Non-abelian Mellin Transformations and Applications

TL;DR

This paper generalizes the Mellin transformation to non-abelian settings, proving t-exactness and constructing new duality spaces, thereby extending foundational work in algebraic geometry and topology.

Contribution

It introduces a non-abelian Mellin transformation with arbitrary coefficients and establishes t-exactness, expanding the scope of previous abelian results.

Findings

Generalization of Mellin transformation to non-abelian groups

Proven t-exactness of the non-abelian Mellin transformation

Constructed new examples of duality spaces

Abstract

We study non-abelian versions of the Mellin transformations, originally introduced by Gabber-Loeser on complex affine tori. Our main result is a generalization to the non-abelian context and with arbitrary coefficients of the t-exactness of Gabber-Loeser's Mellin transformation. As an intermediate step, we obtain vanishing results for the Sabbah specialization functors. Our main application is to construct new examples of duality spaces in the sense of Bieri-Eckmann, generalizing results of Denham-Suciu.