A generalization of Kac polynomials and tensor product of representations of $G{L}_n({\mathbb{F}}_q)$

We study the multiplicities of semisimple split characters in tensor product of semisimple split characters of $G{L}_n({\mathbb{F}}_q)$. We prove that these multiplicities are polynomial in q with non-negative integer coefficients and we obtain a criterion for their non-vanishing. We give moreover an interpretation of these polynomials in terms of the counting of the representations of star-shaped quivers, generalizing a previous result of Hausel, Letellier and Rodriguez-Villegas, who linked multiplicities for generic $k$-tuples of semisimple split characters and Kac polynomials.