Parabolic K-Motivic Hecke Category

• The parabolic K-motivic Hecke category is a monoidal framework built using parabolic Bott-Samelson resolutions and equivariant K-theory.
• It provides a combinatorial description via singular K-theory Soergel bimodules that extends classical Hecke category structures.
• It underpins conjectural quantum K-theoretic derived Satake equivalences and connects to K-motivic Springer theory and the Langlands program.

The parabolic K-motivic Hecke category is a monoidal category associated to a (possibly disconnected) Kac-Moody group, constructed to paper connections among singular K-theory Soergel bimodules, equivariant algebraic K-theory, and geometric representation theory. In the context of the spherical affine case, it provides a categorical framework appearing in conjectural quantum K-theoretic derived Satake equivalences, particularly in relation to the question posed by Cautis-Kamnitzer. This category is built from data arising from parabolic Bott-Samelson resolutions and their associated equivariant K-theories, endowing it with a combinatorial structure homologous to that of singular K-theory Soergel bimodules (Eberhardt et al., 24 Nov 2025).

1. Construction via Parabolic Bott-Samelson Resolutions

The primary construction of the parabolic K-motivic Hecke category employs parabolic Bott-Samelson resolutions, which are iterated spaces resolving singularities corresponding to parabolic subgroups of a Kac-Moody group. Associated to these spaces is their equivariant algebraic K-theory, and the morphisms in the category are derived from correspondences between these resolutions. The composition and monoidal structure is implemented by convolution, reminiscent of classical Hecke categories but adapted to the context of K-theory and parabolic data (Eberhardt et al., 24 Nov 2025).

A plausible implication is that the categorical framework necessitates detailed tracking of parabolic subgroups and their representation in algebraic K-theory, possibly using explicit generators originating from the geometry of Bott-Samelson varieties.

2. Singular K-Theory Soergel Bimodules

A central result is the combinatorial description of the category via singular K-theory Soergel bimodules. These bimodules are constructed from the equivariant K-theory of the Bott-Samelson resolutions and generalize the classic Soergel bimodules, which appear in the categorification of Hecke algebras. The singular aspect stems from the incorporation of the parabolic subgroup, modifying the module actions to accommodate localization at singular parameters. This extends the framework to a broader class of monoidal categories, encompassing both regular and singular cases in the K-theoretic setting (Eberhardt et al., 24 Nov 2025).

This suggests the structural invariants ordinarily found in Soergel bimodule theory—such as indecomposable objects and homological gradings—admit analogues in the K-theoretic and parabolic environment.

3. Applications to Derived Satake Equivalences

In the spherical affine case, the parabolic K-motivic Hecke category is identified as the categorified structure underlying one half of a conjectural quantum K-theoretic derived Satake equivalence. This addresses a specific conjecture of Cautis-Kamnitzer: the expectation that there exists an equivalence between a quantum K-theoretic version of the derived Satake category and a category of modules or sheaves constructed from singular K-theory Soergel bimodules. The categorical identification of the K-motivic Hecke category in this setting provides evidence and categorical context for such an equivalence (Eberhardt et al., 24 Nov 2025).

A plausible implication is that this category may serve as a testbed for further generalizations of the geometric Satake correspondence in quantum and K-theoretic frameworks.

4. Relation to K-Motivic Springer Theory and Broader Hecke Algebras

The parabolic K-motivic Hecke category generalizes categorical structures arising in K-motivic Springer theory as developed in geometric representation theory (Eberhardt, 23 Jan 2024). The construction of reduced K-motives and their six-functor formalism is designed to accommodate linearly reductive stacks, and categories of Springer K-motives are shown to exhibit formality, positioning them as natural candidates for connections with K-theoretic Hecke categories. The formalism developed enables applications and comparisons to both K-theoretic quiver Hecke and Schur algebras, with morphisms and convolution operations induced by parabolic and Springer data (Eberhardt, 23 Jan 2024).

This suggests that the parabolic K-motivic Hecke category encapsulates a spectrum of Hecke-type categories under a unified K-theoretic and motivic lens.

5. Connections and Conjectures in the Langlands Program

A significant aspect of the motivic and K-theoretic categorical constructions is their purported relevance to the Langlands program. By deploying K-motives and formulating categorical Chern character maps, the geometric realization of representations of affine Hecke algebras and split reductive $p$-adic groups is achieved in the motivic context. The broader aim is to situate the parabolic K-motivic Hecke category within the web of relationships suggested by the local Langlands correspondence, extending the applicability of motivic and K-theoretic tools to arithmetic representation theory (Eberhardt, 23 Jan 2024).

A plausible implication is that conjectural extensions and deeper categorical correspondences may emerge by aligning categorical K-motives and Hecke categories with arithmetic phenomena addressed by the Langlands program.

6. Outlook and Open Problems

Open questions remain regarding the explicit construction, equivalences, and representation-theoretic ramifications of the parabolic K-motivic Hecke category. The conjectural quantum K-theoretic derived Satake equivalence remains unproven, and the precise role of singular K-theory Soergel bimodules continues to be an area of active investigation. Further technical elaboration—including base-change, purity arguments, and the behaviour of convolution functors—awaits detailed development within forthcoming foundational manuscripts (Eberhardt et al., 24 Nov 2025).