We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra $\mathcal{ H}_{ \mathbf{ v}}$, in terms of a canonically defined basis $\mathcal{ B}_{ gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\mathcal{ H}$, and to all $\mathbf{ v}\in\mathcal{ Q}$, where $\mathcal{ Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\mathcal{ H}$. By analytic Dirac induction we define for each $b\in \mathcal{ B}_{ gm}$ a continuous (in the sense of [OS2]) family $\mathcal{ Q}^{ reg}_b:=\mathcal{ Q}_b\backslash\mathcal{ Q}_b^{ sing}\ni\mathbf{ v}\to\operatorname{ Ind}_{ D}(b;\mathbf{ v})$, such that $\epsilon(b;\mathbf{ v})\operatorname{ Ind}_{ D}(b;\mathbf{ v})$ (for some $\epsilon(b;\mathbf{ v})\in\{ \pm 1\}$) is an irreducible discrete series character of $\mathcal{ H}_{ \mathbf{ v}}$. Here $\mathcal{ Q}^{ sing}_b\subset\mathcal{ Q}$ is a finite union of hyperplanes in $\mathcal{ Q}$. In the non-simply laced cases we show that the families of virtual discrete series characters $\operatorname{ Ind}_{ D}(b;\mathbf{ v})$ are piecewise rational in the parameters $\mathbf{ v}$. Remarkably, the formal degree of $\operatorname{ Ind}_{ D}(b;\mathbf{ v})$ in such piecewise rational family turns out to be rational. This implies that for each $b\in \mathcal{ B}_{ gm}$ there exists a universal rational constant $d_b$ determining the formal degree in the family of discrete series characters $\epsilon(b;\mathbf{ v})\operatorname{ Ind}_{ D}(b;\mathbf{ v})$. We will compute the canonical constants $d_b$, and the signs $\epsilon(b;\mathbf{ v})$. For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.
Categories: math.RT