# Isotropic cuspidal functions in the Hall algebra of a quiver

## TL;DR

This paper explicitly characterizes isotropic cuspidal functions in the Hall algebra of a quiver, proving conjectures related to their structure, support, and positivity, using elementary methods and group actions.

## Contribution

It provides an explicit description of isotropic cuspidal functions and offers elementary proofs of conjectures by Berenstein and Greenstein, connecting their results with positivity in Hall algebras.

## Key findings

- Explicit linear form on regular cuspidal functions identified
- Isotropic cuspidal dimensions have affine support
- Positivity of absolutely cuspidal polynomials established

## Abstract

From the structure of the category of representations of an affine cycle-free quiver, we determine an explicit linear form on the space of regular cuspidal functions over a finite field: its kernel is exactly the space of cuspidal functions. Moreover, we show that any isotropic cuspidal dimension has an affine support. Brought together, this two results give an explicit description of isotropic cuspidal functions of any quiver. The main theorem together with an appropriate action of some permutation group on the Hall algebra provides a new elementary proof of two conjectures of Berenstein and Greenstein previously proved by Deng and Ruan. We also prove a statement giving non-obvious constraints on the support of the comultiplication of a cuspidal regular function allowing us to connect both mentioned conjectures of Berenstein and Greenstein. Our results imply the positivity conjecture of Bozec and Schiffmann concerning absolutely cuspidal polynomials in isotropic dimensions.