Quantum K-Theoretic Derived Satake Equivalence

• Quantum K-theoretic derived Satake equivalence is a categorical duality that refines the geometric Satake correspondence by incorporating derived and equivariant K-theory with quantum group structures.
• It employs convolution categories and diagrammatic methods, such as the annular spider for SLₙ, to relate geometric representation theory with quantum group modules.
• This framework bridges algebraic K-theory, geometric representation, and quantum group theory, offering insights extendable to semisimple and Kac–Moody groups.

Quantum K-theoretic derived Satake equivalence is a categorical duality at the interface of geometric representation theory, algebraic K-theory, and quantum group theory. It refines the geometric Satake correspondence—which relates perverse sheaves on the affine Grassmannian to the representation theory of a reductive group—by passing to derived and equivariant K-theoretic settings and incorporating the structures of quantum groups via convolution categories. This framework provides a powerful tool for understanding the categorification of the representation theory of quantum groups, especially in type A and, more generally, for all Kac-Moody groups through diagrammatic and motivic formulations. The main cases of interest are those involving the formal loop group, equivariant K-theory, and quantum groups such as $U_q(\mathfrak{g})$ and their associated module categories (Cautis et al., 2015, Eberhardt et al., 24 Nov 2025).

1. Classical and Derived Geometric Satake Correspondence

The geometric Satake correspondence establishes an equivalence between the tensor category of representations of a semisimple complex group $G$ and the category of $G^\vee[[t]]$-equivariant perverse sheaves on the affine Grassmannian $\Gr = G^\vee(\!(t)\!)/G^\vee[[t]]$ of the Langlands dual group $G^\vee$. This abelian correspondence sends the intersection cohomology sheaf $\IC_\lambda$ on $\overline{\Gr^\lambda}$ to the irreducible representation $V(\lambda)$ of $G$ (Mirković–Vilonen) (Cautis et al., 2015).

Bezrukavnikov and Finkelberg constructed a derived enhancement, producing monoidal equivalences: $D^b(\mathcal{O}(\mathfrak{g})\text{-mod}) \simeq D_{G^\vee}(\Gr) \qquad \text{and} \qquad D^b(\mathcal{U}(\mathfrak{g})\text{-mod}) \simeq D_{G^\vee\times\mathbb{C}^\times}(\Gr)$ where the derived category side is either $G^\vee$- or $(G^\vee\times \mathbb{C}^\times)$-equivariant (Cautis et al., 2015). These equivalences are compatible with convolution and tensor product, respectively.

In this context, iterated convolution diagrams and their fibre products $Z(\ul\lambda,\ul\mu)$ encode important structures: their homology spaces are isomorphic to $\Ext^*(\IC_{\ul\lambda}, \IC_{\ul\mu})$ and $\Hom_G(V(\ul\lambda),V(\ul\mu))$ via the work of Ginzburg.

2. Quantum K-Theoretic Convolution Categories

Passing to $G^\vee\times \mathbb{C}^\times$-equivariant algebraic K-theory introduces a quantum parameter $q$. On the algebraic side, $\mathcal{O}(\mathfrak{g})$ and $\mathcal{U}(\mathfrak{g})$ are replaced by the braided quantized coordinate ring $\mathcal{O}_q(G)$ and the quantum group $U_q(\mathfrak{g})$. The primary object is the convolution category $K\Conv(\Gr)$ defined as follows (in the notation of (Cautis et al., 2015)):

• Objects: sequences $\ul\lambda$ of minuscule coweights of $G^\vee$.
• Morphisms: $\Hom_{K\Conv(\Gr)}(\ul\lambda,\ul\mu) = K^{G^\vee\times\mathbb{C}^\times}(Z(\ul\lambda,\ul\mu))$, the equivariant K-theory of the fibre product varieties.
• Composition: via convolution in equivariant K-theory using pullback and pushforward along projection maps of iterated fibre products.

This category is conjectured to be monoidally equivalent to the subcategory $\mathcal{O}_q(G)\text{-mod}_{\min}$ of $U_q(\mathfrak{g})$-equivariant $\mathcal{O}_q(G)$-modules generated by objects of the form $\mathcal{O}_q(G)\otimes V(\lambda)$ with minuscule $\lambda$ (Cautis et al., 2015).

3. Diagrammatic and Combinatorial Realizations: The $SL_n$ Case

A central result is the complete proof of the quantum K-theoretic derived Satake equivalence for $G=SL_n$. This involves a combinatorial/diagrammatic formulation, with key ingredients:

• Spider Category $\Sp_n$:
  • A pivotal monoidal category encoding $\Rep_q(SL_n)$, generated by labeled "web" diagrams subject to local relations.
  • Equivalence: $\Sp_n(q) \cong \Rep_{\min}(U_q(\mathfrak{sl}_n))$ [CKM].
• Annular Spider $\ASp_n$ and Horizontal Trace:
  • The annular spider $\ASp_n(q)$ is defined by allowing webs in an annulus, corresponding to the horizontal trace category $\Sp_n(q)(S^1)$.
  • Morphisms in $\ASp_n(q)$ admit a sum-of-intertwiners interpretation via horizontal trace, yielding a braided monoidal structure.
• Equivalence with $\mathcal{O}_q(SL_n)$-modules:
  • $\ASp_n(q)\cong\mathcal{O}_q(SL_n)\text{-mod}_{\min}$ as monoidal categories.
  • The identification leverages the horizontal trace of the aforementioned spider category (Cautis et al., 2015).
• Quantum Loop Algebras:
  • Fully faithful functors from the idempotented quantum loop algebra $\dot{U}_q(L\mathfrak{gl}_m)$ to both $\ASp_n(q)$ and $K\Conv(\Gr)$.
  • Kernel objects constructed in the equivariant derived category of convolution varieties descend to elements in the relevant K-theories.

