Tensor products of unipotent characters of general linear groups over finite fields

Given unipotent characters U ₁, . . . , U _k of GL _n $ left( {{{mathbb{F}}_q}} right) $ , we prove that $ leftlangle {{U_1} otimes cdots cdots otimes {U_k},1} rightrangle $ is a polynomial in q with non-negative integer coefficients. We study the degree of this polynomial and give a necessary and sufficient condition in terms of the representation theory of symmetric groups and root systems for this polynomial to be non-zero.