Multiplicity of characters of finite reductive groups and Drinfeld doubles

In this paper, we compute the multiplicities of tensor products of almost unipotent characters and Deligne Lusztig characters of a finite reductive group $G^F$, and these multiplicities are related to the ring structure of the complex irreducible characters of $G^F$. In addition, we consider Frobenius Schur indicators of modules over the Drinfeld doubles of finite reductive groups. In the final section, we study the multiplicities of tensor products of almost unipotent characters and pose the question of whether their non vanishing can be detected through the multiplicities of tensor products of irreducible characters of the Weyl group.