Discrete series for the graded Hecke algebra of type $H_4$

This article confirms the prediction that the set of discrete series central character for the graded (affine) Hecke algebra of type $H_4$ coincides with the set of the Heckman-Opdam central characters. Combining with previous cases of Kazhdan-Lusztig, Kriloff, Kriloff-Ram, Opdam-Solleveld, Ciubotaru-Opdam, this completes the classification of discrete series for all the graded Hecke algebras of positive parameters. Main tools include construction of calibrated modules and construction of certain minimally induced modules for discrete series. We also study the anti-sphericiity and Ext-branching laws for some discrete series.