Counting absolutely cuspidals for quivers.

For an arbitrary quiver Q and dimension vector d we prove that the dimension of the space of cuspidal functions on the moduli stack of representations of Q of dimension d over a finite field F_q is given by a polynomial in q with rational coefficients. We define a variant of this polynomial counting absolutely cuspidal functions, prove that it has integral coefficients and conjecture that it has positive coefficients. In the case of totally negative quivers (such as the g-loop quiver for g >1) we provide a closed formula for these polynomials in terms of Kac polynomials.