 2021).

• Convolution and Realization Functors: The six functor formalism for reduced and $K$-motives allows for convolution actions, base change, duality, and the construction of realization functors (e.g., to coherent Springer theory via categorical Chern characters, as in (Eberhardt, 23 Jan 2024)).
• Weight Complex Functors and Dualities: Koszul and Ringel duality are expressed in terms of weight complex functors $t$ (arising from weight structures) and are critical for the “linearization” of the derived categories (Eberhardt et al., 2021, Achar et al., 2016).

5. Geometric and Combinatorial Formulations; Explicit Examples

Several key geometric and combinatorial tools are central:

• Bruhat and Bialynicki–Birula Decompositions: In settings beyond Weyl groups (e.g., wreath products, types $B/C/D$), a new Bruhat decomposition indexed by the wreath product $\Sigma_m \wr \Sigma_d$ is established, leading to Springer-type correspondences beyond Coxeter groups and new geometric realizations of Clifford theory (Hsu et al., 3 Apr 2024).
• Formulas for Decomposition Multiplicities: The decomposition theorem is applied to proper maps (e.g., new resolutions $M \to N$) to express pushforwards as direct sums of intersection cohomology complexes with multiplicities matching Lusztig's predicted representations (Graham et al., 2020).
• Purity and Affine Pavings: The structural purity of Springer fibers and partial quiver flag varieties (secured by affine pavings) is necessary for formality, ensuring that motivic extension algebras have the “tilting” property (Eberhardt, 2018, Eberhardt et al., 2021).
• Combinatorics of Orbits and Basic Sets: In modular settings, decomposable matrices and basic set data relate the closure order of nilpotent orbits and modular reductions to the representation-theoretic content of the derived correspondence (Juteau, 2014, Juteau et al., 2014).

6. Extensions, Categorification, and Future Perspectives

The current theory forms the foundation for several further developments and research directions:

• Categorical and K-theoretic Correspondences: The categorical Chern character bridges reduced $K$-motives and coherent Springer theories, connecting to homological mirror symmetry and the local Langlands correspondence (Eberhardt, 23 Jan 2024).
• Modular and Integral Variants: The integral and modular formalisms clarify the role of reduction and support the computation of Lusztig's conjectural modular character formulas in representation theory of finite and $p$-adic groups (Landim, 22 Sep 2025).
• Applicability to Quiver and Wreath Settings: By extending the categories to quiver settings and wreath products (Eberhardt et al., 2021, Hsu et al., 3 Apr 2024), the general machinery supports categorified Clifford theory and Hecke actions beyond classical types.
• Connection to Langlands Program and Arithmetic Geometry: The six functor formalism for reduced ($K$-)motives on quotient stacks with finitely many orbits supports applications to the arithmetic and local geometric Langlands correspondences (Eberhardt, 23 Jan 2024).

7. Summary Table: Core Correspondences

Geometric/Motivic Category	Algebraic/Representation Category	Key Reference(s)
$D^b(\mathcal{M}^G_{\mathcal{N}})$	$D^b(\mathrm{grMod}\text{--}\mathcal{H})$	(Eberhardt, 2018, Eberhardt et al., 2021)
Reduced $K$-motives $D(\widetilde{\mathcal{N}})$	$\operatorname{Rep}(H_q(W))$	(Eberhardt, 23 Jan 2024)
Blocks $D'(N, A_c)$ (perverse sheaves/motives)	$DG(Q_{\ell}[W(L)] \# S^{*}(z))$	(Rider et al., 2017)

These categorical equivalences provide computational access to representation-theoretic invariants directly from geometric or motivic data and extend the classical Springer framework to settings with integral, modular, or $K$-theoretic coefficients, “reduced” structures, and derived enhancements.

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In conclusion, the derived Springer correspondence for reduced motives unifies geometric, motivic, and representation-theoretic methods into a highly structured and extensible framework. By encoding Hecke algebra and Weyl group representations categorically via reduced, pure, or $K$-theoretic motives on geometrically significant spaces, it enables new approaches to modular representation theory, geometric Langlands correspondences, and the paper of motivic phenomena related to singularities, orbits, and symmetry in algebraic and arithmetic geometry.