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The Parabolic K-motivic Hecke Category

Jens Niklas Eberhardt, Arnaud Eteve

TL;DR

This work defines the parabolic $K$-motivic Hecke category $\mathcal{H}_{P,Q}$ in genuine equivariant $K$-theory and provides a Soergel-type description via singular $K$-theory Soergel bimodules. Under mild hypotheses, there is an equivalence between $\mathcal{H}_{P,Q}$ and chain complexes of singular SBim over the representation rings $R_P,R_Q$, linking motivic categories to algebraic $K$-theory and generalizing known cases. The spherical affine case and the connection to quantum $K$-theory Satake are highlighted, with a detailed discussion of Bott–Samelson objects, purity, and duality, and a programmatic path toward a full quantum Satake correspondence for broader types. The paper also outlines a broader program connecting these motivic Hecke categories to Langlands dual, bi-Whittaker, and coherent-sheaf pictures via weight structures and categorical Chern characters, outlining several future directions.

Abstract

We define and study the parabolic K-motivic Hecke category of a (possibly disconnected) Kac-Moody group. Our main result is a combinatorial description via singular K-theory Soergel bimodules which arise from the equivariant algebraic K-theory of parabolic Bott-Samelson resolutions. In the spherical affine case, the K-motivic Hecke category serves as one side of a conjectural quantum K-theoretic derived Satake equivalence, addressing a conjecture of Cautis-Kamnitzer.

The Parabolic K-motivic Hecke Category

TL;DR

This work defines the parabolic -motivic Hecke category in genuine equivariant -theory and provides a Soergel-type description via singular -theory Soergel bimodules. Under mild hypotheses, there is an equivalence between and chain complexes of singular SBim over the representation rings , linking motivic categories to algebraic -theory and generalizing known cases. The spherical affine case and the connection to quantum -theory Satake are highlighted, with a detailed discussion of Bott–Samelson objects, purity, and duality, and a programmatic path toward a full quantum Satake correspondence for broader types. The paper also outlines a broader program connecting these motivic Hecke categories to Langlands dual, bi-Whittaker, and coherent-sheaf pictures via weight structures and categorical Chern characters, outlining several future directions.

Abstract

We define and study the parabolic K-motivic Hecke category of a (possibly disconnected) Kac-Moody group. Our main result is a combinatorial description via singular K-theory Soergel bimodules which arise from the equivariant algebraic K-theory of parabolic Bott-Samelson resolutions. In the spherical affine case, the K-motivic Hecke category serves as one side of a conjectural quantum K-theoretic derived Satake equivalence, addressing a conjecture of Cautis-Kamnitzer.

Paper Structure

This paper contains 37 sections, 40 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Genuine equivariant $K$-motives
  3. $K$-motivic Hecke category
  4. Singular $K$-theory Soergel bimodules
  5. The spherical case
  6. Quantum Satake for $\operatorname{SL}_n$
  7. Further directions
  8. Acknowledgments
  9. Notations and conventions
  10. Group and root theoretic data
  11. Categories
  12. Algebras and modules
  13. Morita category
  14. Categories of $K$-motives
  15. Reduced $K$-motives
  16. ...and 22 more sections

Key Result

Theorem 1

Assume that the simple roots of $G$ are linearly independent and $\pi_1(G)$ is free. Then there is an equivalence \begin{tikzcd} {\Hecke_{P,Q}} & {\Ch^b(\SBim_{P,Q})} \arrow["\sim", from=1-1, to=1-2] \end{tikzcd}with the category of chain complexes of singular $K$-theory Soergel

Theorems & Definitions (113)

  • Theorem : \ref{['thm:K-is-an-equivalence']}
  • Definition : see \ref{['def:heckecat']}
  • Theorem : \ref{['corol:weight-complex-functor-is-an-equiv']}
  • Theorem : see \ref{['thm:K-is-an-equivalence']}
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • ...and 103 more