Modular principal series representations

Recently, there has been considerable progress in classifying the irreducible representations of Iwahori--Hecke algebras at roots of unity. Here, we present an application of these results to $\ell$-modular Harish--Chandra series for a finite group of Lie type $G(q)$. Under some mild condition on $\ell$, we show that the $\ell$-modular principal series representations of $G(q)$ are naturally parametrized by a {\em subset} of the set of complex irreducible characters of the Weyl group of $G(q)$. We also show that this subset is ``generic'' in a precise sense.