Equivariant motivic Hall algebras

TL;DR

This paper generalizes motivic Hall algebras by integrating parabolic induction and non-archimedean variants, enabling new applications of Harder-Narasimhan recursion formulas through an integration map in various transfer theories.

Contribution

It introduces a unified framework combining Joyce's motivic Hall algebra with Green's parabolic induction and non-archimedean aspects, expanding the algebraic tools available for geometric representation theory.

Findings

Existence of an integration map in various transfer theories including equivariant motivic

Application of the framework to study Harder-Narasimhan recursion formulas

Extension of Joyce's motivic Hall algebra with new algebraic structures

Abstract

We introduce a generalization of Joyce's motivic Hall algebra by combining it with Green's parabolic induction product, as well as a non-archimedean variant of it. In the construction, we follow Dyckerhoff-Kapranov's formalism of 2-Segal objects and their transferred algebra structures. Our main result is the existence of an integration map under any suitable transfer theory, of course including the (analytic) equivariant motivic one. This allows us to study Harder-Narasimhan recursion formulas in new cases.