The theory of lacunas of hyperbolic PDEs was initiated by Petrovsky (1945), and developed further by Leray and Atiyah-Bott-Garding in the late 70's. I will give a brief overview of the theory, which connects PDEs with topology and algebraic geometry. Then I'll introduce Dunkl's "differential-difference" operators related to finite-dimensional complex semi-simple Lie algebras and show that they bring out more lacunas than available in the purely differential case.