  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.

Quantum Algebra