The Parabolic K-motivic Hecke Category

TL;DR

This paper introduces the parabolic K-motivic Hecke category for Kac-Moody groups, providing a combinatorial description via singular K-theory Soergel bimodules, and explores its role in quantum K-theoretic Satake equivalence.

Contribution

It defines the parabolic K-motivic Hecke category and establishes a combinatorial description using singular K-theory Soergel bimodules, advancing the understanding of quantum K-theoretic Satake equivalence.

Findings

Provides a combinatorial description via singular K-theory Soergel bimodules.

Connects the K-motivic Hecke category to conjectural quantum K-theoretic Satake equivalence.

Addresses a conjecture of Cautis-Kamnitzer in the affine case.

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.