The main theorem asserts that this construction yields an equivalence between $\ASp_n(q)$, $\mathcal{O}_q(SL_n)\text{-mod}_{\min}$, and $K\Conv(\Gr^{SL_n})$ as $\mathbb{C}(q)$-linear monoidal categories (Cautis et al., 2015).

4. Singular K-Theoretic Soergel Bimodules and the Parabolic K-Motivic Approach

A further perspective involves the parabolic K-motivic Hecke category, constructed from Bott–Samelson resolutions associated to parabolic subgroups. This approach proceeds by:

• Defining the category via objects corresponding to push-forwards of motivic sheaves along Bott–Samelson resolutions.
• Morphisms are given by the equivariant K-theory $$K^G_0(\mathrm{BS}(\underline{w})\times_{X_I\times X_J}\mathrm{BS}(\underline{w'}))$$.
• The monoidal structure is induced by convolution, making $\oplus_{I,J} H^I_J$ a strict 2-category (Eberhardt et al., 24 Nov 2025).

The singular $K$-theoretic Soergel bimodule category $\mathrm{SBim}(I,J)$, formed as a Karoubi-complete subcategory generated by certain explicit $R$-bimodules, is shown to be canonical equivalent (as a monoidal category) to the parabolic K-motivic Hecke category. The equivalence respects quadratic splittings (rank-one relations), Coxeter braid relations, and polynomial-sliding (Demazure) relations.

5. Spherical Affine Case and Resolution of the Cautis–Kamnitzer Conjecture

Specialization to the spherical affine setting—that is, for the affine Grassmannian $\Gr_G$ of a loop group $LG$ and its positive loop subgroup $L^+G$—produces the convolution category $H_{\mathrm{sph}}$ of $L^+G$-equivariant motives (or perfect complexes). Equivariantization by loop rotation introduces the quantum parameter $q$ intrinsically.

In this context, the category $H_{\mathrm{sph}}$ is equivalent to the singular Soergel bimodule category $\mathrm{SBim}(\varnothing,\varnothing)$. Tannakian reconstruction from this category yields the quantum group $U_q(\check{G})$, and there is a monoidal equivalence: $$H_{\mathrm{sph}} \simeq \mathrm{SBim}(\varnothing,\varnothing) \simeq \mathrm{Rep}_{\mathrm{fin}}(U_q(\check{G}))$$ that sends convolution to tensor product and grading shift by $q$ to weight-twist (Eberhardt et al., 24 Nov 2025). This equivalence realizes the Cautis–Kamnitzer conjecture, establishing that the $K$-theoretic, $\mathbb{G}_m$-equivariant derived Satake category is tensor-equivalent to the category of finite-dimensional integrable modules over $U_q(\check{G})$.

6. Structural and Technical Ingredients

The proofs and constructions rely on several technical notions:

• Property TA: For varieties with $A=SL_n\times\mathbb{C}^\times$ action, vanishing of odd-degree equivariant K-theory and freeness over the representation ring $R(A)$. This ensures that equivariant algebraic K-theory matches topological K-theory and undergirds the explicit equivariant calculations (Cautis et al., 2015).
• Affine Hecke Algebra and Cyclotomic Quotients: The endomorphism algebras in both the diagrammatic and geometric convolution categories receive surjections from the affine Hecke algebra $\mathcal{H}_m(q)$, whose kernel is identified by cyclotomic relations reflecting the quantum Schur–Weyl duality and matching the structure of $K$-theory endomorphism rings.
• Functoriality and Full Faithfulness: Full faithfulness of the constructed functors is established by explicit identification of endomorphism rings and by leveraging the representation theory of quantum groups and Hecke algebras.

7. Extensions and Perspectives for General Semisimple Groups

While the complete, combinatorial proof is established for type A (i.e., $SL_n$), the quantum K-theoretic derived Satake equivalence is conjectured to hold for any semisimple group $G$ (Cautis et al., 2015, Eberhardt et al., 24 Nov 2025). In other types:

• Analogous diagrammatic categories (spiders/webs) and quantum loop algebra actions are expected to model the categories.
• Partial results exist for $G_2$ using Kuperberg web categories, and type $B$ and $C$ in low ranks.
• A plausible implication is that a uniform diagrammatic theory for all reductive types remains an open area for development.

Recent motivic and K-theoretic methods provide a unifying framework through which the quantum Satake equivalence is formulated and proved, with Soergel bimodules and their singular and motivic generalizations playing a central role (Eberhardt et al., 24 Nov 2025). This approach demonstrates the robustness of the equivalence, connecting quantum groups, equivariant motivic sheaves, and algebraic K-theory in a broad, categorical landscape.