Total positivity and conjugacy classes

TL;DR

This paper explores the relationship between total positivity in reductive groups and conjugacy classes, proposing a conjectural Jordan decomposition for the totally positive part and proving it in specific cases.

Contribution

It introduces a new framework connecting total positivity with conjugacy class structure and proposes a conjectural Jordan decomposition for the positive monoid.

Findings

Analysis of how conjugacy classes intersect positive cells

Proposed conjectural Jordan decomposition for $G_{ ext{≥0}}$

Proof of the conjecture in special cases

Abstract

In this paper, we study the interaction between the totally positive monoid $G_{\ge 0}$ attached to a connected reductive group $G$ with a pinning and the conjugacy classes in $G$. In particular, we study how a conjugacy class meets the various cells of $G_{\ge0}$. We also state a conjectural Jordan decomposition for $G_{\ge0}$ and prove it in some special cases.